((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; "},"nextTactic":{"kind":"string","value":"rfl"},"declUpToTactic":{"kind":"string","value":"@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2603_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}},{"rowIdx":207882,"cells":{"state":{"kind":"string","value":"α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns : Set α\nl : Filter β\nf✝ : α → β\nf : β → ↑(Ioo a b)\n⊢ Tendsto f l atTop ↔ Tendsto (fun x => ↑(f x)) l (𝓝[<] b)"},"srcUpToTactic":{"kind":"string","value":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Intervals.Pi\nimport Mathlib.Data.Set.Pointwise.Interval\nimport Mathlib.Order.Filter.Interval\nimport Mathlib.Tactic.TFAE\nimport Mathlib.Topology.Support\nimport Mathlib.Topology.Algebra.Order.LeftRight\n\n#align_import topology.order.basic from \"leanprover-community/mathlib\"@\"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423\"\n\n/-!\n# Theory of topology on ordered spaces\n\n## Main definitions\n\nThe order topology on an ordered space is the topology generated by all open intervals (or\nequivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`.\nHowever, we do *not* register it as an instance (as many existing ordered types already have\ntopologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead,\nwe introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on\nthe type `α` having already a topological space structure and a preorder structure, the topological\nstructure is equal to the order topology.\n\nWe also introduce another (mixin) class `OrderClosedTopology α` saying that the set of points\n`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear\norder with the order topology.\n\nWe prove many basic properties of such topologies.\n\n## Main statements\n\nThis file contains the proofs of the following facts. For exact requirements\n(`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc)\nsee their statements.\n\n### Open / closed sets\n\n* `isOpen_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;\n* `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open;\n* `isClosed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;\n* `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed;\n* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`\n and `{x | f x < g x}` are included by `{x | f x = g x}`;\n* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any\n neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood\n of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.\n\n### Convergence and inequalities\n\n* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually\n `f x ≤ g x`, then `a ≤ b`\n* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`\n (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions\n that assume the inequalities to hold for all `x`.\n\n### Min, max, `sSup` and `sInf`\n\n* `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is\n continuous.\n* `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise\n `min`/`max` tend to `min a b` and `max a b`, respectively.\n* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,\n sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`\n both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.\n\n## Implementation notes\n\nWe do _not_ register the order topology as an instance on a preorder (or even on a linear order).\nIndeed, on many such spaces, a topology has already been constructed in a different way (think\nof the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),\nand is in general not defeq to the one generated by the intervals. We make it available as a\ndefinition `Preorder.topology α` though, that can be registered as an instance when necessary, or\nfor specific types.\n-/\n\n\nopen Set Filter TopologicalSpace Topology Function\n\nopen OrderDual (toDual ofDual)\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is\nclosed for all `a : α`. -/\nclass ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | b ≤ a}` is closed. -/\n isClosed_le' (a : α) : IsClosed { b : α | b ≤ a }\n\nexport ClosedIicTopology (isClosed_le')\n#align is_closed_le' ClosedIicTopology.isClosed_le'\n\n/-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is\nclosed for all `a : α`. -/\nclass ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | a ≤ b}` is closed. -/\n isClosed_ge' (a : α) : IsClosed { b : α | a ≤ b }\n\nexport ClosedIciTopology (isClosed_ge')\n#align is_closed_ge' ClosedIciTopology.isClosed_ge'\n\n/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the\nset of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.\nThis property is satisfied for the order topology on a linear order, but it can be satisfied more\ngenerally, and suffices to derive many interesting properties relating order and topology. -/\nclass OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- The set `{ (x, y) | x ≤ y }` is a closed set. -/\n isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }\n#align order_closed_topology OrderClosedTopology\n\ninstance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h\ninstance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h\n\ntheorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :\n Dense (OrderDual.ofDual ⁻¹' s) :=\n hs\n#align dense.order_dual Dense.orderDual\n\nsection ClosedIicTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIicTopology α]\n\ninstance : ClosedIciTopology αᵒᵈ where\n isClosed_ge' a := isClosed_le' (α := α) a\n\ntheorem isClosed_Iic {a : α} : IsClosed (Iic a) :=\n isClosed_le' a\n#align is_closed_Iic isClosed_Iic\n\n@[simp]\ntheorem closure_Iic (a : α) : closure (Iic a) = Iic a :=\n isClosed_Iic.closure_eq\n#align closure_Iic closure_Iic\n\ntheorem le_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=\n (isClosed_le' b).mem_of_tendsto lim h\n#align le_of_tendsto le_of_tendsto\n\ntheorem le_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, f c ≤ b) : a ≤ b :=\n le_of_tendsto lim (eventually_of_forall h)\n#align le_of_tendsto' le_of_tendsto'\n\nend ClosedIicTopology\n\nsection ClosedIciTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIciTopology α]\n\ninstance : ClosedIicTopology αᵒᵈ where\n isClosed_le' a := isClosed_ge' (α := α) a\n\ntheorem isClosed_Ici {a : α} : IsClosed (Ici a) :=\n isClosed_ge' a\n#align is_closed_Ici isClosed_Ici\n\n@[simp]\ntheorem closure_Ici (a : α) : closure (Ici a) = Ici a :=\n isClosed_Ici.closure_eq\n#align closure_Ici closure_Ici\n\ntheorem ge_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=\n (isClosed_ge' b).mem_of_tendsto lim h\n#align ge_of_tendsto ge_of_tendsto\n\ntheorem ge_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, b ≤ f c) : b ≤ a :=\n ge_of_tendsto lim (eventually_of_forall h)\n#align ge_of_tendsto' ge_of_tendsto'\n\nend ClosedIciTopology\n\nsection OrderClosedTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]\n\nnamespace Subtype\n\n-- todo: add `OrderEmbedding.orderClosedTopology`\ninstance {p : α → Prop} : OrderClosedTopology (Subtype p) :=\n have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Iio_at_top tendsto_comp_coe_Iio_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n "},"nextTactic":{"kind":"string","value":"rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]"},"declUpToTactic":{"kind":"string","value":"@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2609_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}},{"rowIdx":207883,"cells":{"state":{"kind":"string","value":"α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns : Set α\nl : Filter β\nf✝ : α → β\nf : β → ↑(Ioo a b)\n⊢ Tendsto (Subtype.val ∘ f) l (𝓝[<] b) ↔ Tendsto (fun x => ↑(f x)) l (𝓝[<] b)"},"srcUpToTactic":{"kind":"string","value":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Intervals.Pi\nimport Mathlib.Data.Set.Pointwise.Interval\nimport Mathlib.Order.Filter.Interval\nimport Mathlib.Tactic.TFAE\nimport Mathlib.Topology.Support\nimport Mathlib.Topology.Algebra.Order.LeftRight\n\n#align_import topology.order.basic from \"leanprover-community/mathlib\"@\"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423\"\n\n/-!\n# Theory of topology on ordered spaces\n\n## Main definitions\n\nThe order topology on an ordered space is the topology generated by all open intervals (or\nequivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`.\nHowever, we do *not* register it as an instance (as many existing ordered types already have\ntopologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead,\nwe introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on\nthe type `α` having already a topological space structure and a preorder structure, the topological\nstructure is equal to the order topology.\n\nWe also introduce another (mixin) class `OrderClosedTopology α` saying that the set of points\n`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear\norder with the order topology.\n\nWe prove many basic properties of such topologies.\n\n## Main statements\n\nThis file contains the proofs of the following facts. For exact requirements\n(`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc)\nsee their statements.\n\n### Open / closed sets\n\n* `isOpen_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;\n* `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open;\n* `isClosed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;\n* `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed;\n* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`\n and `{x | f x < g x}` are included by `{x | f x = g x}`;\n* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any\n neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood\n of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.\n\n### Convergence and inequalities\n\n* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually\n `f x ≤ g x`, then `a ≤ b`\n* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`\n (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions\n that assume the inequalities to hold for all `x`.\n\n### Min, max, `sSup` and `sInf`\n\n* `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is\n continuous.\n* `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise\n `min`/`max` tend to `min a b` and `max a b`, respectively.\n* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,\n sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`\n both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.\n\n## Implementation notes\n\nWe do _not_ register the order topology as an instance on a preorder (or even on a linear order).\nIndeed, on many such spaces, a topology has already been constructed in a different way (think\nof the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),\nand is in general not defeq to the one generated by the intervals. We make it available as a\ndefinition `Preorder.topology α` though, that can be registered as an instance when necessary, or\nfor specific types.\n-/\n\n\nopen Set Filter TopologicalSpace Topology Function\n\nopen OrderDual (toDual ofDual)\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is\nclosed for all `a : α`. -/\nclass ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | b ≤ a}` is closed. -/\n isClosed_le' (a : α) : IsClosed { b : α | b ≤ a }\n\nexport ClosedIicTopology (isClosed_le')\n#align is_closed_le' ClosedIicTopology.isClosed_le'\n\n/-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is\nclosed for all `a : α`. -/\nclass ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | a ≤ b}` is closed. -/\n isClosed_ge' (a : α) : IsClosed { b : α | a ≤ b }\n\nexport ClosedIciTopology (isClosed_ge')\n#align is_closed_ge' ClosedIciTopology.isClosed_ge'\n\n/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the\nset of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.\nThis property is satisfied for the order topology on a linear order, but it can be satisfied more\ngenerally, and suffices to derive many interesting properties relating order and topology. -/\nclass OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- The set `{ (x, y) | x ≤ y }` is a closed set. -/\n isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }\n#align order_closed_topology OrderClosedTopology\n\ninstance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h\ninstance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h\n\ntheorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :\n Dense (OrderDual.ofDual ⁻¹' s) :=\n hs\n#align dense.order_dual Dense.orderDual\n\nsection ClosedIicTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIicTopology α]\n\ninstance : ClosedIciTopology αᵒᵈ where\n isClosed_ge' a := isClosed_le' (α := α) a\n\ntheorem isClosed_Iic {a : α} : IsClosed (Iic a) :=\n isClosed_le' a\n#align is_closed_Iic isClosed_Iic\n\n@[simp]\ntheorem closure_Iic (a : α) : closure (Iic a) = Iic a :=\n isClosed_Iic.closure_eq\n#align closure_Iic closure_Iic\n\ntheorem le_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=\n (isClosed_le' b).mem_of_tendsto lim h\n#align le_of_tendsto le_of_tendsto\n\ntheorem le_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, f c ≤ b) : a ≤ b :=\n le_of_tendsto lim (eventually_of_forall h)\n#align le_of_tendsto' le_of_tendsto'\n\nend ClosedIicTopology\n\nsection ClosedIciTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIciTopology α]\n\ninstance : ClosedIicTopology αᵒᵈ where\n isClosed_le' a := isClosed_ge' (α := α) a\n\ntheorem isClosed_Ici {a : α} : IsClosed (Ici a) :=\n isClosed_ge' a\n#align is_closed_Ici isClosed_Ici\n\n@[simp]\ntheorem closure_Ici (a : α) : closure (Ici a) = Ici a :=\n isClosed_Ici.closure_eq\n#align closure_Ici closure_Ici\n\ntheorem ge_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=\n (isClosed_ge' b).mem_of_tendsto lim h\n#align ge_of_tendsto ge_of_tendsto\n\ntheorem ge_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, b ≤ f c) : b ≤ a :=\n ge_of_tendsto lim (eventually_of_forall h)\n#align ge_of_tendsto' ge_of_tendsto'\n\nend ClosedIciTopology\n\nsection OrderClosedTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]\n\nnamespace Subtype\n\n-- todo: add `OrderEmbedding.orderClosedTopology`\ninstance {p : α → Prop} : OrderClosedTopology (Subtype p) :=\n have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Iio_at_top tendsto_comp_coe_Iio_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; "},"nextTactic":{"kind":"string","value":"rfl"},"declUpToTactic":{"kind":"string","value":"@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2609_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}},{"rowIdx":207884,"cells":{"state":{"kind":"string","value":"α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns : Set α\nl : Filter β\nf✝ : α → β\nf : β → ↑(Ioo a b)\n⊢ Tendsto f l atBot ↔ Tendsto (fun x => ↑(f x)) l (𝓝[>] a)"},"srcUpToTactic":{"kind":"string","value":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Intervals.Pi\nimport Mathlib.Data.Set.Pointwise.Interval\nimport Mathlib.Order.Filter.Interval\nimport Mathlib.Tactic.TFAE\nimport Mathlib.Topology.Support\nimport Mathlib.Topology.Algebra.Order.LeftRight\n\n#align_import topology.order.basic from \"leanprover-community/mathlib\"@\"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423\"\n\n/-!\n# Theory of topology on ordered spaces\n\n## Main definitions\n\nThe order topology on an ordered space is the topology generated by all open intervals (or\nequivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`.\nHowever, we do *not* register it as an instance (as many existing ordered types already have\ntopologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead,\nwe introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on\nthe type `α` having already a topological space structure and a preorder structure, the topological\nstructure is equal to the order topology.\n\nWe also introduce another (mixin) class `OrderClosedTopology α` saying that the set of points\n`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear\norder with the order topology.\n\nWe prove many basic properties of such topologies.\n\n## Main statements\n\nThis file contains the proofs of the following facts. For exact requirements\n(`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc)\nsee their statements.\n\n### Open / closed sets\n\n* `isOpen_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;\n* `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open;\n* `isClosed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;\n* `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed;\n* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`\n and `{x | f x < g x}` are included by `{x | f x = g x}`;\n* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any\n neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood\n of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.\n\n### Convergence and inequalities\n\n* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually\n `f x ≤ g x`, then `a ≤ b`\n* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`\n (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions\n that assume the inequalities to hold for all `x`.\n\n### Min, max, `sSup` and `sInf`\n\n* `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is\n continuous.\n* `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise\n `min`/`max` tend to `min a b` and `max a b`, respectively.\n* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,\n sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`\n both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.\n\n## Implementation notes\n\nWe do _not_ register the order topology as an instance on a preorder (or even on a linear order).\nIndeed, on many such spaces, a topology has already been constructed in a different way (think\nof the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),\nand is in general not defeq to the one generated by the intervals. We make it available as a\ndefinition `Preorder.topology α` though, that can be registered as an instance when necessary, or\nfor specific types.\n-/\n\n\nopen Set Filter TopologicalSpace Topology Function\n\nopen OrderDual (toDual ofDual)\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is\nclosed for all `a : α`. -/\nclass ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | b ≤ a}` is closed. -/\n isClosed_le' (a : α) : IsClosed { b : α | b ≤ a }\n\nexport ClosedIicTopology (isClosed_le')\n#align is_closed_le' ClosedIicTopology.isClosed_le'\n\n/-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is\nclosed for all `a : α`. -/\nclass ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | a ≤ b}` is closed. -/\n isClosed_ge' (a : α) : IsClosed { b : α | a ≤ b }\n\nexport ClosedIciTopology (isClosed_ge')\n#align is_closed_ge' ClosedIciTopology.isClosed_ge'\n\n/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the\nset of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.\nThis property is satisfied for the order topology on a linear order, but it can be satisfied more\ngenerally, and suffices to derive many interesting properties relating order and topology. -/\nclass OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- The set `{ (x, y) | x ≤ y }` is a closed set. -/\n isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }\n#align order_closed_topology OrderClosedTopology\n\ninstance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h\ninstance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h\n\ntheorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :\n Dense (OrderDual.ofDual ⁻¹' s) :=\n hs\n#align dense.order_dual Dense.orderDual\n\nsection ClosedIicTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIicTopology α]\n\ninstance : ClosedIciTopology αᵒᵈ where\n isClosed_ge' a := isClosed_le' (α := α) a\n\ntheorem isClosed_Iic {a : α} : IsClosed (Iic a) :=\n isClosed_le' a\n#align is_closed_Iic isClosed_Iic\n\n@[simp]\ntheorem closure_Iic (a : α) : closure (Iic a) = Iic a :=\n isClosed_Iic.closure_eq\n#align closure_Iic closure_Iic\n\ntheorem le_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=\n (isClosed_le' b).mem_of_tendsto lim h\n#align le_of_tendsto le_of_tendsto\n\ntheorem le_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, f c ≤ b) : a ≤ b :=\n le_of_tendsto lim (eventually_of_forall h)\n#align le_of_tendsto' le_of_tendsto'\n\nend ClosedIicTopology\n\nsection ClosedIciTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIciTopology α]\n\ninstance : ClosedIicTopology αᵒᵈ where\n isClosed_le' a := isClosed_ge' (α := α) a\n\ntheorem isClosed_Ici {a : α} : IsClosed (Ici a) :=\n isClosed_ge' a\n#align is_closed_Ici isClosed_Ici\n\n@[simp]\ntheorem closure_Ici (a : α) : closure (Ici a) = Ici a :=\n isClosed_Ici.closure_eq\n#align closure_Ici closure_Ici\n\ntheorem ge_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=\n (isClosed_ge' b).mem_of_tendsto lim h\n#align ge_of_tendsto ge_of_tendsto\n\ntheorem ge_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, b ≤ f c) : b ≤ a :=\n ge_of_tendsto lim (eventually_of_forall h)\n#align ge_of_tendsto' ge_of_tendsto'\n\nend ClosedIciTopology\n\nsection OrderClosedTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]\n\nnamespace Subtype\n\n-- todo: add `OrderEmbedding.orderClosedTopology`\ninstance {p : α → Prop} : OrderClosedTopology (Subtype p) :=\n have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Iio_at_top tendsto_comp_coe_Iio_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_top tendsto_Ioo_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n "},"nextTactic":{"kind":"string","value":"rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff]"},"declUpToTactic":{"kind":"string","value":"@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2615_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}},{"rowIdx":207885,"cells":{"state":{"kind":"string","value":"α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns : Set α\nl : Filter β\nf✝ : α → β\nf : β → ↑(Ioo a b)\n⊢ Tendsto (Subtype.val ∘ f) l (𝓝[>] a) ↔ Tendsto (fun x => ↑(f x)) l (𝓝[>] a)"},"srcUpToTactic":{"kind":"string","value":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Intervals.Pi\nimport Mathlib.Data.Set.Pointwise.Interval\nimport Mathlib.Order.Filter.Interval\nimport Mathlib.Tactic.TFAE\nimport Mathlib.Topology.Support\nimport Mathlib.Topology.Algebra.Order.LeftRight\n\n#align_import topology.order.basic from \"leanprover-community/mathlib\"@\"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423\"\n\n/-!\n# Theory of topology on ordered spaces\n\n## Main definitions\n\nThe order topology on an ordered space is the topology generated by all open intervals (or\nequivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`.\nHowever, we do *not* register it as an instance (as many existing ordered types already have\ntopologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead,\nwe introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on\nthe type `α` having already a topological space structure and a preorder structure, the topological\nstructure is equal to the order topology.\n\nWe also introduce another (mixin) class `OrderClosedTopology α` saying that the set of points\n`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear\norder with the order topology.\n\nWe prove many basic properties of such topologies.\n\n## Main statements\n\nThis file contains the proofs of the following facts. For exact requirements\n(`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc)\nsee their statements.\n\n### Open / closed sets\n\n* `isOpen_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;\n* `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open;\n* `isClosed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;\n* `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed;\n* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`\n and `{x | f x < g x}` are included by `{x | f x = g x}`;\n* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any\n neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood\n of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.\n\n### Convergence and inequalities\n\n* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually\n `f x ≤ g x`, then `a ≤ b`\n* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`\n (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions\n that assume the inequalities to hold for all `x`.\n\n### Min, max, `sSup` and `sInf`\n\n* `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is\n continuous.\n* `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise\n `min`/`max` tend to `min a b` and `max a b`, respectively.\n* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,\n sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`\n both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.\n\n## Implementation notes\n\nWe do _not_ register the order topology as an instance on a preorder (or even on a linear order).\nIndeed, on many such spaces, a topology has already been constructed in a different way (think\nof the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),\nand is in general not defeq to the one generated by the intervals. We make it available as a\ndefinition `Preorder.topology α` though, that can be registered as an instance when necessary, or\nfor specific types.\n-/\n\n\nopen Set Filter TopologicalSpace Topology Function\n\nopen OrderDual (toDual ofDual)\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is\nclosed for all `a : α`. -/\nclass ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | b ≤ a}` is closed. -/\n isClosed_le' (a : α) : IsClosed { b : α | b ≤ a }\n\nexport ClosedIicTopology (isClosed_le')\n#align is_closed_le' ClosedIicTopology.isClosed_le'\n\n/-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is\nclosed for all `a : α`. -/\nclass ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | a ≤ b}` is closed. -/\n isClosed_ge' (a : α) : IsClosed { b : α | a ≤ b }\n\nexport ClosedIciTopology (isClosed_ge')\n#align is_closed_ge' ClosedIciTopology.isClosed_ge'\n\n/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the\nset of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.\nThis property is satisfied for the order topology on a linear order, but it can be satisfied more\ngenerally, and suffices to derive many interesting properties relating order and topology. -/\nclass OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- The set `{ (x, y) | x ≤ y }` is a closed set. -/\n isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }\n#align order_closed_topology OrderClosedTopology\n\ninstance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h\ninstance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h\n\ntheorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :\n Dense (OrderDual.ofDual ⁻¹' s) :=\n hs\n#align dense.order_dual Dense.orderDual\n\nsection ClosedIicTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIicTopology α]\n\ninstance : ClosedIciTopology αᵒᵈ where\n isClosed_ge' a := isClosed_le' (α := α) a\n\ntheorem isClosed_Iic {a : α} : IsClosed (Iic a) :=\n isClosed_le' a\n#align is_closed_Iic isClosed_Iic\n\n@[simp]\ntheorem closure_Iic (a : α) : closure (Iic a) = Iic a :=\n isClosed_Iic.closure_eq\n#align closure_Iic closure_Iic\n\ntheorem le_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=\n (isClosed_le' b).mem_of_tendsto lim h\n#align le_of_tendsto le_of_tendsto\n\ntheorem le_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, f c ≤ b) : a ≤ b :=\n le_of_tendsto lim (eventually_of_forall h)\n#align le_of_tendsto' le_of_tendsto'\n\nend ClosedIicTopology\n\nsection ClosedIciTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIciTopology α]\n\ninstance : ClosedIicTopology αᵒᵈ where\n isClosed_le' a := isClosed_ge' (α := α) a\n\ntheorem isClosed_Ici {a : α} : IsClosed (Ici a) :=\n isClosed_ge' a\n#align is_closed_Ici isClosed_Ici\n\n@[simp]\ntheorem closure_Ici (a : α) : closure (Ici a) = Ici a :=\n isClosed_Ici.closure_eq\n#align closure_Ici closure_Ici\n\ntheorem ge_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=\n (isClosed_ge' b).mem_of_tendsto lim h\n#align ge_of_tendsto ge_of_tendsto\n\ntheorem ge_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, b ≤ f c) : b ≤ a :=\n ge_of_tendsto lim (eventually_of_forall h)\n#align ge_of_tendsto' ge_of_tendsto'\n\nend ClosedIciTopology\n\nsection OrderClosedTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]\n\nnamespace Subtype\n\n-- todo: add `OrderEmbedding.orderClosedTopology`\ninstance {p : α → Prop} : OrderClosedTopology (Subtype p) :=\n have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Iio_at_top tendsto_comp_coe_Iio_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_top tendsto_Ioo_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff]; "},"nextTactic":{"kind":"string","value":"rfl"},"declUpToTactic":{"kind":"string","value":"@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff]; "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2615_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}},{"rowIdx":207886,"cells":{"state":{"kind":"string","value":"α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns : Set α\nl : Filter β\nf✝ : α → β\nf : β → ↑(Ioi a)\n⊢ Tendsto f l atBot ↔ Tendsto (fun x => ↑(f x)) l (𝓝[>] a)"},"srcUpToTactic":{"kind":"string","value":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Intervals.Pi\nimport Mathlib.Data.Set.Pointwise.Interval\nimport Mathlib.Order.Filter.Interval\nimport Mathlib.Tactic.TFAE\nimport Mathlib.Topology.Support\nimport Mathlib.Topology.Algebra.Order.LeftRight\n\n#align_import topology.order.basic from \"leanprover-community/mathlib\"@\"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423\"\n\n/-!\n# Theory of topology on ordered spaces\n\n## Main definitions\n\nThe order topology on an ordered space is the topology generated by all open intervals (or\nequivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`.\nHowever, we do *not* register it as an instance (as many existing ordered types already have\ntopologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead,\nwe introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on\nthe type `α` having already a topological space structure and a preorder structure, the topological\nstructure is equal to the order topology.\n\nWe also introduce another (mixin) class `OrderClosedTopology α` saying that the set of points\n`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear\norder with the order topology.\n\nWe prove many basic properties of such topologies.\n\n## Main statements\n\nThis file contains the proofs of the following facts. For exact requirements\n(`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc)\nsee their statements.\n\n### Open / closed sets\n\n* `isOpen_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;\n* `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open;\n* `isClosed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;\n* `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed;\n* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`\n and `{x | f x < g x}` are included by `{x | f x = g x}`;\n* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any\n neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood\n of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.\n\n### Convergence and inequalities\n\n* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually\n `f x ≤ g x`, then `a ≤ b`\n* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`\n (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions\n that assume the inequalities to hold for all `x`.\n\n### Min, max, `sSup` and `sInf`\n\n* `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is\n continuous.\n* `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise\n `min`/`max` tend to `min a b` and `max a b`, respectively.\n* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,\n sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`\n both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.\n\n## Implementation notes\n\nWe do _not_ register the order topology as an instance on a preorder (or even on a linear order).\nIndeed, on many such spaces, a topology has already been constructed in a different way (think\nof the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),\nand is in general not defeq to the one generated by the intervals. We make it available as a\ndefinition `Preorder.topology α` though, that can be registered as an instance when necessary, or\nfor specific types.\n-/\n\n\nopen Set Filter TopologicalSpace Topology Function\n\nopen OrderDual (toDual ofDual)\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is\nclosed for all `a : α`. -/\nclass ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | b ≤ a}` is closed. -/\n isClosed_le' (a : α) : IsClosed { b : α | b ≤ a }\n\nexport ClosedIicTopology (isClosed_le')\n#align is_closed_le' ClosedIicTopology.isClosed_le'\n\n/-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is\nclosed for all `a : α`. -/\nclass ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | a ≤ b}` is closed. -/\n isClosed_ge' (a : α) : IsClosed { b : α | a ≤ b }\n\nexport ClosedIciTopology (isClosed_ge')\n#align is_closed_ge' ClosedIciTopology.isClosed_ge'\n\n/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the\nset of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.\nThis property is satisfied for the order topology on a linear order, but it can be satisfied more\ngenerally, and suffices to derive many interesting properties relating order and topology. -/\nclass OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- The set `{ (x, y) | x ≤ y }` is a closed set. -/\n isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }\n#align order_closed_topology OrderClosedTopology\n\ninstance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h\ninstance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h\n\ntheorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :\n Dense (OrderDual.ofDual ⁻¹' s) :=\n hs\n#align dense.order_dual Dense.orderDual\n\nsection ClosedIicTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIicTopology α]\n\ninstance : ClosedIciTopology αᵒᵈ where\n isClosed_ge' a := isClosed_le' (α := α) a\n\ntheorem isClosed_Iic {a : α} : IsClosed (Iic a) :=\n isClosed_le' a\n#align is_closed_Iic isClosed_Iic\n\n@[simp]\ntheorem closure_Iic (a : α) : closure (Iic a) = Iic a :=\n isClosed_Iic.closure_eq\n#align closure_Iic closure_Iic\n\ntheorem le_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=\n (isClosed_le' b).mem_of_tendsto lim h\n#align le_of_tendsto le_of_tendsto\n\ntheorem le_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, f c ≤ b) : a ≤ b :=\n le_of_tendsto lim (eventually_of_forall h)\n#align le_of_tendsto' le_of_tendsto'\n\nend ClosedIicTopology\n\nsection ClosedIciTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIciTopology α]\n\ninstance : ClosedIicTopology αᵒᵈ where\n isClosed_le' a := isClosed_ge' (α := α) a\n\ntheorem isClosed_Ici {a : α} : IsClosed (Ici a) :=\n isClosed_ge' a\n#align is_closed_Ici isClosed_Ici\n\n@[simp]\ntheorem closure_Ici (a : α) : closure (Ici a) = Ici a :=\n isClosed_Ici.closure_eq\n#align closure_Ici closure_Ici\n\ntheorem ge_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=\n (isClosed_ge' b).mem_of_tendsto lim h\n#align ge_of_tendsto ge_of_tendsto\n\ntheorem ge_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, b ≤ f c) : b ≤ a :=\n ge_of_tendsto lim (eventually_of_forall h)\n#align ge_of_tendsto' ge_of_tendsto'\n\nend ClosedIciTopology\n\nsection OrderClosedTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]\n\nnamespace Subtype\n\n-- todo: add `OrderEmbedding.orderClosedTopology`\ninstance {p : α → Prop} : OrderClosedTopology (Subtype p) :=\n have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Iio_at_top tendsto_comp_coe_Iio_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_top tendsto_Ioo_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_bot tendsto_Ioo_atBot\n\n@[simp]\ntheorem tendsto_Ioi_atBot {f : β → Ioi a} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n "},"nextTactic":{"kind":"string","value":"rw [← comap_coe_Ioi_nhdsWithin_Ioi, tendsto_comap_iff]"},"declUpToTactic":{"kind":"string","value":"@[simp]\ntheorem tendsto_Ioi_atBot {f : β → Ioi a} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2621_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"@[simp]\ntheorem tendsto_Ioi_atBot {f : β → Ioi a} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}},{"rowIdx":207887,"cells":{"state":{"kind":"string","value":"α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns : Set α\nl : Filter β\nf✝ : α → β\nf : β → ↑(Ioi a)\n⊢ Tendsto (Subtype.val ∘ f) l (𝓝[>] a) ↔ Tendsto (fun x => ↑(f x)) l (𝓝[>] a)"},"srcUpToTactic":{"kind":"string","value":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Intervals.Pi\nimport Mathlib.Data.Set.Pointwise.Interval\nimport Mathlib.Order.Filter.Interval\nimport Mathlib.Tactic.TFAE\nimport Mathlib.Topology.Support\nimport Mathlib.Topology.Algebra.Order.LeftRight\n\n#align_import topology.order.basic from \"leanprover-community/mathlib\"@\"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423\"\n\n/-!\n# Theory of topology on ordered spaces\n\n## Main definitions\n\nThe order topology on an ordered space is the topology generated by all open intervals (or\nequivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`.\nHowever, we do *not* register it as an instance (as many existing ordered types already have\ntopologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead,\nwe introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on\nthe type `α` having already a topological space structure and a preorder structure, the topological\nstructure is equal to the order topology.\n\nWe also introduce another (mixin) class `OrderClosedTopology α` saying that the set of points\n`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear\norder with the order topology.\n\nWe prove many basic properties of such topologies.\n\n## Main statements\n\nThis file contains the proofs of the following facts. For exact requirements\n(`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc)\nsee their statements.\n\n### Open / closed sets\n\n* `isOpen_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;\n* `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open;\n* `isClosed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;\n* `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed;\n* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`\n and `{x | f x < g x}` are included by `{x | f x = g x}`;\n* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any\n neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood\n of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.\n\n### Convergence and inequalities\n\n* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually\n `f x ≤ g x`, then `a ≤ b`\n* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`\n (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions\n that assume the inequalities to hold for all `x`.\n\n### Min, max, `sSup` and `sInf`\n\n* `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is\n continuous.\n* `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise\n `min`/`max` tend to `min a b` and `max a b`, respectively.\n* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,\n sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`\n both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.\n\n## Implementation notes\n\nWe do _not_ register the order topology as an instance on a preorder (or even on a linear order).\nIndeed, on many such spaces, a topology has already been constructed in a different way (think\nof the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),\nand is in general not defeq to the one generated by the intervals. We make it available as a\ndefinition `Preorder.topology α` though, that can be registered as an instance when necessary, or\nfor specific types.\n-/\n\n\nopen Set Filter TopologicalSpace Topology Function\n\nopen OrderDual (toDual ofDual)\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is\nclosed for all `a : α`. -/\nclass ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | b ≤ a}` is closed. -/\n isClosed_le' (a : α) : IsClosed { b : α | b ≤ a }\n\nexport ClosedIicTopology (isClosed_le')\n#align is_closed_le' ClosedIicTopology.isClosed_le'\n\n/-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is\nclosed for all `a : α`. -/\nclass ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | a ≤ b}` is closed. -/\n isClosed_ge' (a : α) : IsClosed { b : α | a ≤ b }\n\nexport ClosedIciTopology (isClosed_ge')\n#align is_closed_ge' ClosedIciTopology.isClosed_ge'\n\n/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the\nset of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.\nThis property is satisfied for the order topology on a linear order, but it can be satisfied more\ngenerally, and suffices to derive many interesting properties relating order and topology. -/\nclass OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- The set `{ (x, y) | x ≤ y }` is a closed set. -/\n isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }\n#align order_closed_topology OrderClosedTopology\n\ninstance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h\ninstance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h\n\ntheorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :\n Dense (OrderDual.ofDual ⁻¹' s) :=\n hs\n#align dense.order_dual Dense.orderDual\n\nsection ClosedIicTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIicTopology α]\n\ninstance : ClosedIciTopology αᵒᵈ where\n isClosed_ge' a := isClosed_le' (α := α) a\n\ntheorem isClosed_Iic {a : α} : IsClosed (Iic a) :=\n isClosed_le' a\n#align is_closed_Iic isClosed_Iic\n\n@[simp]\ntheorem closure_Iic (a : α) : closure (Iic a) = Iic a :=\n isClosed_Iic.closure_eq\n#align closure_Iic closure_Iic\n\ntheorem le_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=\n (isClosed_le' b).mem_of_tendsto lim h\n#align le_of_tendsto le_of_tendsto\n\ntheorem le_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, f c ≤ b) : a ≤ b :=\n le_of_tendsto lim (eventually_of_forall h)\n#align le_of_tendsto' le_of_tendsto'\n\nend ClosedIicTopology\n\nsection ClosedIciTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIciTopology α]\n\ninstance : ClosedIicTopology αᵒᵈ where\n isClosed_le' a := isClosed_ge' (α := α) a\n\ntheorem isClosed_Ici {a : α} : IsClosed (Ici a) :=\n isClosed_ge' a\n#align is_closed_Ici isClosed_Ici\n\n@[simp]\ntheorem closure_Ici (a : α) : closure (Ici a) = Ici a :=\n isClosed_Ici.closure_eq\n#align closure_Ici closure_Ici\n\ntheorem ge_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=\n (isClosed_ge' b).mem_of_tendsto lim h\n#align ge_of_tendsto ge_of_tendsto\n\ntheorem ge_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, b ≤ f c) : b ≤ a :=\n ge_of_tendsto lim (eventually_of_forall h)\n#align ge_of_tendsto' ge_of_tendsto'\n\nend ClosedIciTopology\n\nsection OrderClosedTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]\n\nnamespace Subtype\n\n-- todo: add `OrderEmbedding.orderClosedTopology`\ninstance {p : α → Prop} : OrderClosedTopology (Subtype p) :=\n have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Iio_at_top tendsto_comp_coe_Iio_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_top tendsto_Ioo_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_bot tendsto_Ioo_atBot\n\n@[simp]\ntheorem tendsto_Ioi_atBot {f : β → Ioi a} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioi_nhdsWithin_Ioi, tendsto_comap_iff]; "},"nextTactic":{"kind":"string","value":"rfl"},"declUpToTactic":{"kind":"string","value":"@[simp]\ntheorem tendsto_Ioi_atBot {f : β → Ioi a} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioi_nhdsWithin_Ioi, tendsto_comap_iff]; "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2621_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"@[simp]\ntheorem tendsto_Ioi_atBot {f : β → Ioi a} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}},{"rowIdx":207888,"cells":{"state":{"kind":"string","value":"α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns : Set α\nl : Filter β\nf✝ : α → β\nf : β → ↑(Iio a)\n⊢ Tendsto f l atTop ↔ Tendsto (fun x => ↑(f x)) l (𝓝[<] a)"},"srcUpToTactic":{"kind":"string","value":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Intervals.Pi\nimport Mathlib.Data.Set.Pointwise.Interval\nimport Mathlib.Order.Filter.Interval\nimport Mathlib.Tactic.TFAE\nimport Mathlib.Topology.Support\nimport Mathlib.Topology.Algebra.Order.LeftRight\n\n#align_import topology.order.basic from \"leanprover-community/mathlib\"@\"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423\"\n\n/-!\n# Theory of topology on ordered spaces\n\n## Main definitions\n\nThe order topology on an ordered space is the topology generated by all open intervals (or\nequivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`.\nHowever, we do *not* register it as an instance (as many existing ordered types already have\ntopologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead,\nwe introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on\nthe type `α` having already a topological space structure and a preorder structure, the topological\nstructure is equal to the order topology.\n\nWe also introduce another (mixin) class `OrderClosedTopology α` saying that the set of points\n`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear\norder with the order topology.\n\nWe prove many basic properties of such topologies.\n\n## Main statements\n\nThis file contains the proofs of the following facts. For exact requirements\n(`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc)\nsee their statements.\n\n### Open / closed sets\n\n* `isOpen_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;\n* `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open;\n* `isClosed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;\n* `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed;\n* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`\n and `{x | f x < g x}` are included by `{x | f x = g x}`;\n* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any\n neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood\n of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.\n\n### Convergence and inequalities\n\n* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually\n `f x ≤ g x`, then `a ≤ b`\n* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`\n (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions\n that assume the inequalities to hold for all `x`.\n\n### Min, max, `sSup` and `sInf`\n\n* `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is\n continuous.\n* `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise\n `min`/`max` tend to `min a b` and `max a b`, respectively.\n* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,\n sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`\n both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.\n\n## Implementation notes\n\nWe do _not_ register the order topology as an instance on a preorder (or even on a linear order).\nIndeed, on many such spaces, a topology has already been constructed in a different way (think\nof the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),\nand is in general not defeq to the one generated by the intervals. We make it available as a\ndefinition `Preorder.topology α` though, that can be registered as an instance when necessary, or\nfor specific types.\n-/\n\n\nopen Set Filter TopologicalSpace Topology Function\n\nopen OrderDual (toDual ofDual)\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is\nclosed for all `a : α`. -/\nclass ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | b ≤ a}` is closed. -/\n isClosed_le' (a : α) : IsClosed { b : α | b ≤ a }\n\nexport ClosedIicTopology (isClosed_le')\n#align is_closed_le' ClosedIicTopology.isClosed_le'\n\n/-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is\nclosed for all `a : α`. -/\nclass ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | a ≤ b}` is closed. -/\n isClosed_ge' (a : α) : IsClosed { b : α | a ≤ b }\n\nexport ClosedIciTopology (isClosed_ge')\n#align is_closed_ge' ClosedIciTopology.isClosed_ge'\n\n/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the\nset of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.\nThis property is satisfied for the order topology on a linear order, but it can be satisfied more\ngenerally, and suffices to derive many interesting properties relating order and topology. -/\nclass OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- The set `{ (x, y) | x ≤ y }` is a closed set. -/\n isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }\n#align order_closed_topology OrderClosedTopology\n\ninstance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h\ninstance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h\n\ntheorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :\n Dense (OrderDual.ofDual ⁻¹' s) :=\n hs\n#align dense.order_dual Dense.orderDual\n\nsection ClosedIicTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIicTopology α]\n\ninstance : ClosedIciTopology αᵒᵈ where\n isClosed_ge' a := isClosed_le' (α := α) a\n\ntheorem isClosed_Iic {a : α} : IsClosed (Iic a) :=\n isClosed_le' a\n#align is_closed_Iic isClosed_Iic\n\n@[simp]\ntheorem closure_Iic (a : α) : closure (Iic a) = Iic a :=\n isClosed_Iic.closure_eq\n#align closure_Iic closure_Iic\n\ntheorem le_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=\n (isClosed_le' b).mem_of_tendsto lim h\n#align le_of_tendsto le_of_tendsto\n\ntheorem le_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, f c ≤ b) : a ≤ b :=\n le_of_tendsto lim (eventually_of_forall h)\n#align le_of_tendsto' le_of_tendsto'\n\nend ClosedIicTopology\n\nsection ClosedIciTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIciTopology α]\n\ninstance : ClosedIicTopology αᵒᵈ where\n isClosed_le' a := isClosed_ge' (α := α) a\n\ntheorem isClosed_Ici {a : α} : IsClosed (Ici a) :=\n isClosed_ge' a\n#align is_closed_Ici isClosed_Ici\n\n@[simp]\ntheorem closure_Ici (a : α) : closure (Ici a) = Ici a :=\n isClosed_Ici.closure_eq\n#align closure_Ici closure_Ici\n\ntheorem ge_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=\n (isClosed_ge' b).mem_of_tendsto lim h\n#align ge_of_tendsto ge_of_tendsto\n\ntheorem ge_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, b ≤ f c) : b ≤ a :=\n ge_of_tendsto lim (eventually_of_forall h)\n#align ge_of_tendsto' ge_of_tendsto'\n\nend ClosedIciTopology\n\nsection OrderClosedTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]\n\nnamespace Subtype\n\n-- todo: add `OrderEmbedding.orderClosedTopology`\ninstance {p : α → Prop} : OrderClosedTopology (Subtype p) :=\n have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Iio_at_top tendsto_comp_coe_Iio_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_top tendsto_Ioo_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_bot tendsto_Ioo_atBot\n\n@[simp]\ntheorem tendsto_Ioi_atBot {f : β → Ioi a} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioi_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioi_at_bot tendsto_Ioi_atBot\n\n@[simp]\ntheorem tendsto_Iio_atTop {f : β → Iio a} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by\n "},"nextTactic":{"kind":"string","value":"rw [← comap_coe_Iio_nhdsWithin_Iio, tendsto_comap_iff]"},"declUpToTactic":{"kind":"string","value":"@[simp]\ntheorem tendsto_Iio_atTop {f : β → Iio a} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by\n "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2627_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"@[simp]\ntheorem tendsto_Iio_atTop {f : β → Iio a} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}},{"rowIdx":207889,"cells":{"state":{"kind":"string","value":"α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns : Set α\nl : Filter β\nf✝ : α → β\nf : β → ↑(Iio a)\n⊢ Tendsto (Subtype.val ∘ f) l (𝓝[<] a) ↔ Tendsto (fun x => ↑(f x)) l (𝓝[<] a)"},"srcUpToTactic":{"kind":"string","value":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Intervals.Pi\nimport Mathlib.Data.Set.Pointwise.Interval\nimport Mathlib.Order.Filter.Interval\nimport Mathlib.Tactic.TFAE\nimport Mathlib.Topology.Support\nimport Mathlib.Topology.Algebra.Order.LeftRight\n\n#align_import topology.order.basic from \"leanprover-community/mathlib\"@\"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423\"\n\n/-!\n# Theory of topology on ordered spaces\n\n## Main definitions\n\nThe order topology on an ordered space is the topology generated by all open intervals (or\nequivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`.\nHowever, we do *not* register it as an instance (as many existing ordered types already have\ntopologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead,\nwe introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on\nthe type `α` having already a topological space structure and a preorder structure, the topological\nstructure is equal to the order topology.\n\nWe also introduce another (mixin) class `OrderClosedTopology α` saying that the set of points\n`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear\norder with the order topology.\n\nWe prove many basic properties of such topologies.\n\n## Main statements\n\nThis file contains the proofs of the following facts. For exact requirements\n(`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc)\nsee their statements.\n\n### Open / closed sets\n\n* `isOpen_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;\n* `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open;\n* `isClosed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;\n* `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed;\n* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`\n and `{x | f x < g x}` are included by `{x | f x = g x}`;\n* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any\n neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood\n of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.\n\n### Convergence and inequalities\n\n* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually\n `f x ≤ g x`, then `a ≤ b`\n* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`\n (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions\n that assume the inequalities to hold for all `x`.\n\n### Min, max, `sSup` and `sInf`\n\n* `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is\n continuous.\n* `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise\n `min`/`max` tend to `min a b` and `max a b`, respectively.\n* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,\n sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`\n both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.\n\n## Implementation notes\n\nWe do _not_ register the order topology as an instance on a preorder (or even on a linear order).\nIndeed, on many such spaces, a topology has already been constructed in a different way (think\nof the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),\nand is in general not defeq to the one generated by the intervals. We make it available as a\ndefinition `Preorder.topology α` though, that can be registered as an instance when necessary, or\nfor specific types.\n-/\n\n\nopen Set Filter TopologicalSpace Topology Function\n\nopen OrderDual (toDual ofDual)\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is\nclosed for all `a : α`. -/\nclass ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | b ≤ a}` is closed. -/\n isClosed_le' (a : α) : IsClosed { b : α | b ≤ a }\n\nexport ClosedIicTopology (isClosed_le')\n#align is_closed_le' ClosedIicTopology.isClosed_le'\n\n/-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is\nclosed for all `a : α`. -/\nclass ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | a ≤ b}` is closed. -/\n isClosed_ge' (a : α) : IsClosed { b : α | a ≤ b }\n\nexport ClosedIciTopology (isClosed_ge')\n#align is_closed_ge' ClosedIciTopology.isClosed_ge'\n\n/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the\nset of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.\nThis property is satisfied for the order topology on a linear order, but it can be satisfied more\ngenerally, and suffices to derive many interesting properties relating order and topology. -/\nclass OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- The set `{ (x, y) | x ≤ y }` is a closed set. -/\n isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }\n#align order_closed_topology OrderClosedTopology\n\ninstance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h\ninstance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h\n\ntheorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :\n Dense (OrderDual.ofDual ⁻¹' s) :=\n hs\n#align dense.order_dual Dense.orderDual\n\nsection ClosedIicTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIicTopology α]\n\ninstance : ClosedIciTopology αᵒᵈ where\n isClosed_ge' a := isClosed_le' (α := α) a\n\ntheorem isClosed_Iic {a : α} : IsClosed (Iic a) :=\n isClosed_le' a\n#align is_closed_Iic isClosed_Iic\n\n@[simp]\ntheorem closure_Iic (a : α) : closure (Iic a) = Iic a :=\n isClosed_Iic.closure_eq\n#align closure_Iic closure_Iic\n\ntheorem le_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=\n (isClosed_le' b).mem_of_tendsto lim h\n#align le_of_tendsto le_of_tendsto\n\ntheorem le_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, f c ≤ b) : a ≤ b :=\n le_of_tendsto lim (eventually_of_forall h)\n#align le_of_tendsto' le_of_tendsto'\n\nend ClosedIicTopology\n\nsection ClosedIciTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIciTopology α]\n\ninstance : ClosedIicTopology αᵒᵈ where\n isClosed_le' a := isClosed_ge' (α := α) a\n\ntheorem isClosed_Ici {a : α} : IsClosed (Ici a) :=\n isClosed_ge' a\n#align is_closed_Ici isClosed_Ici\n\n@[simp]\ntheorem closure_Ici (a : α) : closure (Ici a) = Ici a :=\n isClosed_Ici.closure_eq\n#align closure_Ici closure_Ici\n\ntheorem ge_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=\n (isClosed_ge' b).mem_of_tendsto lim h\n#align ge_of_tendsto ge_of_tendsto\n\ntheorem ge_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, b ≤ f c) : b ≤ a :=\n ge_of_tendsto lim (eventually_of_forall h)\n#align ge_of_tendsto' ge_of_tendsto'\n\nend ClosedIciTopology\n\nsection OrderClosedTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]\n\nnamespace Subtype\n\n-- todo: add `OrderEmbedding.orderClosedTopology`\ninstance {p : α → Prop} : OrderClosedTopology (Subtype p) :=\n have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Iio_at_top tendsto_comp_coe_Iio_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_top tendsto_Ioo_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_bot tendsto_Ioo_atBot\n\n@[simp]\ntheorem tendsto_Ioi_atBot {f : β → Ioi a} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioi_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioi_at_bot tendsto_Ioi_atBot\n\n@[simp]\ntheorem tendsto_Iio_atTop {f : β → Iio a} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by\n rw [← comap_coe_Iio_nhdsWithin_Iio, tendsto_comap_iff]; "},"nextTactic":{"kind":"string","value":"rfl"},"declUpToTactic":{"kind":"string","value":"@[simp]\ntheorem tendsto_Iio_atTop {f : β → Iio a} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by\n rw [← comap_coe_Iio_nhdsWithin_Iio, tendsto_comap_iff]; "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2627_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"@[simp]\ntheorem tendsto_Iio_atTop {f : β → Iio a} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}},{"rowIdx":207890,"cells":{"state":{"kind":"string","value":"α : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\na b : α\ns : Set α\nl : Filter β\nf : α → β\nx : α\ninst✝ : Nontrivial α\n⊢ NeBot (𝓝[≠] x)"},"srcUpToTactic":{"kind":"string","value":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Intervals.Pi\nimport Mathlib.Data.Set.Pointwise.Interval\nimport Mathlib.Order.Filter.Interval\nimport Mathlib.Tactic.TFAE\nimport Mathlib.Topology.Support\nimport Mathlib.Topology.Algebra.Order.LeftRight\n\n#align_import topology.order.basic from \"leanprover-community/mathlib\"@\"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423\"\n\n/-!\n# Theory of topology on ordered spaces\n\n## Main definitions\n\nThe order topology on an ordered space is the topology generated by all open intervals (or\nequivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`.\nHowever, we do *not* register it as an instance (as many existing ordered types already have\ntopologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead,\nwe introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on\nthe type `α` having already a topological space structure and a preorder structure, the topological\nstructure is equal to the order topology.\n\nWe also introduce another (mixin) class `OrderClosedTopology α` saying that the set of points\n`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear\norder with the order topology.\n\nWe prove many basic properties of such topologies.\n\n## Main statements\n\nThis file contains the proofs of the following facts. For exact requirements\n(`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc)\nsee their statements.\n\n### Open / closed sets\n\n* `isOpen_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;\n* `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open;\n* `isClosed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;\n* `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed;\n* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`\n and `{x | f x < g x}` are included by `{x | f x = g x}`;\n* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any\n neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood\n of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.\n\n### Convergence and inequalities\n\n* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually\n `f x ≤ g x`, then `a ≤ b`\n* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`\n (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions\n that assume the inequalities to hold for all `x`.\n\n### Min, max, `sSup` and `sInf`\n\n* `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is\n continuous.\n* `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise\n `min`/`max` tend to `min a b` and `max a b`, respectively.\n* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,\n sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`\n both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.\n\n## Implementation notes\n\nWe do _not_ register the order topology as an instance on a preorder (or even on a linear order).\nIndeed, on many such spaces, a topology has already been constructed in a different way (think\nof the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),\nand is in general not defeq to the one generated by the intervals. We make it available as a\ndefinition `Preorder.topology α` though, that can be registered as an instance when necessary, or\nfor specific types.\n-/\n\n\nopen Set Filter TopologicalSpace Topology Function\n\nopen OrderDual (toDual ofDual)\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is\nclosed for all `a : α`. -/\nclass ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | b ≤ a}` is closed. -/\n isClosed_le' (a : α) : IsClosed { b : α | b ≤ a }\n\nexport ClosedIicTopology (isClosed_le')\n#align is_closed_le' ClosedIicTopology.isClosed_le'\n\n/-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is\nclosed for all `a : α`. -/\nclass ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | a ≤ b}` is closed. -/\n isClosed_ge' (a : α) : IsClosed { b : α | a ≤ b }\n\nexport ClosedIciTopology (isClosed_ge')\n#align is_closed_ge' ClosedIciTopology.isClosed_ge'\n\n/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the\nset of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.\nThis property is satisfied for the order topology on a linear order, but it can be satisfied more\ngenerally, and suffices to derive many interesting properties relating order and topology. -/\nclass OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- The set `{ (x, y) | x ≤ y }` is a closed set. -/\n isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }\n#align order_closed_topology OrderClosedTopology\n\ninstance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h\ninstance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h\n\ntheorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :\n Dense (OrderDual.ofDual ⁻¹' s) :=\n hs\n#align dense.order_dual Dense.orderDual\n\nsection ClosedIicTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIicTopology α]\n\ninstance : ClosedIciTopology αᵒᵈ where\n isClosed_ge' a := isClosed_le' (α := α) a\n\ntheorem isClosed_Iic {a : α} : IsClosed (Iic a) :=\n isClosed_le' a\n#align is_closed_Iic isClosed_Iic\n\n@[simp]\ntheorem closure_Iic (a : α) : closure (Iic a) = Iic a :=\n isClosed_Iic.closure_eq\n#align closure_Iic closure_Iic\n\ntheorem le_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=\n (isClosed_le' b).mem_of_tendsto lim h\n#align le_of_tendsto le_of_tendsto\n\ntheorem le_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, f c ≤ b) : a ≤ b :=\n le_of_tendsto lim (eventually_of_forall h)\n#align le_of_tendsto' le_of_tendsto'\n\nend ClosedIicTopology\n\nsection ClosedIciTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIciTopology α]\n\ninstance : ClosedIicTopology αᵒᵈ where\n isClosed_le' a := isClosed_ge' (α := α) a\n\ntheorem isClosed_Ici {a : α} : IsClosed (Ici a) :=\n isClosed_ge' a\n#align is_closed_Ici isClosed_Ici\n\n@[simp]\ntheorem closure_Ici (a : α) : closure (Ici a) = Ici a :=\n isClosed_Ici.closure_eq\n#align closure_Ici closure_Ici\n\ntheorem ge_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=\n (isClosed_ge' b).mem_of_tendsto lim h\n#align ge_of_tendsto ge_of_tendsto\n\ntheorem ge_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, b ≤ f c) : b ≤ a :=\n ge_of_tendsto lim (eventually_of_forall h)\n#align ge_of_tendsto' ge_of_tendsto'\n\nend ClosedIciTopology\n\nsection OrderClosedTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]\n\nnamespace Subtype\n\n-- todo: add `OrderEmbedding.orderClosedTopology`\ninstance {p : α → Prop} : OrderClosedTopology (Subtype p) :=\n have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Iio_at_top tendsto_comp_coe_Iio_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_top tendsto_Ioo_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_bot tendsto_Ioo_atBot\n\n@[simp]\ntheorem tendsto_Ioi_atBot {f : β → Ioi a} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioi_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioi_at_bot tendsto_Ioi_atBot\n\n@[simp]\ntheorem tendsto_Iio_atTop {f : β → Iio a} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by\n rw [← comap_coe_Iio_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Iio_at_top tendsto_Iio_atTop\n\ninstance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by\n "},"nextTactic":{"kind":"string","value":"refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_"},"declUpToTactic":{"kind":"string","value":"instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by\n "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2633_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}},{"rowIdx":207891,"cells":{"state":{"kind":"string","value":"α : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\na b : α\ns✝ : Set α\nl : Filter β\nf : α → β\nx : α\ninst✝ : Nontrivial α\ns : Set α\nhs : s ∈ 𝓝[≠] x\n⊢ Set.Nonempty s"},"srcUpToTactic":{"kind":"string","value":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Intervals.Pi\nimport Mathlib.Data.Set.Pointwise.Interval\nimport Mathlib.Order.Filter.Interval\nimport Mathlib.Tactic.TFAE\nimport Mathlib.Topology.Support\nimport Mathlib.Topology.Algebra.Order.LeftRight\n\n#align_import topology.order.basic from \"leanprover-community/mathlib\"@\"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423\"\n\n/-!\n# Theory of topology on ordered spaces\n\n## Main definitions\n\nThe order topology on an ordered space is the topology generated by all open intervals (or\nequivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`.\nHowever, we do *not* register it as an instance (as many existing ordered types already have\ntopologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead,\nwe introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on\nthe type `α` having already a topological space structure and a preorder structure, the topological\nstructure is equal to the order topology.\n\nWe also introduce another (mixin) class `OrderClosedTopology α` saying that the set of points\n`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear\norder with the order topology.\n\nWe prove many basic properties of such topologies.\n\n## Main statements\n\nThis file contains the proofs of the following facts. For exact requirements\n(`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc)\nsee their statements.\n\n### Open / closed sets\n\n* `isOpen_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;\n* `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open;\n* `isClosed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;\n* `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed;\n* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`\n and `{x | f x < g x}` are included by `{x | f x = g x}`;\n* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any\n neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood\n of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.\n\n### Convergence and inequalities\n\n* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually\n `f x ≤ g x`, then `a ≤ b`\n* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`\n (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions\n that assume the inequalities to hold for all `x`.\n\n### Min, max, `sSup` and `sInf`\n\n* `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is\n continuous.\n* `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise\n `min`/`max` tend to `min a b` and `max a b`, respectively.\n* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,\n sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`\n both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.\n\n## Implementation notes\n\nWe do _not_ register the order topology as an instance on a preorder (or even on a linear order).\nIndeed, on many such spaces, a topology has already been constructed in a different way (think\nof the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),\nand is in general not defeq to the one generated by the intervals. We make it available as a\ndefinition `Preorder.topology α` though, that can be registered as an instance when necessary, or\nfor specific types.\n-/\n\n\nopen Set Filter TopologicalSpace Topology Function\n\nopen OrderDual (toDual ofDual)\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is\nclosed for all `a : α`. -/\nclass ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | b ≤ a}` is closed. -/\n isClosed_le' (a : α) : IsClosed { b : α | b ≤ a }\n\nexport ClosedIicTopology (isClosed_le')\n#align is_closed_le' ClosedIicTopology.isClosed_le'\n\n/-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is\nclosed for all `a : α`. -/\nclass ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | a ≤ b}` is closed. -/\n isClosed_ge' (a : α) : IsClosed { b : α | a ≤ b }\n\nexport ClosedIciTopology (isClosed_ge')\n#align is_closed_ge' ClosedIciTopology.isClosed_ge'\n\n/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the\nset of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.\nThis property is satisfied for the order topology on a linear order, but it can be satisfied more\ngenerally, and suffices to derive many interesting properties relating order and topology. -/\nclass OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- The set `{ (x, y) | x ≤ y }` is a closed set. -/\n isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }\n#align order_closed_topology OrderClosedTopology\n\ninstance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h\ninstance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h\n\ntheorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :\n Dense (OrderDual.ofDual ⁻¹' s) :=\n hs\n#align dense.order_dual Dense.orderDual\n\nsection ClosedIicTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIicTopology α]\n\ninstance : ClosedIciTopology αᵒᵈ where\n isClosed_ge' a := isClosed_le' (α := α) a\n\ntheorem isClosed_Iic {a : α} : IsClosed (Iic a) :=\n isClosed_le' a\n#align is_closed_Iic isClosed_Iic\n\n@[simp]\ntheorem closure_Iic (a : α) : closure (Iic a) = Iic a :=\n isClosed_Iic.closure_eq\n#align closure_Iic closure_Iic\n\ntheorem le_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=\n (isClosed_le' b).mem_of_tendsto lim h\n#align le_of_tendsto le_of_tendsto\n\ntheorem le_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, f c ≤ b) : a ≤ b :=\n le_of_tendsto lim (eventually_of_forall h)\n#align le_of_tendsto' le_of_tendsto'\n\nend ClosedIicTopology\n\nsection ClosedIciTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIciTopology α]\n\ninstance : ClosedIicTopology αᵒᵈ where\n isClosed_le' a := isClosed_ge' (α := α) a\n\ntheorem isClosed_Ici {a : α} : IsClosed (Ici a) :=\n isClosed_ge' a\n#align is_closed_Ici isClosed_Ici\n\n@[simp]\ntheorem closure_Ici (a : α) : closure (Ici a) = Ici a :=\n isClosed_Ici.closure_eq\n#align closure_Ici closure_Ici\n\ntheorem ge_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=\n (isClosed_ge' b).mem_of_tendsto lim h\n#align ge_of_tendsto ge_of_tendsto\n\ntheorem ge_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, b ≤ f c) : b ≤ a :=\n ge_of_tendsto lim (eventually_of_forall h)\n#align ge_of_tendsto' ge_of_tendsto'\n\nend ClosedIciTopology\n\nsection OrderClosedTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]\n\nnamespace Subtype\n\n-- todo: add `OrderEmbedding.orderClosedTopology`\ninstance {p : α → Prop} : OrderClosedTopology (Subtype p) :=\n have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Iio_at_top tendsto_comp_coe_Iio_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_top tendsto_Ioo_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_bot tendsto_Ioo_atBot\n\n@[simp]\ntheorem tendsto_Ioi_atBot {f : β → Ioi a} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioi_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioi_at_bot tendsto_Ioi_atBot\n\n@[simp]\ntheorem tendsto_Iio_atTop {f : β → Iio a} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by\n rw [← comap_coe_Iio_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Iio_at_top tendsto_Iio_atTop\n\ninstance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by\n refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_\n "},"nextTactic":{"kind":"string","value":"obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs"},"declUpToTactic":{"kind":"string","value":"instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by\n refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_\n "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2633_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}},{"rowIdx":207892,"cells":{"state":{"kind":"string","value":"case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\na b : α\ns✝ : Set α\nl : Filter β\nf : α → β\nx : α\ninst✝ : Nontrivial α\ns : Set α\nhs : s ∈ 𝓝[≠] x\nu : Set α\nu_open : IsOpen u\nxu : x ∈ u\nus : u ∩ {x}ᶜ ⊆ s\n⊢ Set.Nonempty s"},"srcUpToTactic":{"kind":"string","value":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Intervals.Pi\nimport Mathlib.Data.Set.Pointwise.Interval\nimport Mathlib.Order.Filter.Interval\nimport Mathlib.Tactic.TFAE\nimport Mathlib.Topology.Support\nimport Mathlib.Topology.Algebra.Order.LeftRight\n\n#align_import topology.order.basic from \"leanprover-community/mathlib\"@\"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423\"\n\n/-!\n# Theory of topology on ordered spaces\n\n## Main definitions\n\nThe order topology on an ordered space is the topology generated by all open intervals (or\nequivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`.\nHowever, we do *not* register it as an instance (as many existing ordered types already have\ntopologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead,\nwe introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on\nthe type `α` having already a topological space structure and a preorder structure, the topological\nstructure is equal to the order topology.\n\nWe also introduce another (mixin) class `OrderClosedTopology α` saying that the set of points\n`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear\norder with the order topology.\n\nWe prove many basic properties of such topologies.\n\n## Main statements\n\nThis file contains the proofs of the following facts. For exact requirements\n(`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc)\nsee their statements.\n\n### Open / closed sets\n\n* `isOpen_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;\n* `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open;\n* `isClosed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;\n* `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed;\n* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`\n and `{x | f x < g x}` are included by `{x | f x = g x}`;\n* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any\n neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood\n of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.\n\n### Convergence and inequalities\n\n* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually\n `f x ≤ g x`, then `a ≤ b`\n* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`\n (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions\n that assume the inequalities to hold for all `x`.\n\n### Min, max, `sSup` and `sInf`\n\n* `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is\n continuous.\n* `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise\n `min`/`max` tend to `min a b` and `max a b`, respectively.\n* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,\n sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`\n both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.\n\n## Implementation notes\n\nWe do _not_ register the order topology as an instance on a preorder (or even on a linear order).\nIndeed, on many such spaces, a topology has already been constructed in a different way (think\nof the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),\nand is in general not defeq to the one generated by the intervals. We make it available as a\ndefinition `Preorder.topology α` though, that can be registered as an instance when necessary, or\nfor specific types.\n-/\n\n\nopen Set Filter TopologicalSpace Topology Function\n\nopen OrderDual (toDual ofDual)\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is\nclosed for all `a : α`. -/\nclass ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | b ≤ a}` is closed. -/\n isClosed_le' (a : α) : IsClosed { b : α | b ≤ a }\n\nexport ClosedIicTopology (isClosed_le')\n#align is_closed_le' ClosedIicTopology.isClosed_le'\n\n/-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is\nclosed for all `a : α`. -/\nclass ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | a ≤ b}` is closed. -/\n isClosed_ge' (a : α) : IsClosed { b : α | a ≤ b }\n\nexport ClosedIciTopology (isClosed_ge')\n#align is_closed_ge' ClosedIciTopology.isClosed_ge'\n\n/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the\nset of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.\nThis property is satisfied for the order topology on a linear order, but it can be satisfied more\ngenerally, and suffices to derive many interesting properties relating order and topology. -/\nclass OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- The set `{ (x, y) | x ≤ y }` is a closed set. -/\n isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }\n#align order_closed_topology OrderClosedTopology\n\ninstance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h\ninstance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h\n\ntheorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :\n Dense (OrderDual.ofDual ⁻¹' s) :=\n hs\n#align dense.order_dual Dense.orderDual\n\nsection ClosedIicTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIicTopology α]\n\ninstance : ClosedIciTopology αᵒᵈ where\n isClosed_ge' a := isClosed_le' (α := α) a\n\ntheorem isClosed_Iic {a : α} : IsClosed (Iic a) :=\n isClosed_le' a\n#align is_closed_Iic isClosed_Iic\n\n@[simp]\ntheorem closure_Iic (a : α) : closure (Iic a) = Iic a :=\n isClosed_Iic.closure_eq\n#align closure_Iic closure_Iic\n\ntheorem le_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=\n (isClosed_le' b).mem_of_tendsto lim h\n#align le_of_tendsto le_of_tendsto\n\ntheorem le_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, f c ≤ b) : a ≤ b :=\n le_of_tendsto lim (eventually_of_forall h)\n#align le_of_tendsto' le_of_tendsto'\n\nend ClosedIicTopology\n\nsection ClosedIciTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIciTopology α]\n\ninstance : ClosedIicTopology αᵒᵈ where\n isClosed_le' a := isClosed_ge' (α := α) a\n\ntheorem isClosed_Ici {a : α} : IsClosed (Ici a) :=\n isClosed_ge' a\n#align is_closed_Ici isClosed_Ici\n\n@[simp]\ntheorem closure_Ici (a : α) : closure (Ici a) = Ici a :=\n isClosed_Ici.closure_eq\n#align closure_Ici closure_Ici\n\ntheorem ge_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=\n (isClosed_ge' b).mem_of_tendsto lim h\n#align ge_of_tendsto ge_of_tendsto\n\ntheorem ge_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, b ≤ f c) : b ≤ a :=\n ge_of_tendsto lim (eventually_of_forall h)\n#align ge_of_tendsto' ge_of_tendsto'\n\nend ClosedIciTopology\n\nsection OrderClosedTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]\n\nnamespace Subtype\n\n-- todo: add `OrderEmbedding.orderClosedTopology`\ninstance {p : α → Prop} : OrderClosedTopology (Subtype p) :=\n have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Iio_at_top tendsto_comp_coe_Iio_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_top tendsto_Ioo_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_bot tendsto_Ioo_atBot\n\n@[simp]\ntheorem tendsto_Ioi_atBot {f : β → Ioi a} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioi_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioi_at_bot tendsto_Ioi_atBot\n\n@[simp]\ntheorem tendsto_Iio_atTop {f : β → Iio a} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by\n rw [← comap_coe_Iio_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Iio_at_top tendsto_Iio_atTop\n\ninstance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by\n refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_\n obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs\n "},"nextTactic":{"kind":"string","value":"obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩"},"declUpToTactic":{"kind":"string","value":"instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by\n refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_\n obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs\n "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2633_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}},{"rowIdx":207893,"cells":{"state":{"kind":"string","value":"case intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\na✝ b✝ : α\ns✝ : Set α\nl : Filter β\nf : α → β\nx : α\ninst✝ : Nontrivial α\ns : Set α\nhs : s ∈ 𝓝[≠] x\nu : Set α\nu_open : IsOpen u\nxu : x ∈ u\nus : u ∩ {x}ᶜ ⊆ s\na b : α\na_lt_b : a < b\nhab : Ioo a b ⊆ u\n⊢ Set.Nonempty s"},"srcUpToTactic":{"kind":"string","value":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Intervals.Pi\nimport Mathlib.Data.Set.Pointwise.Interval\nimport Mathlib.Order.Filter.Interval\nimport Mathlib.Tactic.TFAE\nimport Mathlib.Topology.Support\nimport Mathlib.Topology.Algebra.Order.LeftRight\n\n#align_import topology.order.basic from \"leanprover-community/mathlib\"@\"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423\"\n\n/-!\n# Theory of topology on ordered spaces\n\n## Main definitions\n\nThe order topology on an ordered space is the topology generated by all open intervals (or\nequivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`.\nHowever, we do *not* register it as an instance (as many existing ordered types already have\ntopologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead,\nwe introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on\nthe type `α` having already a topological space structure and a preorder structure, the topological\nstructure is equal to the order topology.\n\nWe also introduce another (mixin) class `OrderClosedTopology α` saying that the set of points\n`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear\norder with the order topology.\n\nWe prove many basic properties of such topologies.\n\n## Main statements\n\nThis file contains the proofs of the following facts. For exact requirements\n(`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc)\nsee their statements.\n\n### Open / closed sets\n\n* `isOpen_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;\n* `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open;\n* `isClosed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;\n* `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed;\n* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`\n and `{x | f x < g x}` are included by `{x | f x = g x}`;\n* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any\n neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood\n of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.\n\n### Convergence and inequalities\n\n* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually\n `f x ≤ g x`, then `a ≤ b`\n* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`\n (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions\n that assume the inequalities to hold for all `x`.\n\n### Min, max, `sSup` and `sInf`\n\n* `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is\n continuous.\n* `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise\n `min`/`max` tend to `min a b` and `max a b`, respectively.\n* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,\n sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`\n both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.\n\n## Implementation notes\n\nWe do _not_ register the order topology as an instance on a preorder (or even on a linear order).\nIndeed, on many such spaces, a topology has already been constructed in a different way (think\nof the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),\nand is in general not defeq to the one generated by the intervals. We make it available as a\ndefinition `Preorder.topology α` though, that can be registered as an instance when necessary, or\nfor specific types.\n-/\n\n\nopen Set Filter TopologicalSpace Topology Function\n\nopen OrderDual (toDual ofDual)\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is\nclosed for all `a : α`. -/\nclass ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | b ≤ a}` is closed. -/\n isClosed_le' (a : α) : IsClosed { b : α | b ≤ a }\n\nexport ClosedIicTopology (isClosed_le')\n#align is_closed_le' ClosedIicTopology.isClosed_le'\n\n/-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is\nclosed for all `a : α`. -/\nclass ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | a ≤ b}` is closed. -/\n isClosed_ge' (a : α) : IsClosed { b : α | a ≤ b }\n\nexport ClosedIciTopology (isClosed_ge')\n#align is_closed_ge' ClosedIciTopology.isClosed_ge'\n\n/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the\nset of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.\nThis property is satisfied for the order topology on a linear order, but it can be satisfied more\ngenerally, and suffices to derive many interesting properties relating order and topology. -/\nclass OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- The set `{ (x, y) | x ≤ y }` is a closed set. -/\n isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }\n#align order_closed_topology OrderClosedTopology\n\ninstance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h\ninstance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h\n\ntheorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :\n Dense (OrderDual.ofDual ⁻¹' s) :=\n hs\n#align dense.order_dual Dense.orderDual\n\nsection ClosedIicTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIicTopology α]\n\ninstance : ClosedIciTopology αᵒᵈ where\n isClosed_ge' a := isClosed_le' (α := α) a\n\ntheorem isClosed_Iic {a : α} : IsClosed (Iic a) :=\n isClosed_le' a\n#align is_closed_Iic isClosed_Iic\n\n@[simp]\ntheorem closure_Iic (a : α) : closure (Iic a) = Iic a :=\n isClosed_Iic.closure_eq\n#align closure_Iic closure_Iic\n\ntheorem le_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=\n (isClosed_le' b).mem_of_tendsto lim h\n#align le_of_tendsto le_of_tendsto\n\ntheorem le_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, f c ≤ b) : a ≤ b :=\n le_of_tendsto lim (eventually_of_forall h)\n#align le_of_tendsto' le_of_tendsto'\n\nend ClosedIicTopology\n\nsection ClosedIciTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIciTopology α]\n\ninstance : ClosedIicTopology αᵒᵈ where\n isClosed_le' a := isClosed_ge' (α := α) a\n\ntheorem isClosed_Ici {a : α} : IsClosed (Ici a) :=\n isClosed_ge' a\n#align is_closed_Ici isClosed_Ici\n\n@[simp]\ntheorem closure_Ici (a : α) : closure (Ici a) = Ici a :=\n isClosed_Ici.closure_eq\n#align closure_Ici closure_Ici\n\ntheorem ge_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=\n (isClosed_ge' b).mem_of_tendsto lim h\n#align ge_of_tendsto ge_of_tendsto\n\ntheorem ge_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, b ≤ f c) : b ≤ a :=\n ge_of_tendsto lim (eventually_of_forall h)\n#align ge_of_tendsto' ge_of_tendsto'\n\nend ClosedIciTopology\n\nsection OrderClosedTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]\n\nnamespace Subtype\n\n-- todo: add `OrderEmbedding.orderClosedTopology`\ninstance {p : α → Prop} : OrderClosedTopology (Subtype p) :=\n have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Iio_at_top tendsto_comp_coe_Iio_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_top tendsto_Ioo_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_bot tendsto_Ioo_atBot\n\n@[simp]\ntheorem tendsto_Ioi_atBot {f : β → Ioi a} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioi_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioi_at_bot tendsto_Ioi_atBot\n\n@[simp]\ntheorem tendsto_Iio_atTop {f : β → Iio a} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by\n rw [← comap_coe_Iio_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Iio_at_top tendsto_Iio_atTop\n\ninstance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by\n refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_\n obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs\n obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩\n "},"nextTactic":{"kind":"string","value":"obtain ⟨y, hy⟩ : ∃ y, a < y ∧ y < b := exists_between a_lt_b"},"declUpToTactic":{"kind":"string","value":"instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by\n refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_\n obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs\n obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩\n "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2633_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}},{"rowIdx":207894,"cells":{"state":{"kind":"string","value":"case intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\na✝ b✝ : α\ns✝ : Set α\nl : Filter β\nf : α → β\nx : α\ninst✝ : Nontrivial α\ns : Set α\nhs : s ∈ 𝓝[≠] x\nu : Set α\nu_open : IsOpen u\nxu : x ∈ u\nus : u ∩ {x}ᶜ ⊆ s\na b : α\na_lt_b : a < b\nhab : Ioo a b ⊆ u\ny : α\nhy : a < y ∧ y < b\n⊢ Set.Nonempty s"},"srcUpToTactic":{"kind":"string","value":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Intervals.Pi\nimport Mathlib.Data.Set.Pointwise.Interval\nimport Mathlib.Order.Filter.Interval\nimport Mathlib.Tactic.TFAE\nimport Mathlib.Topology.Support\nimport Mathlib.Topology.Algebra.Order.LeftRight\n\n#align_import topology.order.basic from \"leanprover-community/mathlib\"@\"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423\"\n\n/-!\n# Theory of topology on ordered spaces\n\n## Main definitions\n\nThe order topology on an ordered space is the topology generated by all open intervals (or\nequivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`.\nHowever, we do *not* register it as an instance (as many existing ordered types already have\ntopologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead,\nwe introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on\nthe type `α` having already a topological space structure and a preorder structure, the topological\nstructure is equal to the order topology.\n\nWe also introduce another (mixin) class `OrderClosedTopology α` saying that the set of points\n`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear\norder with the order topology.\n\nWe prove many basic properties of such topologies.\n\n## Main statements\n\nThis file contains the proofs of the following facts. For exact requirements\n(`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc)\nsee their statements.\n\n### Open / closed sets\n\n* `isOpen_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;\n* `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open;\n* `isClosed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;\n* `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed;\n* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`\n and `{x | f x < g x}` are included by `{x | f x = g x}`;\n* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any\n neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood\n of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.\n\n### Convergence and inequalities\n\n* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually\n `f x ≤ g x`, then `a ≤ b`\n* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`\n (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions\n that assume the inequalities to hold for all `x`.\n\n### Min, max, `sSup` and `sInf`\n\n* `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is\n continuous.\n* `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise\n `min`/`max` tend to `min a b` and `max a b`, respectively.\n* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,\n sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`\n both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.\n\n## Implementation notes\n\nWe do _not_ register the order topology as an instance on a preorder (or even on a linear order).\nIndeed, on many such spaces, a topology has already been constructed in a different way (think\nof the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),\nand is in general not defeq to the one generated by the intervals. We make it available as a\ndefinition `Preorder.topology α` though, that can be registered as an instance when necessary, or\nfor specific types.\n-/\n\n\nopen Set Filter TopologicalSpace Topology Function\n\nopen OrderDual (toDual ofDual)\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is\nclosed for all `a : α`. -/\nclass ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | b ≤ a}` is closed. -/\n isClosed_le' (a : α) : IsClosed { b : α | b ≤ a }\n\nexport ClosedIicTopology (isClosed_le')\n#align is_closed_le' ClosedIicTopology.isClosed_le'\n\n/-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is\nclosed for all `a : α`. -/\nclass ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | a ≤ b}` is closed. -/\n isClosed_ge' (a : α) : IsClosed { b : α | a ≤ b }\n\nexport ClosedIciTopology (isClosed_ge')\n#align is_closed_ge' ClosedIciTopology.isClosed_ge'\n\n/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the\nset of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.\nThis property is satisfied for the order topology on a linear order, but it can be satisfied more\ngenerally, and suffices to derive many interesting properties relating order and topology. -/\nclass OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- The set `{ (x, y) | x ≤ y }` is a closed set. -/\n isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }\n#align order_closed_topology OrderClosedTopology\n\ninstance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h\ninstance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h\n\ntheorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :\n Dense (OrderDual.ofDual ⁻¹' s) :=\n hs\n#align dense.order_dual Dense.orderDual\n\nsection ClosedIicTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIicTopology α]\n\ninstance : ClosedIciTopology αᵒᵈ where\n isClosed_ge' a := isClosed_le' (α := α) a\n\ntheorem isClosed_Iic {a : α} : IsClosed (Iic a) :=\n isClosed_le' a\n#align is_closed_Iic isClosed_Iic\n\n@[simp]\ntheorem closure_Iic (a : α) : closure (Iic a) = Iic a :=\n isClosed_Iic.closure_eq\n#align closure_Iic closure_Iic\n\ntheorem le_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=\n (isClosed_le' b).mem_of_tendsto lim h\n#align le_of_tendsto le_of_tendsto\n\ntheorem le_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, f c ≤ b) : a ≤ b :=\n le_of_tendsto lim (eventually_of_forall h)\n#align le_of_tendsto' le_of_tendsto'\n\nend ClosedIicTopology\n\nsection ClosedIciTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIciTopology α]\n\ninstance : ClosedIicTopology αᵒᵈ where\n isClosed_le' a := isClosed_ge' (α := α) a\n\ntheorem isClosed_Ici {a : α} : IsClosed (Ici a) :=\n isClosed_ge' a\n#align is_closed_Ici isClosed_Ici\n\n@[simp]\ntheorem closure_Ici (a : α) : closure (Ici a) = Ici a :=\n isClosed_Ici.closure_eq\n#align closure_Ici closure_Ici\n\ntheorem ge_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=\n (isClosed_ge' b).mem_of_tendsto lim h\n#align ge_of_tendsto ge_of_tendsto\n\ntheorem ge_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, b ≤ f c) : b ≤ a :=\n ge_of_tendsto lim (eventually_of_forall h)\n#align ge_of_tendsto' ge_of_tendsto'\n\nend ClosedIciTopology\n\nsection OrderClosedTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]\n\nnamespace Subtype\n\n-- todo: add `OrderEmbedding.orderClosedTopology`\ninstance {p : α → Prop} : OrderClosedTopology (Subtype p) :=\n have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Iio_at_top tendsto_comp_coe_Iio_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_top tendsto_Ioo_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_bot tendsto_Ioo_atBot\n\n@[simp]\ntheorem tendsto_Ioi_atBot {f : β → Ioi a} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioi_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioi_at_bot tendsto_Ioi_atBot\n\n@[simp]\ntheorem tendsto_Iio_atTop {f : β → Iio a} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by\n rw [← comap_coe_Iio_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Iio_at_top tendsto_Iio_atTop\n\ninstance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by\n refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_\n obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs\n obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩\n obtain ⟨y, hy⟩ : ∃ y, a < y ∧ y < b := exists_between a_lt_b\n "},"nextTactic":{"kind":"string","value":"rcases ne_or_eq x y with (xy | rfl)"},"declUpToTactic":{"kind":"string","value":"instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by\n refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_\n obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs\n obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩\n obtain ⟨y, hy⟩ : ∃ y, a < y ∧ y < b := exists_between a_lt_b\n "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2633_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}},{"rowIdx":207895,"cells":{"state":{"kind":"string","value":"case intro.intro.intro.intro.intro.intro.intro.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\na✝ b✝ : α\ns✝ : Set α\nl : Filter β\nf : α → β\nx : α\ninst✝ : Nontrivial α\ns : Set α\nhs : s ∈ 𝓝[≠] x\nu : Set α\nu_open : IsOpen u\nxu : x ∈ u\nus : u ∩ {x}ᶜ ⊆ s\na b : α\na_lt_b : a < b\nhab : Ioo a b ⊆ u\ny : α\nhy : a < y ∧ y < b\nxy : x ≠ y\n⊢ Set.Nonempty s"},"srcUpToTactic":{"kind":"string","value":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Intervals.Pi\nimport Mathlib.Data.Set.Pointwise.Interval\nimport Mathlib.Order.Filter.Interval\nimport Mathlib.Tactic.TFAE\nimport Mathlib.Topology.Support\nimport Mathlib.Topology.Algebra.Order.LeftRight\n\n#align_import topology.order.basic from \"leanprover-community/mathlib\"@\"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423\"\n\n/-!\n# Theory of topology on ordered spaces\n\n## Main definitions\n\nThe order topology on an ordered space is the topology generated by all open intervals (or\nequivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`.\nHowever, we do *not* register it as an instance (as many existing ordered types already have\ntopologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead,\nwe introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on\nthe type `α` having already a topological space structure and a preorder structure, the topological\nstructure is equal to the order topology.\n\nWe also introduce another (mixin) class `OrderClosedTopology α` saying that the set of points\n`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear\norder with the order topology.\n\nWe prove many basic properties of such topologies.\n\n## Main statements\n\nThis file contains the proofs of the following facts. For exact requirements\n(`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc)\nsee their statements.\n\n### Open / closed sets\n\n* `isOpen_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;\n* `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open;\n* `isClosed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;\n* `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed;\n* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`\n and `{x | f x < g x}` are included by `{x | f x = g x}`;\n* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any\n neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood\n of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.\n\n### Convergence and inequalities\n\n* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually\n `f x ≤ g x`, then `a ≤ b`\n* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`\n (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions\n that assume the inequalities to hold for all `x`.\n\n### Min, max, `sSup` and `sInf`\n\n* `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is\n continuous.\n* `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise\n `min`/`max` tend to `min a b` and `max a b`, respectively.\n* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,\n sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`\n both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.\n\n## Implementation notes\n\nWe do _not_ register the order topology as an instance on a preorder (or even on a linear order).\nIndeed, on many such spaces, a topology has already been constructed in a different way (think\nof the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),\nand is in general not defeq to the one generated by the intervals. We make it available as a\ndefinition `Preorder.topology α` though, that can be registered as an instance when necessary, or\nfor specific types.\n-/\n\n\nopen Set Filter TopologicalSpace Topology Function\n\nopen OrderDual (toDual ofDual)\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is\nclosed for all `a : α`. -/\nclass ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | b ≤ a}` is closed. -/\n isClosed_le' (a : α) : IsClosed { b : α | b ≤ a }\n\nexport ClosedIicTopology (isClosed_le')\n#align is_closed_le' ClosedIicTopology.isClosed_le'\n\n/-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is\nclosed for all `a : α`. -/\nclass ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | a ≤ b}` is closed. -/\n isClosed_ge' (a : α) : IsClosed { b : α | a ≤ b }\n\nexport ClosedIciTopology (isClosed_ge')\n#align is_closed_ge' ClosedIciTopology.isClosed_ge'\n\n/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the\nset of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.\nThis property is satisfied for the order topology on a linear order, but it can be satisfied more\ngenerally, and suffices to derive many interesting properties relating order and topology. -/\nclass OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- The set `{ (x, y) | x ≤ y }` is a closed set. -/\n isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }\n#align order_closed_topology OrderClosedTopology\n\ninstance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h\ninstance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h\n\ntheorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :\n Dense (OrderDual.ofDual ⁻¹' s) :=\n hs\n#align dense.order_dual Dense.orderDual\n\nsection ClosedIicTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIicTopology α]\n\ninstance : ClosedIciTopology αᵒᵈ where\n isClosed_ge' a := isClosed_le' (α := α) a\n\ntheorem isClosed_Iic {a : α} : IsClosed (Iic a) :=\n isClosed_le' a\n#align is_closed_Iic isClosed_Iic\n\n@[simp]\ntheorem closure_Iic (a : α) : closure (Iic a) = Iic a :=\n isClosed_Iic.closure_eq\n#align closure_Iic closure_Iic\n\ntheorem le_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=\n (isClosed_le' b).mem_of_tendsto lim h\n#align le_of_tendsto le_of_tendsto\n\ntheorem le_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, f c ≤ b) : a ≤ b :=\n le_of_tendsto lim (eventually_of_forall h)\n#align le_of_tendsto' le_of_tendsto'\n\nend ClosedIicTopology\n\nsection ClosedIciTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIciTopology α]\n\ninstance : ClosedIicTopology αᵒᵈ where\n isClosed_le' a := isClosed_ge' (α := α) a\n\ntheorem isClosed_Ici {a : α} : IsClosed (Ici a) :=\n isClosed_ge' a\n#align is_closed_Ici isClosed_Ici\n\n@[simp]\ntheorem closure_Ici (a : α) : closure (Ici a) = Ici a :=\n isClosed_Ici.closure_eq\n#align closure_Ici closure_Ici\n\ntheorem ge_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=\n (isClosed_ge' b).mem_of_tendsto lim h\n#align ge_of_tendsto ge_of_tendsto\n\ntheorem ge_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, b ≤ f c) : b ≤ a :=\n ge_of_tendsto lim (eventually_of_forall h)\n#align ge_of_tendsto' ge_of_tendsto'\n\nend ClosedIciTopology\n\nsection OrderClosedTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]\n\nnamespace Subtype\n\n-- todo: add `OrderEmbedding.orderClosedTopology`\ninstance {p : α → Prop} : OrderClosedTopology (Subtype p) :=\n have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Iio_at_top tendsto_comp_coe_Iio_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_top tendsto_Ioo_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_bot tendsto_Ioo_atBot\n\n@[simp]\ntheorem tendsto_Ioi_atBot {f : β → Ioi a} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioi_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioi_at_bot tendsto_Ioi_atBot\n\n@[simp]\ntheorem tendsto_Iio_atTop {f : β → Iio a} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by\n rw [← comap_coe_Iio_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Iio_at_top tendsto_Iio_atTop\n\ninstance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by\n refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_\n obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs\n obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩\n obtain ⟨y, hy⟩ : ∃ y, a < y ∧ y < b := exists_between a_lt_b\n rcases ne_or_eq x y with (xy | rfl)\n · "},"nextTactic":{"kind":"string","value":"exact ⟨y, us ⟨hab hy, xy.symm⟩⟩"},"declUpToTactic":{"kind":"string","value":"instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by\n refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_\n obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs\n obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩\n obtain ⟨y, hy⟩ : ∃ y, a < y ∧ y < b := exists_between a_lt_b\n rcases ne_or_eq x y with (xy | rfl)\n · "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2633_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}},{"rowIdx":207896,"cells":{"state":{"kind":"string","value":"case intro.intro.intro.intro.intro.intro.intro.inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\na✝ b✝ : α\ns✝ : Set α\nl : Filter β\nf : α → β\nx : α\ninst✝ : Nontrivial α\ns : Set α\nhs : s ∈ 𝓝[≠] x\nu : Set α\nu_open : IsOpen u\nxu : x ∈ u\nus : u ∩ {x}ᶜ ⊆ s\na b : α\na_lt_b : a < b\nhab : Ioo a b ⊆ u\nhy : a < x ∧ x < b\n⊢ Set.Nonempty s"},"srcUpToTactic":{"kind":"string","value":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Intervals.Pi\nimport Mathlib.Data.Set.Pointwise.Interval\nimport Mathlib.Order.Filter.Interval\nimport Mathlib.Tactic.TFAE\nimport Mathlib.Topology.Support\nimport Mathlib.Topology.Algebra.Order.LeftRight\n\n#align_import topology.order.basic from \"leanprover-community/mathlib\"@\"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423\"\n\n/-!\n# Theory of topology on ordered spaces\n\n## Main definitions\n\nThe order topology on an ordered space is the topology generated by all open intervals (or\nequivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`.\nHowever, we do *not* register it as an instance (as many existing ordered types already have\ntopologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead,\nwe introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on\nthe type `α` having already a topological space structure and a preorder structure, the topological\nstructure is equal to the order topology.\n\nWe also introduce another (mixin) class `OrderClosedTopology α` saying that the set of points\n`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear\norder with the order topology.\n\nWe prove many basic properties of such topologies.\n\n## Main statements\n\nThis file contains the proofs of the following facts. For exact requirements\n(`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc)\nsee their statements.\n\n### Open / closed sets\n\n* `isOpen_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;\n* `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open;\n* `isClosed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;\n* `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed;\n* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`\n and `{x | f x < g x}` are included by `{x | f x = g x}`;\n* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any\n neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood\n of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.\n\n### Convergence and inequalities\n\n* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually\n `f x ≤ g x`, then `a ≤ b`\n* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`\n (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions\n that assume the inequalities to hold for all `x`.\n\n### Min, max, `sSup` and `sInf`\n\n* `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is\n continuous.\n* `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise\n `min`/`max` tend to `min a b` and `max a b`, respectively.\n* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,\n sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`\n both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.\n\n## Implementation notes\n\nWe do _not_ register the order topology as an instance on a preorder (or even on a linear order).\nIndeed, on many such spaces, a topology has already been constructed in a different way (think\nof the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),\nand is in general not defeq to the one generated by the intervals. We make it available as a\ndefinition `Preorder.topology α` though, that can be registered as an instance when necessary, or\nfor specific types.\n-/\n\n\nopen Set Filter TopologicalSpace Topology Function\n\nopen OrderDual (toDual ofDual)\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is\nclosed for all `a : α`. -/\nclass ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | b ≤ a}` is closed. -/\n isClosed_le' (a : α) : IsClosed { b : α | b ≤ a }\n\nexport ClosedIicTopology (isClosed_le')\n#align is_closed_le' ClosedIicTopology.isClosed_le'\n\n/-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is\nclosed for all `a : α`. -/\nclass ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | a ≤ b}` is closed. -/\n isClosed_ge' (a : α) : IsClosed { b : α | a ≤ b }\n\nexport ClosedIciTopology (isClosed_ge')\n#align is_closed_ge' ClosedIciTopology.isClosed_ge'\n\n/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the\nset of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.\nThis property is satisfied for the order topology on a linear order, but it can be satisfied more\ngenerally, and suffices to derive many interesting properties relating order and topology. -/\nclass OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- The set `{ (x, y) | x ≤ y }` is a closed set. -/\n isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }\n#align order_closed_topology OrderClosedTopology\n\ninstance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h\ninstance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h\n\ntheorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :\n Dense (OrderDual.ofDual ⁻¹' s) :=\n hs\n#align dense.order_dual Dense.orderDual\n\nsection ClosedIicTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIicTopology α]\n\ninstance : ClosedIciTopology αᵒᵈ where\n isClosed_ge' a := isClosed_le' (α := α) a\n\ntheorem isClosed_Iic {a : α} : IsClosed (Iic a) :=\n isClosed_le' a\n#align is_closed_Iic isClosed_Iic\n\n@[simp]\ntheorem closure_Iic (a : α) : closure (Iic a) = Iic a :=\n isClosed_Iic.closure_eq\n#align closure_Iic closure_Iic\n\ntheorem le_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=\n (isClosed_le' b).mem_of_tendsto lim h\n#align le_of_tendsto le_of_tendsto\n\ntheorem le_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, f c ≤ b) : a ≤ b :=\n le_of_tendsto lim (eventually_of_forall h)\n#align le_of_tendsto' le_of_tendsto'\n\nend ClosedIicTopology\n\nsection ClosedIciTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIciTopology α]\n\ninstance : ClosedIicTopology αᵒᵈ where\n isClosed_le' a := isClosed_ge' (α := α) a\n\ntheorem isClosed_Ici {a : α} : IsClosed (Ici a) :=\n isClosed_ge' a\n#align is_closed_Ici isClosed_Ici\n\n@[simp]\ntheorem closure_Ici (a : α) : closure (Ici a) = Ici a :=\n isClosed_Ici.closure_eq\n#align closure_Ici closure_Ici\n\ntheorem ge_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=\n (isClosed_ge' b).mem_of_tendsto lim h\n#align ge_of_tendsto ge_of_tendsto\n\ntheorem ge_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, b ≤ f c) : b ≤ a :=\n ge_of_tendsto lim (eventually_of_forall h)\n#align ge_of_tendsto' ge_of_tendsto'\n\nend ClosedIciTopology\n\nsection OrderClosedTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]\n\nnamespace Subtype\n\n-- todo: add `OrderEmbedding.orderClosedTopology`\ninstance {p : α → Prop} : OrderClosedTopology (Subtype p) :=\n have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Iio_at_top tendsto_comp_coe_Iio_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_top tendsto_Ioo_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_bot tendsto_Ioo_atBot\n\n@[simp]\ntheorem tendsto_Ioi_atBot {f : β → Ioi a} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioi_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioi_at_bot tendsto_Ioi_atBot\n\n@[simp]\ntheorem tendsto_Iio_atTop {f : β → Iio a} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by\n rw [← comap_coe_Iio_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Iio_at_top tendsto_Iio_atTop\n\ninstance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by\n refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_\n obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs\n obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩\n obtain ⟨y, hy⟩ : ∃ y, a < y ∧ y < b := exists_between a_lt_b\n rcases ne_or_eq x y with (xy | rfl)\n · exact ⟨y, us ⟨hab hy, xy.symm⟩⟩\n "},"nextTactic":{"kind":"string","value":"obtain ⟨z, hz⟩ : ∃ z, a < z ∧ z < x := exists_between hy.1"},"declUpToTactic":{"kind":"string","value":"instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by\n refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_\n obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs\n obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩\n obtain ⟨y, hy⟩ : ∃ y, a < y ∧ y < b := exists_between a_lt_b\n rcases ne_or_eq x y with (xy | rfl)\n · exact ⟨y, us ⟨hab hy, xy.symm⟩⟩\n "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2633_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}},{"rowIdx":207897,"cells":{"state":{"kind":"string","value":"case intro.intro.intro.intro.intro.intro.intro.inr.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\na✝ b✝ : α\ns✝ : Set α\nl : Filter β\nf : α → β\nx : α\ninst✝ : Nontrivial α\ns : Set α\nhs : s ∈ 𝓝[≠] x\nu : Set α\nu_open : IsOpen u\nxu : x ∈ u\nus : u ∩ {x}ᶜ ⊆ s\na b : α\na_lt_b : a < b\nhab : Ioo a b ⊆ u\nhy : a < x ∧ x < b\nz : α\nhz : a < z ∧ z < x\n⊢ Set.Nonempty s"},"srcUpToTactic":{"kind":"string","value":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Intervals.Pi\nimport Mathlib.Data.Set.Pointwise.Interval\nimport Mathlib.Order.Filter.Interval\nimport Mathlib.Tactic.TFAE\nimport Mathlib.Topology.Support\nimport Mathlib.Topology.Algebra.Order.LeftRight\n\n#align_import topology.order.basic from \"leanprover-community/mathlib\"@\"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423\"\n\n/-!\n# Theory of topology on ordered spaces\n\n## Main definitions\n\nThe order topology on an ordered space is the topology generated by all open intervals (or\nequivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`.\nHowever, we do *not* register it as an instance (as many existing ordered types already have\ntopologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead,\nwe introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on\nthe type `α` having already a topological space structure and a preorder structure, the topological\nstructure is equal to the order topology.\n\nWe also introduce another (mixin) class `OrderClosedTopology α` saying that the set of points\n`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear\norder with the order topology.\n\nWe prove many basic properties of such topologies.\n\n## Main statements\n\nThis file contains the proofs of the following facts. For exact requirements\n(`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc)\nsee their statements.\n\n### Open / closed sets\n\n* `isOpen_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;\n* `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open;\n* `isClosed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;\n* `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed;\n* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`\n and `{x | f x < g x}` are included by `{x | f x = g x}`;\n* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any\n neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood\n of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.\n\n### Convergence and inequalities\n\n* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually\n `f x ≤ g x`, then `a ≤ b`\n* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`\n (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions\n that assume the inequalities to hold for all `x`.\n\n### Min, max, `sSup` and `sInf`\n\n* `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is\n continuous.\n* `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise\n `min`/`max` tend to `min a b` and `max a b`, respectively.\n* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,\n sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`\n both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.\n\n## Implementation notes\n\nWe do _not_ register the order topology as an instance on a preorder (or even on a linear order).\nIndeed, on many such spaces, a topology has already been constructed in a different way (think\nof the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),\nand is in general not defeq to the one generated by the intervals. We make it available as a\ndefinition `Preorder.topology α` though, that can be registered as an instance when necessary, or\nfor specific types.\n-/\n\n\nopen Set Filter TopologicalSpace Topology Function\n\nopen OrderDual (toDual ofDual)\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is\nclosed for all `a : α`. -/\nclass ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | b ≤ a}` is closed. -/\n isClosed_le' (a : α) : IsClosed { b : α | b ≤ a }\n\nexport ClosedIicTopology (isClosed_le')\n#align is_closed_le' ClosedIicTopology.isClosed_le'\n\n/-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is\nclosed for all `a : α`. -/\nclass ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | a ≤ b}` is closed. -/\n isClosed_ge' (a : α) : IsClosed { b : α | a ≤ b }\n\nexport ClosedIciTopology (isClosed_ge')\n#align is_closed_ge' ClosedIciTopology.isClosed_ge'\n\n/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the\nset of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.\nThis property is satisfied for the order topology on a linear order, but it can be satisfied more\ngenerally, and suffices to derive many interesting properties relating order and topology. -/\nclass OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- The set `{ (x, y) | x ≤ y }` is a closed set. -/\n isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }\n#align order_closed_topology OrderClosedTopology\n\ninstance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h\ninstance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h\n\ntheorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :\n Dense (OrderDual.ofDual ⁻¹' s) :=\n hs\n#align dense.order_dual Dense.orderDual\n\nsection ClosedIicTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIicTopology α]\n\ninstance : ClosedIciTopology αᵒᵈ where\n isClosed_ge' a := isClosed_le' (α := α) a\n\ntheorem isClosed_Iic {a : α} : IsClosed (Iic a) :=\n isClosed_le' a\n#align is_closed_Iic isClosed_Iic\n\n@[simp]\ntheorem closure_Iic (a : α) : closure (Iic a) = Iic a :=\n isClosed_Iic.closure_eq\n#align closure_Iic closure_Iic\n\ntheorem le_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=\n (isClosed_le' b).mem_of_tendsto lim h\n#align le_of_tendsto le_of_tendsto\n\ntheorem le_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, f c ≤ b) : a ≤ b :=\n le_of_tendsto lim (eventually_of_forall h)\n#align le_of_tendsto' le_of_tendsto'\n\nend ClosedIicTopology\n\nsection ClosedIciTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIciTopology α]\n\ninstance : ClosedIicTopology αᵒᵈ where\n isClosed_le' a := isClosed_ge' (α := α) a\n\ntheorem isClosed_Ici {a : α} : IsClosed (Ici a) :=\n isClosed_ge' a\n#align is_closed_Ici isClosed_Ici\n\n@[simp]\ntheorem closure_Ici (a : α) : closure (Ici a) = Ici a :=\n isClosed_Ici.closure_eq\n#align closure_Ici closure_Ici\n\ntheorem ge_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=\n (isClosed_ge' b).mem_of_tendsto lim h\n#align ge_of_tendsto ge_of_tendsto\n\ntheorem ge_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, b ≤ f c) : b ≤ a :=\n ge_of_tendsto lim (eventually_of_forall h)\n#align ge_of_tendsto' ge_of_tendsto'\n\nend ClosedIciTopology\n\nsection OrderClosedTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]\n\nnamespace Subtype\n\n-- todo: add `OrderEmbedding.orderClosedTopology`\ninstance {p : α → Prop} : OrderClosedTopology (Subtype p) :=\n have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Iio_at_top tendsto_comp_coe_Iio_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_top tendsto_Ioo_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_bot tendsto_Ioo_atBot\n\n@[simp]\ntheorem tendsto_Ioi_atBot {f : β → Ioi a} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioi_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioi_at_bot tendsto_Ioi_atBot\n\n@[simp]\ntheorem tendsto_Iio_atTop {f : β → Iio a} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by\n rw [← comap_coe_Iio_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Iio_at_top tendsto_Iio_atTop\n\ninstance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by\n refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_\n obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs\n obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩\n obtain ⟨y, hy⟩ : ∃ y, a < y ∧ y < b := exists_between a_lt_b\n rcases ne_or_eq x y with (xy | rfl)\n · exact ⟨y, us ⟨hab hy, xy.symm⟩⟩\n obtain ⟨z, hz⟩ : ∃ z, a < z ∧ z < x := exists_between hy.1\n "},"nextTactic":{"kind":"string","value":"exact ⟨z, us ⟨hab ⟨hz.1, hz.2.trans hy.2⟩, hz.2.ne⟩⟩"},"declUpToTactic":{"kind":"string","value":"instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by\n refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_\n obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs\n obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩\n obtain ⟨y, hy⟩ : ∃ y, a < y ∧ y < b := exists_between a_lt_b\n rcases ne_or_eq x y with (xy | rfl)\n · exact ⟨y, us ⟨hab hy, xy.symm⟩⟩\n obtain ⟨z, hz⟩ : ∃ z, a < z ∧ z < x := exists_between hy.1\n "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2633_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}},{"rowIdx":207898,"cells":{"state":{"kind":"string","value":"α : Type u\nβ : Type v\nγ : Type w\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\ninst✝² : DenselyOrdered α\na b : α\ns✝ : Set α\nl : Filter β\nf : α → β\ninst✝¹ : Nontrivial α\ns : Set α\ninst✝ : SeparableSpace ↑s\nhs : Dense s\n⊢ ∃ t ⊆ s, Set.Countable t ∧ Dense t ∧ (∀ (x : α), IsBot x → x ∉ t) ∧ ∀ (x : α), IsTop x → x ∉ t"},"srcUpToTactic":{"kind":"string","value":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Intervals.Pi\nimport Mathlib.Data.Set.Pointwise.Interval\nimport Mathlib.Order.Filter.Interval\nimport Mathlib.Tactic.TFAE\nimport Mathlib.Topology.Support\nimport Mathlib.Topology.Algebra.Order.LeftRight\n\n#align_import topology.order.basic from \"leanprover-community/mathlib\"@\"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423\"\n\n/-!\n# Theory of topology on ordered spaces\n\n## Main definitions\n\nThe order topology on an ordered space is the topology generated by all open intervals (or\nequivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`.\nHowever, we do *not* register it as an instance (as many existing ordered types already have\ntopologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead,\nwe introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on\nthe type `α` having already a topological space structure and a preorder structure, the topological\nstructure is equal to the order topology.\n\nWe also introduce another (mixin) class `OrderClosedTopology α` saying that the set of points\n`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear\norder with the order topology.\n\nWe prove many basic properties of such topologies.\n\n## Main statements\n\nThis file contains the proofs of the following facts. For exact requirements\n(`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc)\nsee their statements.\n\n### Open / closed sets\n\n* `isOpen_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;\n* `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open;\n* `isClosed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;\n* `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed;\n* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`\n and `{x | f x < g x}` are included by `{x | f x = g x}`;\n* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any\n neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood\n of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.\n\n### Convergence and inequalities\n\n* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually\n `f x ≤ g x`, then `a ≤ b`\n* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`\n (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions\n that assume the inequalities to hold for all `x`.\n\n### Min, max, `sSup` and `sInf`\n\n* `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is\n continuous.\n* `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise\n `min`/`max` tend to `min a b` and `max a b`, respectively.\n* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,\n sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`\n both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.\n\n## Implementation notes\n\nWe do _not_ register the order topology as an instance on a preorder (or even on a linear order).\nIndeed, on many such spaces, a topology has already been constructed in a different way (think\nof the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),\nand is in general not defeq to the one generated by the intervals. We make it available as a\ndefinition `Preorder.topology α` though, that can be registered as an instance when necessary, or\nfor specific types.\n-/\n\n\nopen Set Filter TopologicalSpace Topology Function\n\nopen OrderDual (toDual ofDual)\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is\nclosed for all `a : α`. -/\nclass ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | b ≤ a}` is closed. -/\n isClosed_le' (a : α) : IsClosed { b : α | b ≤ a }\n\nexport ClosedIicTopology (isClosed_le')\n#align is_closed_le' ClosedIicTopology.isClosed_le'\n\n/-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is\nclosed for all `a : α`. -/\nclass ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | a ≤ b}` is closed. -/\n isClosed_ge' (a : α) : IsClosed { b : α | a ≤ b }\n\nexport ClosedIciTopology (isClosed_ge')\n#align is_closed_ge' ClosedIciTopology.isClosed_ge'\n\n/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the\nset of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.\nThis property is satisfied for the order topology on a linear order, but it can be satisfied more\ngenerally, and suffices to derive many interesting properties relating order and topology. -/\nclass OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- The set `{ (x, y) | x ≤ y }` is a closed set. -/\n isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }\n#align order_closed_topology OrderClosedTopology\n\ninstance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h\ninstance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h\n\ntheorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :\n Dense (OrderDual.ofDual ⁻¹' s) :=\n hs\n#align dense.order_dual Dense.orderDual\n\nsection ClosedIicTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIicTopology α]\n\ninstance : ClosedIciTopology αᵒᵈ where\n isClosed_ge' a := isClosed_le' (α := α) a\n\ntheorem isClosed_Iic {a : α} : IsClosed (Iic a) :=\n isClosed_le' a\n#align is_closed_Iic isClosed_Iic\n\n@[simp]\ntheorem closure_Iic (a : α) : closure (Iic a) = Iic a :=\n isClosed_Iic.closure_eq\n#align closure_Iic closure_Iic\n\ntheorem le_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=\n (isClosed_le' b).mem_of_tendsto lim h\n#align le_of_tendsto le_of_tendsto\n\ntheorem le_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, f c ≤ b) : a ≤ b :=\n le_of_tendsto lim (eventually_of_forall h)\n#align le_of_tendsto' le_of_tendsto'\n\nend ClosedIicTopology\n\nsection ClosedIciTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIciTopology α]\n\ninstance : ClosedIicTopology αᵒᵈ where\n isClosed_le' a := isClosed_ge' (α := α) a\n\ntheorem isClosed_Ici {a : α} : IsClosed (Ici a) :=\n isClosed_ge' a\n#align is_closed_Ici isClosed_Ici\n\n@[simp]\ntheorem closure_Ici (a : α) : closure (Ici a) = Ici a :=\n isClosed_Ici.closure_eq\n#align closure_Ici closure_Ici\n\ntheorem ge_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=\n (isClosed_ge' b).mem_of_tendsto lim h\n#align ge_of_tendsto ge_of_tendsto\n\ntheorem ge_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, b ≤ f c) : b ≤ a :=\n ge_of_tendsto lim (eventually_of_forall h)\n#align ge_of_tendsto' ge_of_tendsto'\n\nend ClosedIciTopology\n\nsection OrderClosedTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]\n\nnamespace Subtype\n\n-- todo: add `OrderEmbedding.orderClosedTopology`\ninstance {p : α → Prop} : OrderClosedTopology (Subtype p) :=\n have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Iio_at_top tendsto_comp_coe_Iio_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_top tendsto_Ioo_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_bot tendsto_Ioo_atBot\n\n@[simp]\ntheorem tendsto_Ioi_atBot {f : β → Ioi a} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioi_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioi_at_bot tendsto_Ioi_atBot\n\n@[simp]\ntheorem tendsto_Iio_atTop {f : β → Iio a} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by\n rw [← comap_coe_Iio_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Iio_at_top tendsto_Iio_atTop\n\ninstance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by\n refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_\n obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs\n obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩\n obtain ⟨y, hy⟩ : ∃ y, a < y ∧ y < b := exists_between a_lt_b\n rcases ne_or_eq x y with (xy | rfl)\n · exact ⟨y, us ⟨hab hy, xy.symm⟩⟩\n obtain ⟨z, hz⟩ : ∃ z, a < z ∧ z < x := exists_between hy.1\n exact ⟨z, us ⟨hab ⟨hz.1, hz.2.trans hy.2⟩, hz.2.ne⟩⟩\n\n/-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a\nseparable space (e.g., if `α` has a second countable topology), then there exists a countable\ndense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/\ntheorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivial α] {s : Set α} [SeparableSpace s]\n (hs : Dense s) :\n ∃ t, t ⊆ s ∧ t.Countable ∧ Dense t ∧ (∀ x, IsBot x → x ∉ t) ∧ ∀ x, IsTop x → x ∉ t := by\n "},"nextTactic":{"kind":"string","value":"rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, htd⟩"},"declUpToTactic":{"kind":"string","value":"/-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a\nseparable space (e.g., if `α` has a second countable topology), then there exists a countable\ndense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/\ntheorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivial α] {s : Set α} [SeparableSpace s]\n (hs : Dense s) :\n ∃ t, t ⊆ s ∧ t.Countable ∧ Dense t ∧ (∀ x, IsBot x → x ∉ t) ∧ ∀ x, IsTop x → x ∉ t := by\n "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2643_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"/-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a\nseparable space (e.g., if `α` has a second countable topology), then there exists a countable\ndense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/\ntheorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivial α] {s : Set α} [SeparableSpace s]\n (hs : Dense s) :\n ∃ t, t ⊆ s ∧ t.Countable ∧ Dense t ∧ (∀ x, IsBot x → x ∉ t) ∧ ∀ x, IsTop x → x ∉ t "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}},{"rowIdx":207899,"cells":{"state":{"kind":"string","value":"case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\ninst✝² : DenselyOrdered α\na b : α\ns✝ : Set α\nl : Filter β\nf : α → β\ninst✝¹ : Nontrivial α\ns : Set α\ninst✝ : SeparableSpace ↑s\nhs : Dense s\nt : Set α\nhts : t ⊆ s\nhtc : Set.Countable t\nhtd : Dense t\n⊢ ∃ t ⊆ s, Set.Countable t ∧ Dense t ∧ (∀ (x : α), IsBot x → x ∉ t) ∧ ∀ (x : α), IsTop x → x ∉ t"},"srcUpToTactic":{"kind":"string","value":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Intervals.Pi\nimport Mathlib.Data.Set.Pointwise.Interval\nimport Mathlib.Order.Filter.Interval\nimport Mathlib.Tactic.TFAE\nimport Mathlib.Topology.Support\nimport Mathlib.Topology.Algebra.Order.LeftRight\n\n#align_import topology.order.basic from \"leanprover-community/mathlib\"@\"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423\"\n\n/-!\n# Theory of topology on ordered spaces\n\n## Main definitions\n\nThe order topology on an ordered space is the topology generated by all open intervals (or\nequivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`.\nHowever, we do *not* register it as an instance (as many existing ordered types already have\ntopologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead,\nwe introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on\nthe type `α` having already a topological space structure and a preorder structure, the topological\nstructure is equal to the order topology.\n\nWe also introduce another (mixin) class `OrderClosedTopology α` saying that the set of points\n`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear\norder with the order topology.\n\nWe prove many basic properties of such topologies.\n\n## Main statements\n\nThis file contains the proofs of the following facts. For exact requirements\n(`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc)\nsee their statements.\n\n### Open / closed sets\n\n* `isOpen_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;\n* `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open;\n* `isClosed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;\n* `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed;\n* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`\n and `{x | f x < g x}` are included by `{x | f x = g x}`;\n* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any\n neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood\n of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.\n\n### Convergence and inequalities\n\n* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually\n `f x ≤ g x`, then `a ≤ b`\n* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`\n (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions\n that assume the inequalities to hold for all `x`.\n\n### Min, max, `sSup` and `sInf`\n\n* `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is\n continuous.\n* `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise\n `min`/`max` tend to `min a b` and `max a b`, respectively.\n* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,\n sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`\n both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.\n\n## Implementation notes\n\nWe do _not_ register the order topology as an instance on a preorder (or even on a linear order).\nIndeed, on many such spaces, a topology has already been constructed in a different way (think\nof the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),\nand is in general not defeq to the one generated by the intervals. We make it available as a\ndefinition `Preorder.topology α` though, that can be registered as an instance when necessary, or\nfor specific types.\n-/\n\n\nopen Set Filter TopologicalSpace Topology Function\n\nopen OrderDual (toDual ofDual)\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is\nclosed for all `a : α`. -/\nclass ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | b ≤ a}` is closed. -/\n isClosed_le' (a : α) : IsClosed { b : α | b ≤ a }\n\nexport ClosedIicTopology (isClosed_le')\n#align is_closed_le' ClosedIicTopology.isClosed_le'\n\n/-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is\nclosed for all `a : α`. -/\nclass ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- For any `a`, the set `{b | a ≤ b}` is closed. -/\n isClosed_ge' (a : α) : IsClosed { b : α | a ≤ b }\n\nexport ClosedIciTopology (isClosed_ge')\n#align is_closed_ge' ClosedIciTopology.isClosed_ge'\n\n/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the\nset of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.\nThis property is satisfied for the order topology on a linear order, but it can be satisfied more\ngenerally, and suffices to derive many interesting properties relating order and topology. -/\nclass OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where\n /-- The set `{ (x, y) | x ≤ y }` is a closed set. -/\n isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 }\n#align order_closed_topology OrderClosedTopology\n\ninstance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h\ninstance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h\n\ntheorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :\n Dense (OrderDual.ofDual ⁻¹' s) :=\n hs\n#align dense.order_dual Dense.orderDual\n\nsection ClosedIicTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIicTopology α]\n\ninstance : ClosedIciTopology αᵒᵈ where\n isClosed_ge' a := isClosed_le' (α := α) a\n\ntheorem isClosed_Iic {a : α} : IsClosed (Iic a) :=\n isClosed_le' a\n#align is_closed_Iic isClosed_Iic\n\n@[simp]\ntheorem closure_Iic (a : α) : closure (Iic a) = Iic a :=\n isClosed_Iic.closure_eq\n#align closure_Iic closure_Iic\n\ntheorem le_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=\n (isClosed_le' b).mem_of_tendsto lim h\n#align le_of_tendsto le_of_tendsto\n\ntheorem le_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, f c ≤ b) : a ≤ b :=\n le_of_tendsto lim (eventually_of_forall h)\n#align le_of_tendsto' le_of_tendsto'\n\nend ClosedIicTopology\n\nsection ClosedIciTopology\n\nvariable [TopologicalSpace α] [Preorder α] [t : ClosedIciTopology α]\n\ninstance : ClosedIicTopology αᵒᵈ where\n isClosed_le' a := isClosed_ge' (α := α) a\n\ntheorem isClosed_Ici {a : α} : IsClosed (Ici a) :=\n isClosed_ge' a\n#align is_closed_Ici isClosed_Ici\n\n@[simp]\ntheorem closure_Ici (a : α) : closure (Ici a) = Ici a :=\n isClosed_Ici.closure_eq\n#align closure_Ici closure_Ici\n\ntheorem ge_of_tendsto {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=\n (isClosed_ge' b).mem_of_tendsto lim h\n#align ge_of_tendsto ge_of_tendsto\n\ntheorem ge_of_tendsto' {f : β → α} {a b : α} {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))\n (h : ∀ c, b ≤ f c) : b ≤ a :=\n ge_of_tendsto lim (eventually_of_forall h)\n#align ge_of_tendsto' ge_of_tendsto'\n\nend ClosedIciTopology\n\nsection OrderClosedTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α]\n\nnamespace Subtype\n\n-- todo: add `OrderEmbedding.orderClosedTopology`\ninstance {p : α → Prop} : OrderClosedTopology (Subtype p) :=\n have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=\n continuous_subtype_val.prod_map continuous_subtype_val\n OrderClosedTopology.mk (t.isClosed_le'.preimage this)\n\nend Subtype\n\ntheorem isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=\n t.isClosed_le'\n#align is_closed_le_prod isClosed_le_prod\n\ntheorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsClosed { b | f b ≤ g b } :=\n continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_le_prod\n#align is_closed_le isClosed_le\n\ninstance : ClosedIicTopology α where\n isClosed_le' _ := isClosed_le continuous_id continuous_const\n\ninstance : ClosedIciTopology α where\n isClosed_ge' _ := isClosed_le continuous_const continuous_id\n\ninstance : OrderClosedTopology αᵒᵈ :=\n ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩\n\ntheorem isClosed_Icc {a b : α} : IsClosed (Icc a b) :=\n IsClosed.inter isClosed_Ici isClosed_Iic\n#align is_closed_Icc isClosed_Icc\n\n@[simp]\ntheorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=\n isClosed_Icc.closure_eq\n#align closure_Icc closure_Icc\n\ntheorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=\n have : Tendsto (fun b => (f b, g b)) b (𝓝 (a₁, a₂)) := hf.prod_mk_nhds hg\n show (a₁, a₂) ∈ { p : α × α | p.1 ≤ p.2 } from t.isClosed_le'.mem_of_tendsto this h\n#align le_of_tendsto_of_tendsto le_of_tendsto_of_tendsto\n\nalias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto\n#align tendsto_le_of_eventually_le tendsto_le_of_eventuallyLE\n\ntheorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]\n (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=\n le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)\n#align le_of_tendsto_of_tendsto' le_of_tendsto_of_tendsto'\n\n@[simp]\ntheorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n closure { b | f b ≤ g b } = { b | f b ≤ g b } :=\n (isClosed_le hf hg).closure_eq\n#align closure_le_eq closure_le_eq\n\ntheorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)\n (hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=\n (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg\n#align closure_lt_subset_le closure_lt_subset_le\n\ntheorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}\n (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)\n (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=\n show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from\n OrderClosedTopology.isClosed_le'.closure_subset ((hf.prod hg).mem_closure hx h)\n#align continuous_within_at.closure_le ContinuousWithinAt.closure_le\n\n/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,\nthen the set `{x ∈ s | f x ≤ g x}` is a closed set. -/\ntheorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=\n (hf.prod hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le'\n#align is_closed.is_closed_le IsClosed.isClosed_le\n\ntheorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)\n (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :\n f x ≤ g x :=\n have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩\n (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2\n#align le_on_closure le_on_closure\n\ntheorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=\n (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd\n#align is_closed.epigraph IsClosed.epigraph\n\ntheorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)\n (hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=\n (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl)\n#align is_closed.hypograph IsClosed.hypograph\n\nend Preorder\n\nsection PartialOrder\n\nvariable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α]\n\n-- see Note [lower instance priority]\ninstance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α :=\n t2_iff_isClosed_diagonal.2 <| by\n simpa only [diagonal, le_antisymm_iff] using\n t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)\n#align order_closed_topology.to_t2_space OrderClosedTopology.to_t2Space\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]\n\ntheorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :\n IsOpen { b | f b < g b } := by\n simpa only [lt_iff_not_le] using (isClosed_le hg hf).isOpen_compl\n#align is_open_lt isOpen_lt\n\ntheorem isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=\n isOpen_lt continuous_fst continuous_snd\n#align is_open_lt_prod isOpen_lt_prod\n\nvariable {a b : α}\n\ntheorem isOpen_Iio : IsOpen (Iio a) :=\n isOpen_lt continuous_id continuous_const\n#align is_open_Iio isOpen_Iio\n\ntheorem isOpen_Ioi : IsOpen (Ioi a) :=\n isOpen_lt continuous_const continuous_id\n#align is_open_Ioi isOpen_Ioi\n\ntheorem isOpen_Ioo : IsOpen (Ioo a b) :=\n IsOpen.inter isOpen_Ioi isOpen_Iio\n#align is_open_Ioo isOpen_Ioo\n\n@[simp]\ntheorem interior_Ioi : interior (Ioi a) = Ioi a :=\n isOpen_Ioi.interior_eq\n#align interior_Ioi interior_Ioi\n\n@[simp]\ntheorem interior_Iio : interior (Iio a) = Iio a :=\n isOpen_Iio.interior_eq\n#align interior_Iio interior_Iio\n\n@[simp]\ntheorem interior_Ioo : interior (Ioo a b) = Ioo a b :=\n isOpen_Ioo.interior_eq\n#align interior_Ioo interior_Ioo\n\ntheorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by\n simp only [interior_Ioo, subset_closure]\n#align Ioo_subset_closure_interior Ioo_subset_closure_interior\n\ntheorem Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=\n IsOpen.mem_nhds isOpen_Iio h\n#align Iio_mem_nhds Iio_mem_nhds\n\ntheorem Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=\n IsOpen.mem_nhds isOpen_Ioi h\n#align Ioi_mem_nhds Ioi_mem_nhds\n\ntheorem Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=\n mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self\n#align Iic_mem_nhds Iic_mem_nhds\n\ntheorem Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=\n mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self\n#align Ici_mem_nhds Ici_mem_nhds\n\ntheorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=\n IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩\n#align Ioo_mem_nhds Ioo_mem_nhds\n\ntheorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds Ioc_mem_nhds\n\ntheorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self\n#align Ico_mem_nhds Ico_mem_nhds\n\ntheorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=\n mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self\n#align Icc_mem_nhds Icc_mem_nhds\n\ntheorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=\n tendsto_nhds.1 h (· < u) isOpen_Iio hv\n#align eventually_lt_of_tendsto_lt eventually_lt_of_tendsto_lt\n\ntheorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=\n tendsto_nhds.1 h (· > u) isOpen_Ioi hv\n#align eventually_gt_of_tendsto_gt eventually_gt_of_tendsto_gt\n\ntheorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=\n (eventually_lt_of_tendsto_lt hv h).mono fun _ => le_of_lt\n#align eventually_le_of_tendsto_lt eventually_le_of_tendsto_lt\n\ntheorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)\n (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=\n (eventually_gt_of_tendsto_gt hv h).mono fun _ => le_of_lt\n#align eventually_ge_of_tendsto_gt eventually_ge_of_tendsto_gt\n\nvariable [TopologicalSpace γ]\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderClosedTopology`\n\nLimits to the left and to the right of real functions are defined in terms of neighborhoods to\nthe left and to the right, either open or closed, i.e., members of `𝓝[>] a` and\n`𝓝[≥] a` on the right, and similarly on the left. Here we simply prove that all\nright-neighborhoods of a point are equal, and we'll prove later other useful characterizations which\nrequire the stronger hypothesis `OrderTopology α` -/\n\n\n/-!\n#### Right neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩\n#align Ioo_mem_nhds_within_Ioi Ioo_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[>] a :=\n Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Covby.nhdsWithin_Ioi {a b : α} (h : a ⋖ b) : 𝓝[>] a = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Ioi' h.1\n\ntheorem Ioc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ioi Ioc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[>] a :=\n Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Ico_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Ioi Ico_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[>] a :=\n Ico_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ioi {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ioi H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Ioi Icc_mem_nhdsWithin_Ioi\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Icc_mem_nhdsWithin_Ioi' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[>] a :=\n Icc_mem_nhdsWithin_Ioi ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Ioc_eq_nhds_within_Ioi nhdsWithin_Ioc_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Ioi {a b : α} (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ioo_eq_nhds_within_Ioi nhdsWithin_Ioo_eq_nhdsWithin_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioc_iff_Ioi continuousWithinAt_Ioc_iff_Ioi\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Ioi [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]\n#align continuous_within_at_Ioo_iff_Ioi continuousWithinAt_Ioo_iff_Ioi\n\n/-!\n#### Left neighborhoods, point excluded\n-/\n\ntheorem Ioo_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := by\n simpa only [dual_Ioo] using\n Ioo_mem_nhdsWithin_Ioi (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioo_mem_nhds_within_Iio Ioo_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioo_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioo a b ∈ 𝓝[<] b :=\n Ioo_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Covby.nhdsWithin_Iio {a b : α} (h : a ⋖ b) : 𝓝[<] b = ⊥ :=\n empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsWithin_Iio' h.1\n\ntheorem Ico_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iio Ico_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ico_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[<] b :=\n Ico_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Ioc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Iio Ioc_mem_nhdsWithin_Iio\n\n-- porting note: new lemma; todo: swap `'`?\ntheorem Ioc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[<] b :=\n Ioc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iio {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iio H) Ioo_subset_Icc_self\n#align Icc_mem_nhds_within_Iio Icc_mem_nhdsWithin_Iio\n\ntheorem Icc_mem_nhdsWithin_Iio' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[<] b :=\n Icc_mem_nhdsWithin_Iio ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := by\n simpa only [dual_Ioc] using nhdsWithin_Ioc_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ico_eq_nhds_within_Iio nhdsWithin_Ico_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem nhdsWithin_Ioo_eq_nhdsWithin_Iio {a b : α} (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := by\n simpa only [dual_Ioo] using nhdsWithin_Ioo_eq_nhdsWithin_Ioi h.dual\n#align nhds_within_Ioo_eq_nhds_within_Iio nhdsWithin_Ioo_eq_nhdsWithin_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ico_iff_Iio continuousWithinAt_Ico_iff_Iio\n\n@[simp]\ntheorem continuousWithinAt_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :\n ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Iio h]\n#align continuous_within_at_Ioo_iff_Iio continuousWithinAt_Ioo_iff_Iio\n\n/-!\n#### Right neighborhoods, point included\n-/\n\ntheorem Ioo_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Ici Ioo_mem_nhdsWithin_Ici\n\ntheorem Ioc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Ici H) Ioo_subset_Ioc_self\n#align Ioc_mem_nhds_within_Ici Ioc_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=\n mem_nhdsWithin.2\n ⟨Iio c, isOpen_Iio, H.2, by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩\n#align Ico_mem_nhds_within_Ici Ico_mem_nhdsWithin_Ici\n\ntheorem Ico_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Ico a b ∈ 𝓝[≥] a :=\n Ico_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\ntheorem Icc_mem_nhdsWithin_Ici {a b c : α} (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=\n mem_of_superset (Ico_mem_nhdsWithin_Ici H) Ico_subset_Icc_self\n#align Icc_mem_nhds_within_Ici Icc_mem_nhdsWithin_Ici\n\ntheorem Icc_mem_nhdsWithin_Ici' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≥] a :=\n Icc_mem_nhdsWithin_Ici ⟨le_rfl, H⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h\n#align nhds_within_Icc_eq_nhds_within_Ici nhdsWithin_Icc_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem nhdsWithin_Ico_eq_nhdsWithin_Ici {a b : α} (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=\n nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h\n#align nhds_within_Ico_eq_nhds_within_Ici nhdsWithin_Ico_eq_nhdsWithin_Ici\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Icc_iff_Ici continuousWithinAt_Icc_iff_Ici\n\n@[simp]\ntheorem continuousWithinAt_Ico_iff_Ici [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by\n simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsWithin_Ici h]\n#align continuous_within_at_Ico_iff_Ici continuousWithinAt_Ico_iff_Ici\n\n/-!\n#### Left neighborhoods, point included\n-/\n\n\ntheorem Ioo_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=\n mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds isOpen_Ioo H\n#align Ioo_mem_nhds_within_Iic Ioo_mem_nhdsWithin_Iic\n\ntheorem Ico_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioo_mem_nhdsWithin_Iic H) Ioo_subset_Ico_self\n#align Ico_mem_nhds_within_Iic Ico_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := by\n simpa only [dual_Ico] using\n Ico_mem_nhdsWithin_Ici (show toDual b ∈ Ico (toDual c) (toDual a) from H.symm)\n#align Ioc_mem_nhds_within_Iic Ioc_mem_nhdsWithin_Iic\n\ntheorem Ioc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=\n Ioc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\ntheorem Icc_mem_nhdsWithin_Iic {a b c : α} (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=\n mem_of_superset (Ioc_mem_nhdsWithin_Iic H) Ioc_subset_Icc_self\n#align Icc_mem_nhds_within_Iic Icc_mem_nhdsWithin_Iic\n\ntheorem Icc_mem_nhdsWithin_Iic' {a b : α} (H : a < b) : Icc a b ∈ 𝓝[≤] b :=\n Icc_mem_nhdsWithin_Iic ⟨H, le_rfl⟩\n\n@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := by\n simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Icc_eq_nhds_within_Iic nhdsWithin_Icc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem nhdsWithin_Ioc_eq_nhdsWithin_Iic {a b : α} (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := by\n simpa only [dual_Ico] using nhdsWithin_Ico_eq_nhdsWithin_Ici h.dual\n#align nhds_within_Ioc_eq_nhds_within_Iic nhdsWithin_Ioc_eq_nhdsWithin_Iic\n\n@[simp]\ntheorem continuousWithinAt_Icc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Icc_iff_Iic continuousWithinAt_Icc_iff_Iic\n\n@[simp]\ntheorem continuousWithinAt_Ioc_iff_Iic [TopologicalSpace β] {a b : α} {f : α → β} (h : a < b) :\n ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by\n simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]\n#align continuous_within_at_Ioc_iff_Iic continuousWithinAt_Ioc_iff_Iic\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α}\n\nsection\n\nvariable [TopologicalSpace β]\n\ntheorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :\n { b | f b < g b } ⊆ interior { b | f b ≤ g b } :=\n (interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg\n#align lt_subset_interior_le lt_subset_interior_le\n\ntheorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by\n rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]\n rintro b ⟨hb₁, hb₂⟩\n refine' le_antisymm hb₁ (closure_lt_subset_le hg hf _)\n convert hb₂ using 2; simp only [not_le.symm]; rfl\n#align frontier_le_subset_eq frontier_le_subset_eq\n\ntheorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=\n frontier_le_subset_eq (@continuous_id α _) continuous_const\n#align frontier_Iic_subset frontier_Iic_subset\n\ntheorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=\n frontier_Iic_subset (α := αᵒᵈ) _\n#align frontier_Ici_subset frontier_Ici_subset\n\ntheorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :\n frontier { b | f b < g b } ⊆ { b | f b = g b } := by\n simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf\n#align frontier_lt_subset_eq frontier_lt_subset_eq\n\ntheorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })\n (hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :\n Continuous fun x => if f x ≤ g x then f' x else g' x := by\n refine' continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) _ (hg'.mono _)\n · rwa [(isClosed_le hf hg).closure_eq]\n · simp only [not_le]\n exact closure_lt_subset_le hg hf\n#align continuous_if_le continuous_if_le\n\ntheorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}\n (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)\n (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=\n continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg\n#align continuous.if_le Continuous.if_le\n\ntheorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))\n (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=\n let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi\n (hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>\n h _ h₁ _ h₂\n#align tendsto.eventually_lt Filter.Tendsto.eventually_lt\n\nnonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)\n (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=\n hf.eventually_lt hg hfg\n#align continuous_at.eventually_lt ContinuousAt.eventually_lt\n\n@[continuity]\nprotected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => min (f b) (g b) := by\n simp only [min_def]\n exact hf.if_le hg hf hg fun x => id\n#align continuous.min Continuous.min\n\n@[continuity]\nprotected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) :\n Continuous fun b => max (f b) (g b) :=\n Continuous.min (α := αᵒᵈ) hf hg\n#align continuous.max Continuous.max\n\nend\n\ntheorem continuous_min : Continuous fun p : α × α => min p.1 p.2 :=\n continuous_fst.min continuous_snd\n#align continuous_min continuous_min\n\ntheorem continuous_max : Continuous fun p : α × α => max p.1 p.2 :=\n continuous_fst.max continuous_snd\n#align continuous_max continuous_max\n\nprotected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=\n (continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.max Filter.Tendsto.max\n\nprotected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))\n (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=\n (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)\n#align filter.tendsto.min Filter.Tendsto.min\n\nprotected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max a (f i)) l (𝓝 a) := by\n convert ((continuous_max.comp (@Continuous.Prod.mk α α _ _ a)).tendsto a).comp h\n simp\n#align filter.tendsto.max_right Filter.Tendsto.max_right\n\nprotected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => max (f i) a) l (𝓝 a) := by\n simp_rw [max_comm _ a]\n exact h.max_right\n#align filter.tendsto.max_left Filter.Tendsto.max_left\n\ntheorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by\n obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h\n exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩\n#align filter.tendsto_nhds_max_right Filter.tendsto_nhds_max_right\n\ntheorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :\n Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by\n simp_rw [max_comm _ a]\n exact Filter.tendsto_nhds_max_right h\n#align filter.tendsto_nhds_max_left Filter.tendsto_nhds_max_left\n\ntheorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min a (f i)) l (𝓝 a) :=\n Filter.Tendsto.max_right (α := αᵒᵈ) h\n#align filter.tendsto.min_right Filter.Tendsto.min_right\n\ntheorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :\n Tendsto (fun i => min (f i) a) l (𝓝 a) :=\n Filter.Tendsto.max_left (α := αᵒᵈ) h\n#align filter.tendsto.min_left Filter.Tendsto.min_left\n\ntheorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_right (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_right Filter.tendsto_nhds_min_right\n\ntheorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :\n Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=\n Filter.tendsto_nhds_max_left (α := αᵒᵈ) h\n#align filter.tendsto_nhds_min_left Filter.tendsto_nhds_min_left\n\nprotected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y < x :=\n hs.exists_mem_open isOpen_Iio (exists_lt x)\n#align dense.exists_lt Dense.exists_lt\n\nprotected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x < y :=\n hs.orderDual.exists_lt x\n#align dense.exists_gt Dense.exists_gt\n\nprotected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, y ≤ x :=\n (hs.exists_lt x).imp fun _ h => ⟨h.1, h.2.le⟩\n#align dense.exists_le Dense.exists_le\n\nprotected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le x\n#align dense.exists_ge Dense.exists_ge\n\ntheorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :\n ∃ y ∈ s, y ≤ x := by\n by_cases hx : IsBot x\n · exact ⟨x, hbot x hx, le_rfl⟩\n · simp only [IsBot, not_forall, not_le] at hx\n rcases hs.exists_mem_open isOpen_Iio hx with ⟨y, hys, hy : y < x⟩\n exact ⟨y, hys, hy.le⟩\n#align dense.exists_le' Dense.exists_le'\n\ntheorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :\n ∃ y ∈ s, x ≤ y :=\n hs.orderDual.exists_le' htop x\n#align dense.exists_ge' Dense.exists_ge'\n\ntheorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :\n ∃ z ∈ s, z ∈ Ioo x y :=\n hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h)\n#align dense.exists_between Dense.exists_between\n\ntheorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by\n refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))\n rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩\n exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩\n\ntheorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :\n Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=\n Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x\n\nend LinearOrder\n\nend OrderClosedTopology\n\ninstance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β]\n [OrderClosedTopology β] : OrderClosedTopology (α × β) :=\n ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter\n (isClosed_le continuous_fst.snd continuous_snd.snd)⟩\n\ninstance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]\n [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by\n constructor\n simp only [Pi.le_def, setOf_forall]\n exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd'\n\ninstance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :\n OrderClosedTopology (α → β) :=\n inferInstance\n#align pi.order_closed_topology' Pi.orderClosedTopology'\n\n-- porting note: todo: define `Preorder.topology` before `OrderTopology` and reuse the def\n/-- The order topology on an ordered type is the topology generated by open intervals. We register\nit on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.\nWe define it as a mixin. If you want to introduce the order topology on a preorder, use\n`Preorder.topology`. -/\nclass OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where\n /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/\n topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }\n#align order_topology OrderTopology\n\n/-- (Order) topology on a partial order `α` generated by the subbase of open intervals\n`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an\ninstance as many ordered sets are already endowed with the same topology, most often in a non-defeq\nway though. Register as a local instance when necessary. -/\ndef Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α :=\n generateFrom { s : Set α | ∃ a : α, s = { b : α | a < b } ∨ s = { b : α | b < a } }\n#align preorder.topology Preorder.topology\n\nsection OrderTopology\n\nsection Preorder\n\nvariable [TopologicalSpace α] [Preorder α] [t : OrderTopology α]\n\ninstance : OrderTopology αᵒᵈ :=\n ⟨by\n convert OrderTopology.topology_eq_generate_intervals (α := α) using 6\n apply or_comm⟩\n\ntheorem isOpen_iff_generate_intervals {s : Set α} :\n IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by\n rw [t.topology_eq_generate_intervals]; rfl\n#align is_open_iff_generate_intervals isOpen_iff_generate_intervals\n\ntheorem isOpen_lt' (a : α) : IsOpen { b : α | a < b } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩\n#align is_open_lt' isOpen_lt'\n\ntheorem isOpen_gt' (a : α) : IsOpen { b : α | b < a } :=\n isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inr rfl⟩\n#align is_open_gt' isOpen_gt'\n\ntheorem lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=\n (isOpen_lt' _).mem_nhds h\n#align lt_mem_nhds lt_mem_nhds\n\ntheorem le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=\n (lt_mem_nhds h).mono fun _ => le_of_lt\n#align le_mem_nhds le_mem_nhds\n\ntheorem gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=\n (isOpen_gt' _).mem_nhds h\n#align gt_mem_nhds gt_mem_nhds\n\ntheorem ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=\n (gt_mem_nhds h).mono fun _ => le_of_lt\n#align ge_mem_nhds ge_mem_nhds\n\ntheorem nhds_eq_order (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by\n rw [t.topology_eq_generate_intervals, nhds_generateFrom]\n simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]\n#align nhds_eq_order nhds_eq_order\n\ntheorem tendsto_order {f : β → α} {a : α} {x : Filter β} :\n Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by\n simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl\n#align tendsto_order tendsto_order\n\ninstance tendstoIccClassNhds (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by\n simp only [nhds_eq_order, iInf_subtype']\n refine\n ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass\n fun s _ => ?_\n refine' ((ordConnected_biInter _).inter (ordConnected_biInter _)).out <;> intro _ _\n exacts [ordConnected_Ioi, ordConnected_Iio]\n#align tendsto_Icc_class_nhds tendstoIccClassNhds\n\ninstance tendstoIcoClassNhds (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self\n#align tendsto_Ico_class_nhds tendstoIcoClassNhds\n\ninstance tendstoIocClassNhds (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self\n#align tendsto_Ioc_class_nhds tendstoIocClassNhds\n\ninstance tendstoIooClassNhds (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) :=\n tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self\n#align tendsto_Ioo_class_nhds tendstoIooClassNhds\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold eventually for the filter. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)\n (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) :=\n (hg.Icc hh).of_smallSets <| hgf.and hfh\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_of_tendsto_of_tendsto_of_le_of_le'\n\n/-- **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities\nhold everywhere. -/\ntheorem tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : Filter β} {a : α}\n (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :\n Tendsto f b (𝓝 a) :=\n tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (eventually_of_forall hgf)\n (eventually_of_forall hfh)\n#align tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_of_tendsto_of_tendsto_of_le_of_le\n\ntheorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :\n 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by\n simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl\n#align nhds_order_unbounded nhds_order_unbounded\n\ntheorem tendsto_order_unbounded {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u)\n (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :\n Tendsto f x (𝓝 a) := by\n simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]\n exact fun l hl u => h l u hl\n#align tendsto_order_unbounded tendsto_order_unbounded\n\nend Preorder\n\ninstance tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}\n {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :\n TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) :=\n Filter.tendstoIxxClass_inf\n#align tendsto_Ixx_nhds_within tendstoIxxNhdsWithin\n\ninstance tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]\n [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :\n TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by\n constructor\n conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]\n simp only [smallSets_iInf, smallSets_comap, tendsto_iInf, tendsto_lift', (· ∘ ·),\n mem_powerset_iff]\n intro i s hs\n have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f\n refine' (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _\n exact fun p hp g hg => hp ⟨hg.1 _, hg.2 _⟩\n#align tendsto_Icc_class_nhds_pi tendstoIccClassNhdsPi\n\n-- porting note: new lemma\ntheorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :\n induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_of_nhds_le_nhds fun x => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]\n refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)\n exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha]\n\n-- porting note: new lemma\ntheorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]\n [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)\n (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :\n induced f ‹TopologicalSpace β› = Preorder.topology α := by\n let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩\n refine le_antisymm (induced_topology_le_preorder hf) ?_\n refine le_of_nhds_le_nhds fun a => ?_\n simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]\n refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)\n · rcases em (∃ x, ¬(b < f x)) with (⟨x, hx⟩ | hb)\n · rcases H₁ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz))\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n · rcases em (∃ x, ¬(f x < b)) with (⟨x, hx⟩ | hb)\n · rcases H₂ hb hx with ⟨y, hya, hyb⟩\n exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb)\n · push_neg at hb\n exact le_principal_iff.2 (univ_mem' hb)\n\ntheorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :\n @OrderTopology _ (induced f ta) _ :=\n let _ := induced f ta\n ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩\n#align induced_order_topology' induced_orderTopology'\n\ntheorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]\n [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)\n (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ :=\n induced_orderTopology' f (hf)\n (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)\n fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩\n#align induced_order_topology induced_orderTopology\n\n/-- The topology induced by a strictly monotone function with order-connected range is the preorder\ntopology. -/\nnonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]\n [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by\n refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩\n exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩\n\n/-- A strictly monotone function between linear orders with order topology is a topological\nembedding provided that the range of `f` is order-connected. -/\ntheorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]\n [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}\n (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f :=\n ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩\n\n/-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the\norder is the same as the restriction to the subset of the order topology. -/\ninstance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]\n [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t :=\n ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder $ by\n rwa [← @Subtype.range_val _ t] at ht⟩\n#align order_topology_of_ord_connected orderTopology_of_ordConnected\n\ntheorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by\n rw [nhdsWithin, nhds_eq_order]\n refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)\n exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)\n#align nhds_within_Ici_eq'' nhdsWithin_Ici_eq''\n\ntheorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :\n 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) :=\n nhdsWithin_Ici_eq'' (toDual a)\n#align nhds_within_Iic_eq'' nhdsWithin_Iic_eq''\n\ntheorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by\n simp only [nhdsWithin_Ici_eq'', biInf_inf ha, inf_principal, Iio_inter_Ici]\n#align nhds_within_Ici_eq' nhdsWithin_Ici_eq'\n\ntheorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by\n simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]\n#align nhds_within_Iic_eq' nhdsWithin_Iic_eq'\n\ntheorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n (nhdsWithin_Ici_eq' ha).symm ▸\n hasBasis_biInf_principal\n (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),\n Ico_subset_Ico_right (min_le_right _ _)⟩)\n ha\n#align nhds_within_Ici_basis' nhdsWithin_Ici_basis'\n\ntheorem nhdsWithin_Iic_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}\n (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by\n convert nhdsWithin_Ici_basis' (α := αᵒᵈ) ha using 2\n exact dual_Ico.symm\n#align nhds_within_Iic_basis' nhdsWithin_Iic_basis'\n\ntheorem nhdsWithin_Ici_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α]\n (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u :=\n nhdsWithin_Ici_basis' (exists_gt a)\n#align nhds_within_Ici_basis nhdsWithin_Ici_basis\n\ntheorem nhdsWithin_Iic_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α]\n (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a :=\n nhdsWithin_Iic_basis' (exists_lt a)\n#align nhds_within_Iic_basis nhdsWithin_Iic_basis\n\ntheorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :\n 𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)]\n#align nhds_top_order nhds_top_order\n\ntheorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] :\n 𝓝 (⊥ : α) = ⨅ (l) (h₂ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)]\n#align nhds_bot_order nhds_bot_order\n\ntheorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by\n have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top\n simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this\n#align nhds_top_basis nhds_top_basis\n\ntheorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a :=\n nhds_top_basis (α := αᵒᵈ)\n#align nhds_bot_basis nhds_bot_basis\n\ntheorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=\n nhds_top_basis.to_hasBasis\n (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)\n fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩\n#align nhds_top_basis_Ici nhds_top_basis_Ici\n\ntheorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α]\n [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic :=\n nhds_top_basis_Ici (α := αᵒᵈ)\n#align nhds_bot_basis_Iic nhds_bot_basis_Iic\n\ntheorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by\n simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢\n intro x hx\n filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le\n#align tendsto_nhds_top_mono tendsto_nhds_top_mono\n\ntheorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_top_mono (β := βᵒᵈ) hf hg\n#align tendsto_nhds_bot_mono tendsto_nhds_bot_mono\n\ntheorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) :=\n tendsto_nhds_top_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_top_mono' tendsto_nhds_top_mono'\n\ntheorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β]\n {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) :=\n tendsto_nhds_bot_mono hf (eventually_of_forall hg)\n#align tendsto_nhds_bot_mono' tendsto_nhds_bot_mono'\n\nsection LinearOrder\n\nvariable [TopologicalSpace α] [LinearOrder α]\n\nsection OrderClosedTopology\n\nvariable [OrderClosedTopology α] {a b : α}\n\ntheorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab\n#align eventually_le_nhds eventually_le_nhds\n\ntheorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab\n#align eventually_lt_nhds eventually_lt_nhds\n\ntheorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab\n#align eventually_ge_nhds eventually_ge_nhds\n\ntheorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab\n#align eventually_gt_nhds eventually_gt_nhds\n\nend OrderClosedTopology\n\nsection OrderTopology\n\nvariable [OrderTopology α]\n\ntheorem order_separated {a₁ a₂ : α} (h : a₁ < a₂) :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ :=\n let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi\n ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩\n#align order_separated order_separated\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OrderTopology.to_orderClosedTopology : OrderClosedTopology α where\n isClosed_le' := isOpen_compl_iff.1 <| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) =>\n have h : a₂ < a₁ := lt_of_not_ge h\n let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h\n ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <| h b₂ h₂ b₁ h₁⟩\n#align order_topology.to_order_closed_topology OrderTopology.to_orderClosedTopology\n\ntheorem exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :\n ∃ l < a, Ioc l a ⊆ s :=\n (nhdsWithin_Iic_basis' h).mem_iff.mp (nhdsWithin_le_nhds hs)\n#align exists_Ioc_subset_of_mem_nhds exists_Ioc_subset_of_mem_nhds\n\ntheorem exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :\n ∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=\n let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩\n ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,\n (Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩\n#align exists_Ioc_subset_of_mem_nhds' exists_Ioc_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :\n ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by\n simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual\n#align exists_Ico_subset_of_mem_nhds' exists_Ico_subset_of_mem_nhds'\n\ntheorem exists_Ico_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :\n ∃ u, a < u ∧ Ico a u ⊆ s :=\n let ⟨_l', hl'⟩ := h;\n let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl'\n ⟨l, hl.1.1, hl.2⟩\n#align exists_Ico_subset_of_mem_nhds exists_Ico_subset_of_mem_nhds\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Ici {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :\n ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by\n rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)\n · use a\n simpa [ha.Ici_eq] using hs\n · rcases (nhdsWithin_Ici_basis' ha).mem_iff.mp hs with ⟨b, hab, hbs⟩\n rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hac, hcb⟩)\n · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty]\n exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsWithin_Ici' hab, hbs⟩⟩\n · refine' ⟨c, hac.le, Icc_mem_nhdsWithin_Ici' hac, _⟩\n exact (Icc_subset_Ico_right hcb).trans hbs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Ici exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\n\ntheorem exists_Icc_mem_subset_of_mem_nhdsWithin_Iic {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) :\n ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by\n simpa only [dual_Icc, toDual.surjective.exists] using\n exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (α := αᵒᵈ) (a := toDual a) hs\n#align exists_Icc_mem_subset_of_mem_nhds_within_Iic exists_Icc_mem_subset_of_mem_nhdsWithin_Iic\n\ntheorem exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :\n ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩\n rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩\n refine' ⟨b, c, ⟨hba, hac⟩, _⟩\n rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]\n exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩\n#align exists_Icc_mem_subset_of_mem_nhds exists_Icc_mem_subset_of_mem_nhds\n\ntheorem IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :\n ∃ a b, a < b ∧ Ioo a b ⊆ s := by\n obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h\n obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x\n rcases lt_trichotomy x y with (H | rfl | H)\n · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=\n exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩\n · exact (hy rfl).elim\n · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=\n exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩\n exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩\n#align is_open.exists_Ioo_subset IsOpen.exists_Ioo_subset\n\ntheorem dense_of_exists_between [Nontrivial α] {s : Set α}\n (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by\n refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_\n obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty\n obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab\n exact ⟨x, ⟨H hx, xs⟩⟩\n#align dense_of_exists_between dense_of_exists_between\n\n/-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only\nif for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass\nassumptions. -/\ntheorem dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :\n Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b :=\n ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩\n#align dense_iff_exists_between dense_iff_exists_between\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,\nprovided `a` is neither a bottom element nor a top element. -/\ntheorem mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by\n constructor\n · intro h\n rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩\n rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩\n exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩\n · rintro ⟨l, u, ha, h⟩\n apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h\n#align mem_nhds_iff_exists_Ioo_subset' mem_nhds_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.\n-/\ntheorem mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=\n mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a)\n#align mem_nhds_iff_exists_Ioo_subset mem_nhds_iff_exists_Ioo_subset\n\ntheorem nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩\n#align nhds_basis_Ioo' nhds_basis_Ioo'\n\ntheorem nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :\n (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 :=\n nhds_basis_Ioo' (exists_lt a) (exists_gt a)\n#align nhds_basis_Ioo nhds_basis_Ioo\n\ntheorem Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop}\n (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } :=\n mem_nhds_iff_exists_Ioo_subset.1 hp\n#align filter.eventually.exists_Ioo_subset Filter.Eventually.exists_Ioo_subset\n\ntheorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :\n ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by\n refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_\n refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)\n · simp only [union_subset_iff, image_subset_iff]\n exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩\n · rintro _ ⟨a, rfl | rfl⟩\n · rw [hs.Ioi_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1\n · rw [hs.Iio_eq_biUnion]\n let _ := generateFrom (Ioi '' s ∪ Iio '' s)\n exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1\n\nvariable (α)\n\n/-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then\nit has second countable topology. Note that the \"densely ordered\" assumption cannot be dropped, see\n[double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/\ntheorem SecondCountableTopology.of_separableSpace_orderTopology [DenselyOrdered α]\n [SeparableSpace α] : SecondCountableTopology α := by\n rcases exists_countable_dense α with ⟨s, hc, hd⟩\n refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩\n exact (hc.image _).union (hc.image _)\n\nvariable {α}\n\n-- porting note: new lemma\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_right [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x ⋖ y } := by\n nontriviality α\n let s := { x : α | ∃ y, x ⋖ y }\n have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id\n choose! y hy using this\n have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt\n suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a }\n · have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by\n rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩\n exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩\n refine Set.Countable.mono this ?_\n refine' Countable.biUnion (countable_countableBasis α) fun a ha => H _ _\n exact isOpen_of_mem_countableBasis ha\n intro a ha\n suffices H : Set.Countable { x | (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x }\n · exact H.of_diff (subsingleton_isBot α).countable\n simp only [and_assoc]\n let t := { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x }\n have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by\n intro x hx\n apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1)\n simpa only [IsBot, not_forall, not_le] using hx.right.right.right\n choose! z hz h'z using this\n have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by\n rcases hxx'.lt_or_lt with (h' | h')\n · refine' disjoint_left.2 fun u ux ux' => xt.2.2.1 _\n refine' h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h')\n · refine' disjoint_left.2 fun u ux ux' => x't.2.2.1 _\n refine' h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), _⟩\n by_contra! H\n exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h')\n refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩\n suffices H : Ioc (z x) x = Ioo (z x) (y x)\n · rw [H]\n exact isOpen_Ioo\n exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\n@[deprecated countable_setOf_covby_right]\ntheorem countable_of_isolated_right' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right\n#align countable_of_isolated_right countable_of_isolated_right'\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_covby_left [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y ⋖ x } := by\n convert countable_setOf_covby_right (α := αᵒᵈ) using 5\n exact toDual_covby_toDual_iff.symm\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_of_isolated_left' [SecondCountableTopology α] :\n Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by\n simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left\n#align countable_of_isolated_left countable_of_isolated_left'\n\n/-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space.\nThen the family is countable.\nThis is not a straightforward consequence of second-countability as some of these intervals might be\nempty (but in fact this can happen only for countably many of them). -/\ntheorem Set.PairwiseDisjoint.countable_of_Ioo [SecondCountableTopology α] {y : α → α} {s : Set α}\n (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable :=\n have : (s \\ { x | ∃ y, x ⋖ y }).Countable :=\n (h.subset (diff_subset _ _)).countable_of_isOpen (fun _ _ => isOpen_Ioo)\n fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2)\n this.of_diff countable_setOf_covby_right\n#align set.pairwise_disjoint.countable_of_Ioo Set.PairwiseDisjoint.countable_of_Ioo\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by\n /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)`\n which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a\n family can only be countable as `α` is second-countable. -/\n nontriviality β\n have : Nonempty α := Nonempty.map f (by infer_instance)\n let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y}\n have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx\n -- choose `z x` such that `f` does not take the values in `(f x, z x)`.\n choose! z hz using this\n have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy\n -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals\n -- (the intervals `(f x, z x)`) is at most countable.\n have fs_count : (f '' s).Countable := by\n have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by\n rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv\n wlog hle : u ≤ v generalizing u v\n · exact (this v vs u us huv.symm (le_of_not_le hle)).symm\n have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)\n apply disjoint_iff_forall_ne.2\n rintro a ha b hb rfl\n simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb\n exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)\n apply Set.PairwiseDisjoint.countable_of_Ioo A\n rintro _ ⟨y, ys, rfl⟩\n simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1\n exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/\ntheorem countable_image_lt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} :=\n countable_image_lt_image_Ioi (β := βᵒᵈ) f\n\n/-- For a function taking values in a second countable space, the set of points `x` for\nwhich the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/\ntheorem countable_image_gt_image_Iio [LinearOrder β] (f : β → α) [SecondCountableTopology α] :\n Set.Countable {x | ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} :=\n countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f\n\ninstance instIsCountablyGenerated_atTop [SecondCountableTopology α] :\n IsCountablyGenerated (atTop : Filter α) := by\n by_cases h : ∃ (x : α), IsTop x\n · rcases h with ⟨x, hx⟩\n rw [atTop_eq_pure_of_isTop hx]\n exact isCountablyGenerated_pure x\n · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩\n have : Countable b := by exact Iff.mpr countable_coe_iff b_count\n have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by\n intro s\n have : (s : Set α) ≠ ∅ := by\n intro H\n apply b_ne\n convert s.2\n exact H.symm\n exact Iff.mp nmem_singleton_empty this\n choose a ha using A\n have : (atTop : Filter α) = (generate (Ici '' (range a))) := by\n apply atTop_eq_generate_of_not_bddAbove\n intro ⟨x, hx⟩\n simp only [IsTop, not_exists, not_forall, not_le] at h\n rcases h x with ⟨y, hy⟩\n obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=\n hb.exists_subset_of_mem_open hy isOpen_Ioi\n have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)\n have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)\n exact lt_irrefl _ (I.trans_lt J)\n rw [this]\n exact ⟨_, (countable_range _).image _, rfl⟩\n\ninstance instIsCountablyGenerated_atBot [SecondCountableTopology α] :\n IsCountablyGenerated (atBot : Filter α) :=\n @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _\n\nsection Pi\n\n/-!\n### Intervals in `Π i, π i` belong to `𝓝 x`\n\nFor each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because\nsometimes Lean fails to unify different instances while trying to apply the dependent version to,\ne.g., `ι → ℝ`.\n-/\n\nvariable {ι : Type*} {π : ι → Type*} [Finite ι] [∀ i, LinearOrder (π i)]\n [∀ i, TopologicalSpace (π i)] [∀ i, OrderTopology (π i)] {a b x : ∀ i, π i} {a' b' x' : ι → α}\n\ntheorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=\n pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _)\n#align pi_Iic_mem_nhds pi_Iic_mem_nhds\n\ntheorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=\n pi_Iic_mem_nhds ha\n#align pi_Iic_mem_nhds' pi_Iic_mem_nhds'\n\ntheorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=\n pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _)\n#align pi_Ici_mem_nhds pi_Ici_mem_nhds\n\ntheorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=\n pi_Ici_mem_nhds ha\n#align pi_Ici_mem_nhds' pi_Ici_mem_nhds'\n\ntheorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=\n pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _)\n#align pi_Icc_mem_nhds pi_Icc_mem_nhds\n\ntheorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=\n pi_Icc_mem_nhds ha hb\n#align pi_Icc_mem_nhds' pi_Icc_mem_nhds'\n\nvariable [Nonempty ι]\n\ntheorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)\n exact Iio_mem_nhds (ha i)\n#align pi_Iio_mem_nhds pi_Iio_mem_nhds\n\ntheorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=\n pi_Iio_mem_nhds ha\n#align pi_Iio_mem_nhds' pi_Iio_mem_nhds'\n\ntheorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=\n @pi_Iio_mem_nhds ι (fun i => (π i)ᵒᵈ) _ _ _ _ _ _ _ ha\n#align pi_Ioi_mem_nhds pi_Ioi_mem_nhds\n\ntheorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=\n pi_Ioi_mem_nhds ha\n#align pi_Ioi_mem_nhds' pi_Ioi_mem_nhds'\n\ntheorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)\n exact Ioc_mem_nhds (ha i) (hb i)\n#align pi_Ioc_mem_nhds pi_Ioc_mem_nhds\n\ntheorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=\n pi_Ioc_mem_nhds ha hb\n#align pi_Ioc_mem_nhds' pi_Ioc_mem_nhds'\n\ntheorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)\n exact Ico_mem_nhds (ha i) (hb i)\n#align pi_Ico_mem_nhds pi_Ico_mem_nhds\n\ntheorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=\n pi_Ico_mem_nhds ha hb\n#align pi_Ico_mem_nhds' pi_Ico_mem_nhds'\n\ntheorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by\n refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)\n exact Ioo_mem_nhds (ha i) (hb i)\n#align pi_Ioo_mem_nhds pi_Ioo_mem_nhds\n\ntheorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=\n pi_Ioo_mem_nhds ha hb\n#align pi_Ioo_mem_nhds' pi_Ioo_mem_nhds'\n\nend Pi\n\ntheorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by\n rcases exists_gt x with ⟨y, hy : x < y⟩\n refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)\n exact disjoint_left.mpr fun z => not_le.2\n#align disjoint_nhds_at_top disjoint_nhds_atTop\n\n@[simp]\ntheorem inf_nhds_atTop [NoMaxOrder α] (x : α) : 𝓝 x ⊓ atTop = ⊥ :=\n disjoint_iff.1 (disjoint_nhds_atTop x)\n#align inf_nhds_at_top inf_nhds_atTop\n\ntheorem disjoint_nhds_atBot [NoMinOrder α] (x : α) : Disjoint (𝓝 x) atBot :=\n disjoint_nhds_atTop (α := αᵒᵈ) x\n#align disjoint_nhds_at_bot disjoint_nhds_atBot\n\n@[simp]\ntheorem inf_nhds_atBot [NoMinOrder α] (x : α) : 𝓝 x ⊓ atBot = ⊥ :=\n inf_nhds_atTop (α := αᵒᵈ) x\n#align inf_nhds_at_bot inf_nhds_atBot\n\ntheorem not_tendsto_nhds_of_tendsto_atTop [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atTop) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atTop x).symm\n#align not_tendsto_nhds_of_tendsto_at_top not_tendsto_nhds_of_tendsto_atTop\n\ntheorem not_tendsto_atTop_of_tendsto_nhds [NoMaxOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atTop :=\n hf.not_tendsto (disjoint_nhds_atTop x)\n#align not_tendsto_at_top_of_tendsto_nhds not_tendsto_atTop_of_tendsto_nhds\n\ntheorem not_tendsto_nhds_of_tendsto_atBot [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n (hf : Tendsto f F atBot) (x : α) : ¬Tendsto f F (𝓝 x) :=\n hf.not_tendsto (disjoint_nhds_atBot x).symm\n#align not_tendsto_nhds_of_tendsto_at_bot not_tendsto_nhds_of_tendsto_atBot\n\ntheorem not_tendsto_atBot_of_tendsto_nhds [NoMinOrder α] {F : Filter β} [NeBot F] {f : β → α}\n {x : α} (hf : Tendsto f F (𝓝 x)) : ¬Tendsto f F atBot :=\n hf.not_tendsto (disjoint_nhds_atBot x)\n#align not_tendsto_at_bot_of_tendsto_nhds not_tendsto_atBot_of_tendsto_nhds\n\n/-!\n### Neighborhoods to the left and to the right on an `OrderTopology`\n\nWe've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`.\nIn an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable\nintervals to the right or to the left of `a`. We give now these characterizations. -/\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `(a, +∞)`;\n1. `s` is a neighborhood of `a` within `(a, b]`;\n2. `s` is a neighborhood of `a` within `(a, b)`;\n3. `s` includes `(a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `(a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[>] a,\n s ∈ 𝓝[Ioc a b] a,\n s ∈ 𝓝[Ioo a b] a,\n ∃ u ∈ Ioc a b, Ioo a u ⊆ s,\n ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 1\n · rintro ⟨u, hau, hu⟩\n exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu\n tfae_have 1 → 4\n · intro h\n rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩\n rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩\n refine' ⟨u, au, fun x hx => _⟩\n refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩\n exact hx.1\n tfae_finish\n#align tfae_mem_nhds_within_Ioi TFAE_mem_nhdsWithin_Ioi\n\ntheorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 3\n#align mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ioi hu' s).out 0 4\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset' mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'\n\ntheorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' h⟩\n\nlemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=\n nhdsWithin_Ioi_basis' <| exists_gt a\n\ntheorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by\n by_cases ha : IsTop a\n · simp [ha, ha.isMax.Ioi_eq]\n · simp only [ha, false_or]\n rw [isTop_iff_isMax, not_isMax_iff] at ha\n simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq]\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s :=\n let ⟨_u', hu'⟩ := exists_gt a\n mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' hu'\n#align mem_nhds_within_Ioi_iff_exists_Ioo_subset mem_nhdsWithin_Ioi_iff_exists_Ioo_subset\n\n/-- The set of points which are isolated on the right is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_right [SecondCountableTopology α] :\n { x : α | 𝓝[>] x = ⊥ }.Countable := by\n simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]\n exact (subsingleton_isTop α).countable.union countable_setOf_covby_right\n\n/-- The set of points which are isolated on the left is countable when the space is\nsecond-countable. -/\ntheorem countable_setOf_isolated_left [SecondCountableTopology α] :\n { x : α | 𝓝[<] x = ⊥ }.Countable :=\n countable_setOf_isolated_right (α := αᵒᵈ)\n\n/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by\n rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩\n#align mem_nhds_within_Ioi_iff_exists_Ioc_subset mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b)`\n1. `s` is a neighborhood of `b` within `[a, b)`\n2. `s` is a neighborhood of `b` within `(a, b)`\n3. `s` includes `(l, b)` for some `l ∈ [a, b)`\n4. `s` includes `(l, b)` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`\n s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`\n s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`\n ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using\n TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iio TFAE_mem_nhdsWithin_Iio\n\ntheorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 3\n#align mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iio hl' s).out 0 4\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset' mem_nhdsWithin_Iio_iff_exists_Ioo_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s :=\n let ⟨_, h⟩ := exists_lt a\n mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h\n#align mem_nhds_within_Iio_iff_exists_Ioo_subset mem_nhdsWithin_Iio_iff_exists_Ioo_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by\n have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset\n simpa only [OrderDual.exists, exists_prop, dual_Ioc] using this\n#align mem_nhds_within_Iio_iff_exists_Ico_subset mem_nhdsWithin_Iio_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=\n let ⟨_, h⟩ := h\n ⟨fun _ => mem_nhdsWithin_Iio_iff_exists_Ioo_subset' h⟩\n\ntheorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by\n convert (config := {preTransparency := .default})\n nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4\n exact ofDual_covby_ofDual_iff\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `a` within `[a, +∞)`;\n1. `s` is a neighborhood of `a` within `[a, b]`;\n2. `s` is a neighborhood of `a` within `[a, b)`;\n3. `s` includes `[a, u)` for some `u ∈ (a, b]`;\n4. `s` includes `[a, u)` for some `u > a`.\n-/\ntheorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≥] a,\n s ∈ 𝓝[Icc a b] a,\n s ∈ 𝓝[Ico a b] a,\n ∃ u ∈ Ioc a b, Ico a u ⊆ s,\n ∃ u ∈ Ioi a , Ico a u ⊆ s] := by\n tfae_have 1 ↔ 2\n · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 3\n · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]\n tfae_have 1 ↔ 5\n · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff\n tfae_have 4 → 5\n · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩\n tfae_have 5 → 4\n · rintro ⟨u, hua, hus⟩\n exact\n ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,\n (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩\n tfae_finish\n#align tfae_mem_nhds_within_Ici TFAE_mem_nhdsWithin_Ici\n\ntheorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u < u'`, provided `a` is not a top element. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n (TFAE_mem_nhdsWithin_Ici hu' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Ici_iff_exists_Ico_subset' mem_nhdsWithin_Ici_iff_exists_Ico_subset'\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s :=\n let ⟨_, hu'⟩ := exists_gt a\n mem_nhdsWithin_Ici_iff_exists_Ico_subset' hu'\n#align mem_nhds_within_Ici_iff_exists_Ico_subset mem_nhdsWithin_Ici_iff_exists_Ico_subset\n\ntheorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :\n (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=\n ⟨fun _ => mem_nhdsWithin_Ici_iff_exists_Ico_subset⟩\n#align nhds_within_Ici_basis_Ico nhdsWithin_Ici_basis_Ico\n\n/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`\nwith `a < u`. -/\ntheorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := by\n rw [mem_nhdsWithin_Ici_iff_exists_Ico_subset]\n constructor\n · rintro ⟨u, au, as⟩\n rcases exists_between au with ⟨v, hv⟩\n exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩\n · rintro ⟨u, au, as⟩\n exact ⟨u, au, Subset.trans Ico_subset_Icc_self as⟩\n#align mem_nhds_within_Ici_iff_exists_Icc_subset mem_nhdsWithin_Ici_iff_exists_Icc_subset\n\nopen List in\n/-- The following statements are equivalent:\n\n0. `s` is a neighborhood of `b` within `(-∞, b]`\n1. `s` is a neighborhood of `b` within `[a, b]`\n2. `s` is a neighborhood of `b` within `(a, b]`\n3. `s` includes `(l, b]` for some `l ∈ [a, b)`\n4. `s` includes `(l, b]` for some `l < b` -/\ntheorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :\n TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`\n s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`\n s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`\n ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`\n ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`\n simpa only [exists_prop, OrderDual.exists, dual_Ici, dual_Ioc, dual_Icc, dual_Ico] using\n TFAE_mem_nhdsWithin_Ici h.dual (ofDual ⁻¹' s)\n#align tfae_mem_nhds_within_Iic TFAE_mem_nhdsWithin_Iic\n\ntheorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 3 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`, provided `a` is not a bottom element. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n (TFAE_mem_nhdsWithin_Iic hl' s).out 0 4 (by norm_num) (by norm_num)\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset' mem_nhdsWithin_Iic_iff_exists_Ioc_subset'\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :\n s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s :=\n let ⟨_, hl'⟩ := exists_lt a\n mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hl'\n#align mem_nhds_within_Iic_iff_exists_Ioc_subset mem_nhdsWithin_Iic_iff_exists_Ioc_subset\n\n/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`\nwith `l < a`. -/\ntheorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}\n {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=\n calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl\n _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=\n mem_nhdsWithin_Ici_iff_exists_Icc_subset\n _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp only [dual_Icc]; rfl\n#align mem_nhds_within_Iic_iff_exists_Icc_subset mem_nhdsWithin_Iic_iff_exists_Icc_subset\n\nend OrderTopology\n\nend LinearOrder\n\nsection LinearOrderedAddCommGroup\n\nvariable [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α]\n\nvariable {l : Filter β} {f g : β → α}\n\ntheorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } := by\n simp only [nhds_eq_order, abs_lt, setOf_and, ← inf_principal, iInf_inf_eq]\n refine (congr_arg₂ _ ?_ ?_).trans inf_comm\n · refine (Equiv.subLeft a).iInf_congr fun x => ?_; simp [Ioi]\n · refine (Equiv.subRight a).iInf_congr fun x => ?_; simp [Iio]\n#align nhds_eq_infi_abs_sub nhds_eq_iInf_abs_sub\n\ntheorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]\n (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α := by\n refine' ⟨eq_of_nhds_eq_nhds fun a => _⟩\n rw [h_nhds]\n letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩\n exact (nhds_eq_iInf_abs_sub a).symm\n#align order_topology_of_nhds_abs orderTopology_of_nhds_abs\n\ntheorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :\n Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε := by\n simp [nhds_eq_iInf_abs_sub, abs_sub_comm a]\n#align linear_ordered_add_comm_group.tendsto_nhds LinearOrderedAddCommGroup.tendsto_nhds\n\ntheorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=\n (nhds_eq_iInf_abs_sub a).symm ▸\n mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [abs_sub_comm, mem_principal_self])\n#align eventually_abs_sub_lt eventually_abs_sub_lt\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atTop` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :\n Tendsto (fun x => f x + g x) l atTop := by\n nontriviality α\n obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C\n refine' tendsto_atTop_add_left_of_le' _ C' _ hg\n exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt\n#align filter.tendsto.add_at_top Filter.Tendsto.add_atTop\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`\nand `g` tends to `atBot` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :\n Tendsto (fun x => f x + g x) l atBot :=\n Filter.Tendsto.add_atTop (α := αᵒᵈ) hf hg\n#align filter.tendsto.add_at_bot Filter.Tendsto.add_atBot\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/\ntheorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atTop := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atTop hf\n#align filter.tendsto.at_top_add Filter.Tendsto.atTop_add\n\n/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to\n`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/\ntheorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :\n Tendsto (fun x => f x + g x) l atBot := by\n conv in _ + _ => rw [add_comm]\n exact hg.add_atBot hf\n#align filter.tendsto.at_bot_add Filter.Tendsto.atBot_add\n\ntheorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } := by\n simp only [nhds_eq_iInf_abs_sub, abs_sub_comm (a := a)]\n refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)\n exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),\n fun _ hz => hz.trans_le (min_le_right _ _)⟩\n#align nhds_basis_abs_sub_lt nhds_basis_abs_sub_lt\n\ntheorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by\n convert nhds_basis_abs_sub_lt a\n simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]\n#align nhds_basis_Ioo_pos nhds_basis_Ioo_pos\n\ntheorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :\n (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).to_hasBasis\n (fun _ε ε₀ ↦ let ⟨δ, δ₀, δε⟩ := exists_between ε₀\n ⟨δ, δ₀, Icc_subset_Ioo (sub_lt_sub_left δε _) (add_lt_add_left δε _)⟩)\n (fun ε ε₀ ↦ ⟨ε, ε₀, Ioo_subset_Icc_self⟩)\n\nvariable (α)\n\ntheorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :\n (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } := by\n simpa using nhds_basis_abs_sub_lt (0 : α)\n#align nhds_basis_zero_abs_sub_lt nhds_basis_zero_abs_sub_lt\n\nvariable {α}\n\n/-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/\ntheorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :\n (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) :=\n (nhds_basis_Ioo_pos a).restrict fun ε hε => ⟨min a ε, lt_min ha hε, min_le_left _ _,\n Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _) (add_le_add_left (min_le_right _ _) _)⟩\n#align nhds_basis_Ioo_pos_of_pos nhds_basis_Ioo_pos_of_pos\n\nend LinearOrderedAddCommGroup\n\n@[deprecated image_neg]\ntheorem preimage_neg [AddGroup α] : preimage (Neg.neg : α → α) = image (Neg.neg : α → α) :=\n funext fun _ => image_neg.symm\n#align preimage_neg preimage_neg\n\n@[deprecated] -- use `Filter.map_neg` from `Mathlib.Order.Filter.Pointwise`\ntheorem Filter.map_neg_eq_comap_neg [AddGroup α] :\n map (Neg.neg : α → α) = comap (Neg.neg : α → α) :=\n funext fun _ => map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)\n#align filter.map_neg_eq_comap_neg Filter.map_neg_eq_comap_neg\n\nsection OrderTopology\n\nvariable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]\n [OrderTopology β]\n\ntheorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≤] a, x ∈ s := by\n rcases hs with ⟨a', ha'⟩\n intro h\n rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)\n · exact h.self_of_nhdsWithin le_rfl ha'\n · rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩\n rcases ha.exists_between hba with ⟨b', hb's, hb'⟩\n exact hb hb' hb's\n#align is_lub.frequently_mem IsLUB.frequently_mem\n\ntheorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_lub.frequently_nhds_mem IsLUB.frequently_nhds_mem\n\ntheorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝[≥] a, x ∈ s :=\n IsLUB.frequently_mem (α := αᵒᵈ) ha hs\n#align is_glb.frequently_mem IsGLB.frequently_mem\n\ntheorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :\n ∃ᶠ x in 𝓝 a, x ∈ s :=\n (ha.frequently_mem hs).filter_mono inf_le_left\n#align is_glb.frequently_nhds_mem IsGLB.frequently_nhds_mem\n\ntheorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_lub.mem_closure IsLUB.mem_closure\n\ntheorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=\n (ha.frequently_nhds_mem hs).mem_closure\n#align is_glb.mem_closure IsGLB.mem_closure\n\ntheorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :\n NeBot (𝓝[s] a) :=\n mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)\n#align is_lub.nhds_within_ne_bot IsLUB.nhdsWithin_neBot\n\ntheorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=\n IsLUB.nhdsWithin_neBot (α := αᵒᵈ)\n#align is_glb.nhds_within_ne_bot IsGLB.nhdsWithin_neBot\n\ntheorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)\n [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=\n ⟨hsa, fun b hb =>\n not_lt.1 fun hba =>\n have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)\n let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this\n have : b < b := lt_of_lt_of_le hxb <| hb hxs\n lt_irrefl b this⟩\n#align is_lub_of_mem_nhds isLUB_of_mem_nhds\n\ntheorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :\n IsLUB s a := by\n rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf\n exact isLUB_of_mem_nhds hsa (mem_principal_self s)\n#align is_lub_of_mem_closure isLUB_of_mem_closure\n\ntheorem isGLB_of_mem_nhds :\n ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=\n isLUB_of_mem_nhds (α := αᵒᵈ)\n#align is_glb_of_mem_nhds isGLB_of_mem_nhds\n\ntheorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :\n IsGLB s a :=\n isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf\n#align is_glb_of_mem_closure isGLB_of_mem_closure\n\ntheorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by\n rintro _ ⟨x, hx, rfl⟩\n replace ha := ha.inter_Ici_of_mem hx\n haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩\n refine' ge_of_tendsto (hb.mono_left (nhdsWithin_mono _ (inter_subset_left s (Ici x)))) _\n exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2\n#align is_lub.mem_upper_bounds_of_tendsto IsLUB.mem_upperBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α`\n-- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop`\ntheorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=\n haveI := ha.nhdsWithin_neBot hs\n ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>\n le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩\n#align is_lub.is_lub_of_tendsto IsLUB.isLUB_of_tendsto\n\ntheorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb\n#align is_glb.mem_lower_bounds_of_tendsto IsGLB.mem_lowerBounds_of_tendsto\n\n-- For a version of this theorem in which the convergence considered on the domain `α` is as\n-- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see\n-- `isGLB_of_tendsto_atBot`\ntheorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}\n {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :\n IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual\n#align is_glb.is_glb_of_tendsto IsGLB.isGLB_of_tendsto\n\ntheorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=\n IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_lub.mem_lower_bounds_of_tendsto IsLUB.mem_lowerBounds_of_tendsto\n\ntheorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=\n IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)\n#align is_lub.is_glb_of_tendsto IsLUB.isGLB_of_tendsto\n\ntheorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]\n {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)\n (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=\n IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb\n#align is_glb.mem_upper_bounds_of_tendsto IsGLB.mem_upperBounds_of_tendsto\n\ntheorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :\n ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},\n AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=\n IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)\n#align is_glb.is_lub_of_tendsto IsGLB.isLUB_of_tendsto\n\ntheorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_lub.mem_of_is_closed IsLUB.mem_of_isClosed\n\nalias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed\n#align is_closed.is_lub_mem IsClosed.isLUB_mem\n\ntheorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)\n (sc : IsClosed s) : a ∈ s :=\n sc.closure_subset <| ha.mem_closure hs\n#align is_glb.mem_of_is_closed IsGLB.mem_of_isClosed\n\nalias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed\n#align is_closed.is_glb_mem IsClosed.isGLB_mem\n\n/-!\n### Existence of sequences tending to `sInf` or `sSup` of a given set\n-/\n\ntheorem IsLUB.exists_seq_strictMono_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)\n replace hvx := hvx.mono_right nhdsWithin_le_nhds\n have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) not_mem)\n have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')\n choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists\n refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',\n hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩\n · rw [iterate_succ_apply']; exact hvN _\n · rw [iterate_succ_apply']; exact hN _\n#align is_lub.exists_seq_strict_mono_tendsto_of_not_mem IsLUB.exists_seq_strictMono_tendsto_of_not_mem\n\ntheorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsLUB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by\n by_cases h : x ∈ t\n · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩\n · rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩\n exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩\n#align is_lub.exists_seq_monotone_tendsto IsLUB.exists_seq_monotone_tendsto\n\ntheorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]\n [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by\n have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim\n have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy\n rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2.symm⟩\n#align exists_seq_strict_mono_tendsto' exists_seq_strictMono_tendsto'\n\ntheorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by\n obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x\n rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩\n exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩\n#align exists_seq_strict_mono_tendsto exists_seq_strictMono_tendsto\n\ntheorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=\n let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x\n ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩\n#align exists_seq_strict_mono_tendsto_nhds_within exists_seq_strictMono_tendsto_nhdsWithin\n\ntheorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by\n rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩\n exact ⟨u, hu.1, hu.2.2⟩\n#align exists_seq_tendsto_Sup exists_seq_tendsto_sSup\n\ntheorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}\n [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_strictMono_tendsto_of_not_mem (α := αᵒᵈ) htx not_mem ht\n#align is_glb.exists_seq_strict_anti_tendsto_of_not_mem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem\n\ntheorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]\n (htx : IsGLB t x) (ht : t.Nonempty) :\n ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=\n IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht\n#align is_glb.exists_seq_antitone_tendsto IsGLB.exists_seq_antitone_tendsto\n\ntheorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}\n (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by\n simpa only [dual_Ioo]\n using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy)\n#align exists_seq_strict_anti_tendsto' exists_seq_strictAnti_tendsto'\n\ntheorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]\n (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=\n exists_seq_strictMono_tendsto (α := αᵒᵈ) x\n#align exists_seq_strict_anti_tendsto exists_seq_strictAnti_tendsto\n\ntheorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]\n [FirstCountableTopology α] (x : α) :\n ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=\n exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _\n#align exists_seq_strict_anti_tendsto_nhds_within exists_seq_strictAnti_tendsto_nhdsWithin\n\ntheorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]\n {x y : α} (h : x < y) :\n ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧\n (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by\n rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩\n rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩\n exact\n ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,\n fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩\n#align exists_seq_strict_anti_strict_mono_tendsto exists_seq_strictAnti_strictMono_tendsto\n\ntheorem exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]\n [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)\n (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=\n exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS'\n#align exists_seq_tendsto_Inf exists_seq_tendsto_sInf\n\nend OrderTopology\n\nsection DenselyOrdered\n\nvariable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}\n {s : Set α}\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top\nelement. -/\ntheorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by\n apply Subset.antisymm\n · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici\n · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]\n exact isGLB_Ioi.mem_closure h\n#align closure_Ioi' closure_Ioi'\n\n/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/\n@[simp]\ntheorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=\n closure_Ioi' nonempty_Ioi\n#align closure_Ioi closure_Ioi\n\n/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom\nelement. -/\ntheorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=\n closure_Ioi' (α := αᵒᵈ) h\n#align closure_Iio' closure_Iio'\n\n/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/\n@[simp]\ntheorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=\n closure_Iio' nonempty_Iio\n#align closure_Iio closure_Iio\n\n/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc\n · cases' hab.lt_or_lt with hab hab\n · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]\n have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab\n simp only [insert_subset_iff, singleton_subset_iff]\n exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩\n · rw [Icc_eq_empty_of_lt hab]\n exact empty_subset _\n#align closure_Ioo closure_Ioo\n\n/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)\n rw [closure_Ioo hab]\n#align closure_Ioc closure_Ioc\n\n/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/\n@[simp]\ntheorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by\n apply Subset.antisymm\n · exact closure_minimal Ico_subset_Icc_self isClosed_Icc\n · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)\n rw [closure_Ioo hab]\n#align closure_Ico closure_Ico\n\n@[simp]\ntheorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by\n rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]\n#align interior_Ici' interior_Ici'\n\ntheorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=\n interior_Ici' nonempty_Iio\n#align interior_Ici interior_Ici\n\n@[simp]\ntheorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=\n interior_Ici' (α := αᵒᵈ) ha\n#align interior_Iic' interior_Iic'\n\ntheorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=\n interior_Iic' nonempty_Ioi\n#align interior_Iic interior_Iic\n\n@[simp]\ntheorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by\n rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]\n#align interior_Icc interior_Icc\n\n@[simp]\ntheorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by\n rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]\n#align interior_Ico interior_Ico\n\n@[simp]\ntheorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by\n rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]\n#align interior_Ioc interior_Ioc\n\ntheorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=\n (closure_minimal interior_subset isClosed_Icc).antisymm <|\n calc\n Icc a b = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Icc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo)\n#align closure_interior_Icc closure_interior_Icc\n\ntheorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by\n rcases eq_or_ne a b with (rfl | h)\n · simp\n · calc\n Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self\n _ = closure (Ioo a b) := (closure_Ioo h).symm\n _ ⊆ closure (interior (Ioc a b)) :=\n closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)\n#align Ioc_subset_closure_interior Ioc_subset_closure_interior\n\ntheorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by\n simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)\n#align Ico_subset_closure_interior Ico_subset_closure_interior\n\n@[simp]\ntheorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by\n simp [frontier, ha]\n#align frontier_Ici' frontier_Ici'\n\ntheorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=\n frontier_Ici' nonempty_Iio\n#align frontier_Ici frontier_Ici\n\n@[simp]\ntheorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by\n simp [frontier, ha]\n#align frontier_Iic' frontier_Iic'\n\ntheorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=\n frontier_Iic' nonempty_Ioi\n#align frontier_Iic frontier_Iic\n\n@[simp]\ntheorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by\n simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Ioi' frontier_Ioi'\n\ntheorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=\n frontier_Ioi' nonempty_Ioi\n#align frontier_Ioi frontier_Ioi\n\n@[simp]\ntheorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by\n simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]\n#align frontier_Iio' frontier_Iio'\n\ntheorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=\n frontier_Iio' nonempty_Iio\n#align frontier_Iio frontier_Iio\n\n@[simp]\ntheorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :\n frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]\n#align frontier_Icc frontier_Icc\n\n@[simp]\ntheorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by\n rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]\n#align frontier_Ioo frontier_Ioo\n\n@[simp]\ntheorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by\n rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]\n#align frontier_Ico frontier_Ico\n\n@[simp]\ntheorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by\n rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]\n#align frontier_Ioc frontier_Ioc\n\ntheorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :\n NeBot (𝓝[Ioi a] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁]\n#align nhds_within_Ioi_ne_bot' nhdsWithin_Ioi_neBot'\n\ntheorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=\n nhdsWithin_Ioi_neBot' nonempty_Ioi H\n#align nhds_within_Ioi_ne_bot nhdsWithin_Ioi_neBot\n\ntheorem nhdsWithin_Ioi_self_neBot' {a : α} (H : (Ioi a).Nonempty) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot' H (le_refl a)\n#align nhds_within_Ioi_self_ne_bot' nhdsWithin_Ioi_self_neBot'\n\ninstance nhdsWithin_Ioi_self_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) :=\n nhdsWithin_Ioi_neBot (le_refl a)\n#align nhds_within_Ioi_self_ne_bot nhdsWithin_Ioi_self_neBot\n\ntheorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :\n NeBot (𝓝[Iio c] b) :=\n mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁]\n#align nhds_within_Iio_ne_bot' nhdsWithin_Iio_neBot'\n\ntheorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=\n nhdsWithin_Iio_neBot' nonempty_Iio H\n#align nhds_within_Iio_ne_bot nhdsWithin_Iio_neBot\n\ntheorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=\n nhdsWithin_Iio_neBot' H (le_refl b)\n#align nhds_within_Iio_self_ne_bot' nhdsWithin_Iio_self_neBot'\n\ninstance nhdsWithin_Iio_self_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) :=\n nhdsWithin_Iio_neBot (le_refl a)\n#align nhds_within_Iio_self_ne_bot nhdsWithin_Iio_self_neBot\n\ntheorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=\n (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H)\n#align right_nhds_within_Ico_ne_bot right_nhdsWithin_Ico_neBot\n\ntheorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=\n (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H)\n#align left_nhds_within_Ioc_ne_bot left_nhdsWithin_Ioc_neBot\n\ntheorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=\n (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align left_nhds_within_Ioo_ne_bot left_nhdsWithin_Ioo_neBot\n\ntheorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=\n (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H)\n#align right_nhds_within_Ioo_ne_bot right_nhdsWithin_Ioo_neBot\n\ntheorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)\n (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by\n nontriviality\n haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›\n rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩\n ext u; constructor\n · rintro ⟨t, ht, hts⟩\n obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=\n (mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht\n obtain ⟨y, hxy, hyb⟩ := exists_between hxb\n refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _\n rintro ⟨z, hzs⟩ (hyz : y ≤ z)\n exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)\n · intro hu\n obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu\n exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩\n#align comap_coe_nhds_within_Iio_of_Ioo_subset comap_coe_nhdsWithin_Iio_of_Ioo_subset\n\ntheorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)\n (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by\n simpa only [OrderDual.exists, dual_Ioo] using hs h\n#align comap_coe_nhds_within_Ioi_of_Ioo_subset comap_coe_nhdsWithin_Ioi_of_Ioo_subset\n\ntheorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :\n map ((↑) : s → α) atTop = 𝓝[<] b := by\n rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)\n · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩\n rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]\n · rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]\n rw [Subtype.range_val]\n exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha)\n#align map_coe_at_top_of_Ioo_subset map_coe_atTop_of_Ioo_subset\n\ntheorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :\n map ((↑) : s → α) atBot = 𝓝[>] a := by\n -- the elaborator gets stuck without `(... : _)`\n refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)\n fun b' hb' => _ : _)\n simpa only [OrderDual.exists, dual_Ioo] using hs b' hb'\n#align map_coe_at_bot_of_Ioo_subset map_coe_atBot_of_Ioo_subset\n\n/-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at\nthe right endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Iio (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=\n comap_coe_nhdsWithin_Iio_of_Ioo_subset Ioo_subset_Iio_self fun h =>\n ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Iio comap_coe_Ioo_nhdsWithin_Iio\n\n/-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at\nthe left endpoint in the ambient order. -/\ntheorem comap_coe_Ioo_nhdsWithin_Ioi (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset Ioo_subset_Ioi_self fun h =>\n ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩\n#align comap_coe_Ioo_nhds_within_Ioi comap_coe_Ioo_nhdsWithin_Ioi\n\ntheorem comap_coe_Ioi_nhdsWithin_Ioi (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=\n comap_coe_nhdsWithin_Ioi_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩\n#align comap_coe_Ioi_nhds_within_Ioi comap_coe_Ioi_nhdsWithin_Ioi\n\ntheorem comap_coe_Iio_nhdsWithin_Iio (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=\n comap_coe_Ioi_nhdsWithin_Ioi (α := αᵒᵈ) a\n#align comap_coe_Iio_nhds_within_Iio comap_coe_Iio_nhdsWithin_Iio\n\n@[simp]\ntheorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=\n map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_top map_coe_Ioo_atTop\n\n@[simp]\ntheorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩\n#align map_coe_Ioo_at_bot map_coe_Ioo_atBot\n\n@[simp]\ntheorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=\n map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩\n#align map_coe_Ioi_at_bot map_coe_Ioi_atBot\n\n@[simp]\ntheorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=\n map_coe_Ioi_atBot (α := αᵒᵈ) _\n#align map_coe_Iio_at_top map_coe_Iio_atTop\n\nvariable {l : Filter β} {f : α → β}\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atTop (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by\n rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_top tendsto_comp_coe_Ioo_atTop\n\n@[simp]\ntheorem tendsto_comp_coe_Ioo_atBot (h : a < b) :\n Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioo_at_bot tendsto_comp_coe_Ioo_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Ioi_atBot :\n Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by\n rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Ioi_at_bot tendsto_comp_coe_Ioi_atBot\n\n-- porting note: todo: `simpNF` claims that `simp` can't use this lemma to simplify LHS but it can\n@[simp, nolint simpNF]\ntheorem tendsto_comp_coe_Iio_atTop :\n Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by\n rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl\n#align tendsto_comp_coe_Iio_at_top tendsto_comp_coe_Iio_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atTop {f : β → Ioo a b} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by\n rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_top tendsto_Ioo_atTop\n\n@[simp]\ntheorem tendsto_Ioo_atBot {f : β → Ioo a b} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioo_at_bot tendsto_Ioo_atBot\n\n@[simp]\ntheorem tendsto_Ioi_atBot {f : β → Ioi a} :\n Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by\n rw [← comap_coe_Ioi_nhdsWithin_Ioi, tendsto_comap_iff]; rfl\n#align tendsto_Ioi_at_bot tendsto_Ioi_atBot\n\n@[simp]\ntheorem tendsto_Iio_atTop {f : β → Iio a} :\n Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by\n rw [← comap_coe_Iio_nhdsWithin_Iio, tendsto_comap_iff]; rfl\n#align tendsto_Iio_at_top tendsto_Iio_atTop\n\ninstance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by\n refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_\n obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs\n obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩\n obtain ⟨y, hy⟩ : ∃ y, a < y ∧ y < b := exists_between a_lt_b\n rcases ne_or_eq x y with (xy | rfl)\n · exact ⟨y, us ⟨hab hy, xy.symm⟩⟩\n obtain ⟨z, hz⟩ : ∃ z, a < z ∧ z < x := exists_between hy.1\n exact ⟨z, us ⟨hab ⟨hz.1, hz.2.trans hy.2⟩, hz.2.ne⟩⟩\n\n/-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a\nseparable space (e.g., if `α` has a second countable topology), then there exists a countable\ndense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/\ntheorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivial α] {s : Set α} [SeparableSpace s]\n (hs : Dense s) :\n ∃ t, t ⊆ s ∧ t.Countable ∧ Dense t ∧ (∀ x, IsBot x → x ∉ t) ∧ ∀ x, IsTop x → x ∉ t := by\n rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, htd⟩\n "},"nextTactic":{"kind":"string","value":"refine' ⟨t \\ ({ x | IsBot x } ∪ { x | IsTop x }), _, _, _, fun x hx => _, fun x hx => _⟩"},"declUpToTactic":{"kind":"string","value":"/-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a\nseparable space (e.g., if `α` has a second countable topology), then there exists a countable\ndense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/\ntheorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivial α] {s : Set α} [SeparableSpace s]\n (hs : Dense s) :\n ∃ t, t ⊆ s ∧ t.Countable ∧ Dense t ∧ (∀ x, IsBot x → x ∉ t) ∧ ∀ x, IsTop x → x ∉ t := by\n rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, htd⟩\n "},"declId":{"kind":"string","value":"Mathlib.Topology.Order.Basic.2643_0.Npdof1X5b8sCkZ2"},"decl":{"kind":"string","value":"/-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a\nseparable space (e.g., if `α` has a second countable topology), then there exists a countable\ndense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/\ntheorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivial α] {s : Set α} [SeparableSpace s]\n (hs : Dense s) :\n ∃ t, t ⊆ s ∧ t.Countable ∧ Dense t ∧ (∀ x, IsBot x → x ∉ t) ∧ ∀ x, IsTop x → x ∉ t "},"file_tag":{"kind":"string","value":"Mathlib_Topology_Order_Basic"}}}],"truncated":false,"partial":true},"paginationData":{"pageIndex":2078,"numItemsPerPage":100,"numTotalItems":213206,"offset":207800,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4Njk4NDg5NSwianRpIjoiODBhZWY4NGMtYjNjYy00OWM0LWJjMTQtOTcwODNkMGM0YTRhIiwic3ViIjoiL2RhdGFzZXRzL2wzbGFiL250cC1tYXRobGliIiwiZXhwIjoxNzg2OTg4NDk1LCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.MPxUdixBD7RFAGBtjmRL2AKEEtO1PV-KeBcYxCKX0CveFyJcnfleuKRw7k0ugtJbSY2wPw8Nht8xPmPJafH2Dw","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":213206,"numBytesParquet":1087420658}]},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
α✝ : Type u
β : Type v
γ : Type w
inst✝¹⁰ : TopologicalSpace α✝
inst✝⁹ : TopologicalSpace β
inst✝⁸ : LinearOrder α✝
inst✝⁷ : LinearOrder β
inst✝⁶ : OrderTopology α✝
inst✝⁵ : OrderTopology β
α : Type u_1
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : DenselyOrdered α
inst✝¹ : OrderTopology α
inst✝ : FirstCo... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy | theorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]
[DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :
∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by
have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).e... | Mathlib.Topology.Order.Basic.2194_0.Npdof1X5b8sCkZ2 | theorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]
[DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :
∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) | Mathlib_Topology_Order_Basic |
α✝ : Type u
β : Type v
γ : Type w
inst✝¹⁰ : TopologicalSpace α✝
inst✝⁹ : TopologicalSpace β
inst✝⁸ : LinearOrder α✝
inst✝⁷ : LinearOrder β
inst✝⁶ : OrderTopology α✝
inst✝⁵ : OrderTopology β
α : Type u_1
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : DenselyOrdered α
inst✝¹ : OrderTopology α
inst✝ : FirstCo... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_not_mem hx ht with ⟨u, hu⟩ | theorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]
[DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :
∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by
have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).e... | Mathlib.Topology.Order.Basic.2194_0.Npdof1X5b8sCkZ2 | theorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]
[DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :
∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) | Mathlib_Topology_Order_Basic |
case intro
α✝ : Type u
β : Type v
γ : Type w
inst✝¹⁰ : TopologicalSpace α✝
inst✝⁹ : TopologicalSpace β
inst✝⁸ : LinearOrder α✝
inst✝⁷ : LinearOrder β
inst✝⁶ : OrderTopology α✝
inst✝⁵ : OrderTopology β
α : Type u_1
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : DenselyOrdered α
inst✝¹ : OrderTopology α
inst... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact ⟨u, hu.1, hu.2.2.symm⟩ | theorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]
[DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :
∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by
have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).e... | Mathlib.Topology.Order.Basic.2194_0.Npdof1X5b8sCkZ2 | theorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]
[DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :
∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁸ : TopologicalSpace α
inst✝⁷ : TopologicalSpace β
inst✝⁶ : LinearOrder α
inst✝⁵ : LinearOrder β
inst✝⁴ : OrderTopology α
inst✝³ : OrderTopology β
inst✝² : DenselyOrdered α
inst✝¹ : NoMinOrder α
inst✝ : FirstCountableTopology α
x : α
⊢ ∃ u, StrictMono u ∧ (∀ (n : ℕ), u n < x) ∧ Ten... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x | theorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]
(x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by
| Mathlib.Topology.Order.Basic.2203_0.Npdof1X5b8sCkZ2 | theorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]
(x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) | Mathlib_Topology_Order_Basic |
case intro
α : Type u
β : Type v
γ : Type w
inst✝⁸ : TopologicalSpace α
inst✝⁷ : TopologicalSpace β
inst✝⁶ : LinearOrder α
inst✝⁵ : LinearOrder β
inst✝⁴ : OrderTopology α
inst✝³ : OrderTopology β
inst✝² : DenselyOrdered α
inst✝¹ : NoMinOrder α
inst✝ : FirstCountableTopology α
x y : α
hy : y < x
⊢ ∃ u, StrictMono u ∧ (∀... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩ | theorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]
(x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by
obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x
| Mathlib.Topology.Order.Basic.2203_0.Npdof1X5b8sCkZ2 | theorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]
(x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) | Mathlib_Topology_Order_Basic |
case intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝⁸ : TopologicalSpace α
inst✝⁷ : TopologicalSpace β
inst✝⁶ : LinearOrder α
inst✝⁵ : LinearOrder β
inst✝⁴ : OrderTopology α
inst✝³ : OrderTopology β
inst✝² : DenselyOrdered α
inst✝¹ : NoMinOrder α
inst✝ : FirstCountableTopology α
x y : α
hy : y < x
u : ℕ ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩ | theorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]
(x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by
obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x
rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩
| Mathlib.Topology.Order.Basic.2203_0.Npdof1X5b8sCkZ2 | theorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]
(x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) | Mathlib_Topology_Order_Basic |
α✝ : Type u
β : Type v
γ : Type w
inst✝⁹ : TopologicalSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : LinearOrder α✝
inst✝⁶ : LinearOrder β
inst✝⁵ : OrderTopology α✝
inst✝⁴ : OrderTopology β
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : FirstCountabl... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩ | theorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]
[TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)
(hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by
| Mathlib.Topology.Order.Basic.2217_0.Npdof1X5b8sCkZ2 | theorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]
[TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)
(hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S | Mathlib_Topology_Order_Basic |
case intro
α✝ : Type u
β : Type v
γ : Type w
inst✝⁹ : TopologicalSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : LinearOrder α✝
inst✝⁶ : LinearOrder β
inst✝⁵ : OrderTopology α✝
inst✝⁴ : OrderTopology β
α : Type u_1
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : Fi... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact ⟨u, hu.1, hu.2.2⟩ | theorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]
[TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)
(hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by
rcases (isLUB_csSup hS hS').exists_... | Mathlib.Topology.Order.Basic.2217_0.Npdof1X5b8sCkZ2 | theorem exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]
[TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)
(hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : LinearOrder α
inst✝⁴ : LinearOrder β
inst✝³ : OrderTopology α
inst✝² : OrderTopology β
inst✝¹ : DenselyOrdered α
inst✝ : FirstCountableTopology α
x y : α
hy : x < y
⊢ ∃ u, StrictAnti u ∧ (∀ (n : ℕ), u n ∈ Ioo x y) ∧ Tendst... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simpa only [dual_Ioo]
using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy) | theorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}
(hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by
| Mathlib.Topology.Order.Basic.2236_0.Npdof1X5b8sCkZ2 | theorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}
(hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : LinearOrder α
inst✝⁴ : LinearOrder β
inst✝³ : OrderTopology α
inst✝² : OrderTopology β
inst✝¹ : DenselyOrdered α
inst✝ : FirstCountableTopology α
x y : α
h : x < y
⊢ ∃ u v,
StrictAnti u ∧
StrictMono v ∧
(... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩ | theorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]
{x y : α} (h : x < y) :
∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧
(∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by
| Mathlib.Topology.Order.Basic.2253_0.Npdof1X5b8sCkZ2 | theorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]
{x y : α} (h : x < y) :
∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧
(∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) | Mathlib_Topology_Order_Basic |
case intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : LinearOrder α
inst✝⁴ : LinearOrder β
inst✝³ : OrderTopology α
inst✝² : OrderTopology β
inst✝¹ : DenselyOrdered α
inst✝ : FirstCountableTopology α
x y : α
h : x < y
u : ℕ → α
hu_anti : StrictAnti u
hu... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩ | theorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]
{x y : α} (h : x < y) :
∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧
(∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by
rcases exists_seq_s... | Mathlib.Topology.Order.Basic.2253_0.Npdof1X5b8sCkZ2 | theorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]
{x y : α} (h : x < y) :
∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧
(∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) | Mathlib_Topology_Order_Basic |
case intro.intro.intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : LinearOrder α
inst✝⁴ : LinearOrder β
inst✝³ : OrderTopology α
inst✝² : OrderTopology β
inst✝¹ : DenselyOrdered α
inst✝ : FirstCountableTopology α
x y : α
h : x < y
u : ℕ → α
hu_anti... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact
⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,
fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩ | theorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]
{x y : α} (h : x < y) :
∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧
(∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by
rcases exists_seq_s... | Mathlib.Topology.Order.Basic.2253_0.Npdof1X5b8sCkZ2 | theorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]
{x y : α} (h : x < y) :
∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧
(∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
a : α
h : Set.Nonempty (Ioi a)
⊢ closure (Ioi a) = Ici a | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | apply Subset.antisymm | /-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top
element. -/
theorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by
| Mathlib.Topology.Order.Basic.2277_0.Npdof1X5b8sCkZ2 | /-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top
element. -/
theorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a | Mathlib_Topology_Order_Basic |
case h₁
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
a : α
h : Set.Nonempty (Ioi a)
⊢ closure (Ioi a) ⊆ Ici a | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact closure_minimal Ioi_subset_Ici_self isClosed_Ici | /-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top
element. -/
theorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by
apply Subset.antisymm
· | Mathlib.Topology.Order.Basic.2277_0.Npdof1X5b8sCkZ2 | /-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top
element. -/
theorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a | Mathlib_Topology_Order_Basic |
case h₂
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
a : α
h : Set.Nonempty (Ioi a)
⊢ Ici a ⊆ closure (Ioi a) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff] | /-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top
element. -/
theorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by
apply Subset.antisymm
· exact closure_minimal Ioi_subset_Ici_self isClosed_Ici
· | Mathlib.Topology.Order.Basic.2277_0.Npdof1X5b8sCkZ2 | /-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top
element. -/
theorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a | Mathlib_Topology_Order_Basic |
case h₂
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
a : α
h : Set.Nonempty (Ioi a)
⊢ a ∈ closure (Ioi a) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact isGLB_Ioi.mem_closure h | /-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top
element. -/
theorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by
apply Subset.antisymm
· exact closure_minimal Ioi_subset_Ici_self isClosed_Ici
· rw [← diff_subset_closure_iff, Ici_diff_I... | Mathlib.Topology.Order.Basic.2277_0.Npdof1X5b8sCkZ2 | /-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top
element. -/
theorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab : a ≠ b
⊢ closure (Ioo a b) = Icc a b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | apply Subset.antisymm | /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by
| Mathlib.Topology.Order.Basic.2304_0.Npdof1X5b8sCkZ2 | /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b | Mathlib_Topology_Order_Basic |
case h₁
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab : a ≠ b
⊢ closure (Ioo a b) ⊆ Icc a b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact closure_minimal Ioo_subset_Icc_self isClosed_Icc | /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by
apply Subset.antisymm
· | Mathlib.Topology.Order.Basic.2304_0.Npdof1X5b8sCkZ2 | /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b | Mathlib_Topology_Order_Basic |
case h₂
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab : a ≠ b
⊢ Icc a b ⊆ closure (Ioo a b) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | cases' hab.lt_or_lt with hab hab | /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by
apply Subset.antisymm
· exact closure_minimal Ioo_subset_Icc_self isClosed_Icc
· | Mathlib.Topology.Order.Basic.2304_0.Npdof1X5b8sCkZ2 | /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b | Mathlib_Topology_Order_Basic |
case h₂.inl
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab✝ : a ≠ b
hab : a < b
⊢ Icc a b ⊆ closure (Ioo a b) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le] | /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by
apply Subset.antisymm
· exact closure_minimal Ioo_subset_Icc_self isClosed_Icc
· cases' hab.lt_or_lt with hab hab
· | Mathlib.Topology.Order.Basic.2304_0.Npdof1X5b8sCkZ2 | /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b | Mathlib_Topology_Order_Basic |
case h₂.inl
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab✝ : a ≠ b
hab : a < b
⊢ {a, b} ⊆ closure (Ioo a b) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab | /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by
apply Subset.antisymm
· exact closure_minimal Ioo_subset_Icc_self isClosed_Icc
· cases' hab.lt_or_lt with hab hab
· rw [← diff_subset_closur... | Mathlib.Topology.Order.Basic.2304_0.Npdof1X5b8sCkZ2 | /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b | Mathlib_Topology_Order_Basic |
case h₂.inl
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab✝ : a ≠ b
hab : a < b
hab' : Set.Nonempty (Ioo a b)
⊢ {a, b} ⊆ closure (Ioo a b) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simp only [insert_subset_iff, singleton_subset_iff] | /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by
apply Subset.antisymm
· exact closure_minimal Ioo_subset_Icc_self isClosed_Icc
· cases' hab.lt_or_lt with hab hab
· rw [← diff_subset_closur... | Mathlib.Topology.Order.Basic.2304_0.Npdof1X5b8sCkZ2 | /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b | Mathlib_Topology_Order_Basic |
case h₂.inl
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab✝ : a ≠ b
hab : a < b
hab' : Set.Nonempty (Ioo a b)
⊢ a ∈ closure (Ioo a b) ∧ b ∈ closure (Ioo a b) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩ | /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by
apply Subset.antisymm
· exact closure_minimal Ioo_subset_Icc_self isClosed_Icc
· cases' hab.lt_or_lt with hab hab
· rw [← diff_subset_closur... | Mathlib.Topology.Order.Basic.2304_0.Npdof1X5b8sCkZ2 | /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b | Mathlib_Topology_Order_Basic |
case h₂.inr
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab✝ : a ≠ b
hab : b < a
⊢ Icc a b ⊆ closure (Ioo a b) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [Icc_eq_empty_of_lt hab] | /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by
apply Subset.antisymm
· exact closure_minimal Ioo_subset_Icc_self isClosed_Icc
· cases' hab.lt_or_lt with hab hab
· rw [← diff_subset_closur... | Mathlib.Topology.Order.Basic.2304_0.Npdof1X5b8sCkZ2 | /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b | Mathlib_Topology_Order_Basic |
case h₂.inr
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab✝ : a ≠ b
hab : b < a
⊢ ∅ ⊆ closure (Ioo a b) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact empty_subset _ | /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by
apply Subset.antisymm
· exact closure_minimal Ioo_subset_Icc_self isClosed_Icc
· cases' hab.lt_or_lt with hab hab
· rw [← diff_subset_closur... | Mathlib.Topology.Order.Basic.2304_0.Npdof1X5b8sCkZ2 | /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab : a ≠ b
⊢ closure (Ioc a b) = Icc a b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | apply Subset.antisymm | /-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by
| Mathlib.Topology.Order.Basic.2318_0.Npdof1X5b8sCkZ2 | /-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b | Mathlib_Topology_Order_Basic |
case h₁
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab : a ≠ b
⊢ closure (Ioc a b) ⊆ Icc a b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact closure_minimal Ioc_subset_Icc_self isClosed_Icc | /-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by
apply Subset.antisymm
· | Mathlib.Topology.Order.Basic.2318_0.Npdof1X5b8sCkZ2 | /-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b | Mathlib_Topology_Order_Basic |
case h₂
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab : a ≠ b
⊢ Icc a b ⊆ closure (Ioc a b) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self) | /-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by
apply Subset.antisymm
· exact closure_minimal Ioc_subset_Icc_self isClosed_Icc
· | Mathlib.Topology.Order.Basic.2318_0.Npdof1X5b8sCkZ2 | /-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab : a ≠ b
⊢ Icc a b ⊆ closure (Ioo a b) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [closure_Ioo hab] | /-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by
apply Subset.antisymm
· exact closure_minimal Ioc_subset_Icc_self isClosed_Icc
· apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)
| Mathlib.Topology.Order.Basic.2318_0.Npdof1X5b8sCkZ2 | /-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab : a ≠ b
⊢ closure (Ico a b) = Icc a b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | apply Subset.antisymm | /-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by
| Mathlib.Topology.Order.Basic.2327_0.Npdof1X5b8sCkZ2 | /-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b | Mathlib_Topology_Order_Basic |
case h₁
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab : a ≠ b
⊢ closure (Ico a b) ⊆ Icc a b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact closure_minimal Ico_subset_Icc_self isClosed_Icc | /-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by
apply Subset.antisymm
· | Mathlib.Topology.Order.Basic.2327_0.Npdof1X5b8sCkZ2 | /-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b | Mathlib_Topology_Order_Basic |
case h₂
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab : a ≠ b
⊢ Icc a b ⊆ closure (Ico a b) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | apply Subset.trans _ (closure_mono Ioo_subset_Ico_self) | /-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by
apply Subset.antisymm
· exact closure_minimal Ico_subset_Icc_self isClosed_Icc
· | Mathlib.Topology.Order.Basic.2327_0.Npdof1X5b8sCkZ2 | /-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab : a ≠ b
⊢ Icc a b ⊆ closure (Ioo a b) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [closure_Ioo hab] | /-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by
apply Subset.antisymm
· exact closure_minimal Ico_subset_Icc_self isClosed_Icc
· apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)
| Mathlib.Topology.Order.Basic.2327_0.Npdof1X5b8sCkZ2 | /-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/
@[simp]
theorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
a : α
ha : Set.Nonempty (Iio a)
⊢ interior (Ici a) = Ioi a | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic] | @[simp]
theorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by
| Mathlib.Topology.Order.Basic.2336_0.Npdof1X5b8sCkZ2 | @[simp]
theorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
a✝ b✝ : α
s : Set α
inst✝¹ : NoMinOrder α
inst✝ : NoMaxOrder α
a b : α
⊢ interior (Icc a b) = Ioo a b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio] | @[simp]
theorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by
| Mathlib.Topology.Order.Basic.2354_0.Npdof1X5b8sCkZ2 | @[simp]
theorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : DenselyOrdered α
a✝ b✝ : α
s : Set α
inst✝ : NoMinOrder α
a b : α
⊢ interior (Ico a b) = Ioo a b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio] | @[simp]
theorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by
| Mathlib.Topology.Order.Basic.2359_0.Npdof1X5b8sCkZ2 | @[simp]
theorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : DenselyOrdered α
a✝ b✝ : α
s : Set α
inst✝ : NoMaxOrder α
a b : α
⊢ interior (Ioc a b) = Ioo a b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio] | @[simp]
theorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by
| Mathlib.Topology.Order.Basic.2364_0.Npdof1X5b8sCkZ2 | @[simp]
theorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
⊢ Ioc a b ⊆ closure (interior (Ioc a b)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases eq_or_ne a b with (rfl | h) | theorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by
| Mathlib.Topology.Order.Basic.2377_0.Npdof1X5b8sCkZ2 | theorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) | Mathlib_Topology_Order_Basic |
case inl
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
a : α
⊢ Ioc a a ⊆ closure (interior (Ioc a a)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simp | theorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by
rcases eq_or_ne a b with (rfl | h)
· | Mathlib.Topology.Order.Basic.2377_0.Npdof1X5b8sCkZ2 | theorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) | Mathlib_Topology_Order_Basic |
case inr
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
h : a ≠ b
⊢ Ioc a b ⊆ closure (interior (Ioc a b)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | calc
Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self
_ = closure (Ioo a b) := (closure_Ioo h).symm
_ ⊆ closure (interior (Ioc a b)) :=
closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo) | theorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by
rcases eq_or_ne a b with (rfl | h)
· simp
· | Mathlib.Topology.Order.Basic.2377_0.Npdof1X5b8sCkZ2 | theorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
⊢ Ico a b ⊆ closure (interior (Ico a b)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a) | theorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by
| Mathlib.Topology.Order.Basic.2387_0.Npdof1X5b8sCkZ2 | theorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
a : α
ha : Set.Nonempty (Iio a)
⊢ frontier (Ici a) = {a} | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simp [frontier, ha] | @[simp]
theorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by
| Mathlib.Topology.Order.Basic.2391_0.Npdof1X5b8sCkZ2 | @[simp]
theorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
a : α
ha : Set.Nonempty (Ioi a)
⊢ frontier (Iic a) = {a} | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simp [frontier, ha] | @[simp]
theorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by
| Mathlib.Topology.Order.Basic.2400_0.Npdof1X5b8sCkZ2 | @[simp]
theorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
a : α
ha : Set.Nonempty (Ioi a)
⊢ frontier (Ioi a) = {a} | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simp [frontier, closure_Ioi' ha, Iic_diff_Iio, Icc_self] | @[simp]
theorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by
| Mathlib.Topology.Order.Basic.2409_0.Npdof1X5b8sCkZ2 | @[simp]
theorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
a : α
ha : Set.Nonempty (Iio a)
⊢ frontier (Iio a) = {a} | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self] | @[simp]
theorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by
| Mathlib.Topology.Order.Basic.2418_0.Npdof1X5b8sCkZ2 | @[simp]
theorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
a✝ b✝ : α
s : Set α
inst✝¹ : NoMinOrder α
inst✝ : NoMaxOrder α
a b : α
h : a ≤ b
⊢ frontier (Icc a b) = {a, b} | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simp [frontier, h, Icc_diff_Ioo_same] | @[simp]
theorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :
frontier (Icc a b) = {a, b} := by | Mathlib.Topology.Order.Basic.2427_0.Npdof1X5b8sCkZ2 | @[simp]
theorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :
frontier (Icc a b) = {a, b} | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
h : a < b
⊢ frontier (Ioo a b) = {a, b} | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le] | @[simp]
theorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by
| Mathlib.Topology.Order.Basic.2432_0.Npdof1X5b8sCkZ2 | @[simp]
theorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : DenselyOrdered α
a✝ b✝ : α
s : Set α
inst✝ : NoMinOrder α
a b : α
h : a < b
⊢ frontier (Ico a b) = {a, b} | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le] | @[simp]
theorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by
| Mathlib.Topology.Order.Basic.2437_0.Npdof1X5b8sCkZ2 | @[simp]
theorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : DenselyOrdered α
a✝ b✝ : α
s : Set α
inst✝ : NoMaxOrder α
a b : α
h : a < b
⊢ frontier (Ioc a b) = {a, b} | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le] | @[simp]
theorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by
| Mathlib.Topology.Order.Basic.2442_0.Npdof1X5b8sCkZ2 | @[simp]
theorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
H₁ : Set.Nonempty (Ioi a)
H₂ : a ≤ b
⊢ b ∈ closure (Ioi a) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rwa [closure_Ioi' H₁] | theorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :
NeBot (𝓝[Ioi a] b) :=
mem_closure_iff_nhdsWithin_neBot.1 <| by | Mathlib.Topology.Order.Basic.2447_0.Npdof1X5b8sCkZ2 | theorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :
NeBot (𝓝[Ioi a] b) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b✝ : α
s : Set α
b c : α
H₁ : Set.Nonempty (Iio c)
H₂ : b ≤ c
⊢ b ∈ closure (Iio c) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rwa [closure_Iio' H₁] | theorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :
NeBot (𝓝[Iio c] b) :=
mem_closure_iff_nhdsWithin_neBot.1 <| by | Mathlib.Topology.Order.Basic.2464_0.Npdof1X5b8sCkZ2 | theorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :
NeBot (𝓝[Iio c] b) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
hb : s ⊆ Iio b
hs : Set.Nonempty s → ∃ a < b, Ioo a b ⊆ s
⊢ comap Subtype.val (𝓝[<] b) = atTop | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | nontriviality | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by
| Mathlib.Topology.Order.Basic.2497_0.Npdof1X5b8sCkZ2 | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
hb : s ⊆ Iio b
hs : Set.Nonempty s → ∃ a < b, Ioo a b ⊆ s
✝ : Nontrivial (Filter { x // x ∈ s })
⊢ comap Subtype.val (𝓝[<] b) = atTop | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_› | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by
nontriviality
| Mathlib.Topology.Order.Basic.2497_0.Npdof1X5b8sCkZ2 | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
hb : s ⊆ Iio b
hs : Set.Nonempty s → ∃ a < b, Ioo a b ⊆ s
✝ : Nontrivial (Filter { x // x ∈ s })
this : Nonempty ↑s
⊢ comap Subtype.val (𝓝[<] b) = atTop | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩ | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by
nontriviality
haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›
| Mathlib.Topology.Order.Basic.2497_0.Npdof1X5b8sCkZ2 | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop | Mathlib_Topology_Order_Basic |
case intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
hb : s ⊆ Iio b
hs✝ : Set.Nonempty s → ∃ a < b, Ioo a b ⊆ s
✝ : Nontrivial (Filter { x // x ∈ s })
this : Nonempty ↑s
a : α
h : a < b
hs : Ioo a b ⊆ s
... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | ext u | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by
nontriviality
haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›
rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩
| Mathlib.Topology.Order.Basic.2497_0.Npdof1X5b8sCkZ2 | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop | Mathlib_Topology_Order_Basic |
case intro.intro.a
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
hb : s ⊆ Iio b
hs✝ : Set.Nonempty s → ∃ a < b, Ioo a b ⊆ s
✝ : Nontrivial (Filter { x // x ∈ s })
this : Nonempty ↑s
a : α
h : a < b
hs : Ioo a b ⊆ ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | constructor | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by
nontriviality
haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›
rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩
ext u; | Mathlib.Topology.Order.Basic.2497_0.Npdof1X5b8sCkZ2 | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop | Mathlib_Topology_Order_Basic |
case intro.intro.a.mp
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
hb : s ⊆ Iio b
hs✝ : Set.Nonempty s → ∃ a < b, Ioo a b ⊆ s
✝ : Nontrivial (Filter { x // x ∈ s })
this : Nonempty ↑s
a : α
h : a < b
hs : Ioo a b... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rintro ⟨t, ht, hts⟩ | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by
nontriviality
haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›
rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩
ext u; constructor
· | Mathlib.Topology.Order.Basic.2497_0.Npdof1X5b8sCkZ2 | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop | Mathlib_Topology_Order_Basic |
case intro.intro.a.mp.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
hb : s ⊆ Iio b
hs✝ : Set.Nonempty s → ∃ a < b, Ioo a b ⊆ s
✝ : Nontrivial (Filter { x // x ∈ s })
this : Nonempty ↑s
a : α
h : a < b
... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=
(mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by
nontriviality
haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›
rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩
ext u; constructor
· rintro ⟨t, ht, ht... | Mathlib.Topology.Order.Basic.2497_0.Npdof1X5b8sCkZ2 | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop | Mathlib_Topology_Order_Basic |
case intro.intro.a.mp.intro.intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
hb : s ⊆ Iio b
hs✝ : Set.Nonempty s → ∃ a < b, Ioo a b ⊆ s
✝ : Nontrivial (Filter { x // x ∈ s })
this : Nonempty ↑... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | obtain ⟨y, hxy, hyb⟩ := exists_between hxb | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by
nontriviality
haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›
rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩
ext u; constructor
· rintro ⟨t, ht, ht... | Mathlib.Topology.Order.Basic.2497_0.Npdof1X5b8sCkZ2 | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop | Mathlib_Topology_Order_Basic |
case intro.intro.a.mp.intro.intro.intro.intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
hb : s ⊆ Iio b
hs✝ : Set.Nonempty s → ∃ a < b, Ioo a b ⊆ s
✝ : Nontrivial (Filter { x // x ∈ s })
this ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | refine' mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _ | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by
nontriviality
haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›
rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩
ext u; constructor
· rintro ⟨t, ht, ht... | Mathlib.Topology.Order.Basic.2497_0.Npdof1X5b8sCkZ2 | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop | Mathlib_Topology_Order_Basic |
case intro.intro.a.mp.intro.intro.intro.intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
hb : s ⊆ Iio b
hs✝ : Set.Nonempty s → ∃ a < b, Ioo a b ⊆ s
✝ : Nontrivial (Filter { x // x ∈ s })
this ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rintro ⟨z, hzs⟩ (hyz : y ≤ z) | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by
nontriviality
haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›
rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩
ext u; constructor
· rintro ⟨t, ht, ht... | Mathlib.Topology.Order.Basic.2497_0.Npdof1X5b8sCkZ2 | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop | Mathlib_Topology_Order_Basic |
case intro.intro.a.mp.intro.intro.intro.intro.intro.intro.intro.mk
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
hb : s ⊆ Iio b
hs✝ : Set.Nonempty s → ∃ a < b, Ioo a b ⊆ s
✝ : Nontrivial (Filter { x // x ∈ s })
th... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩) | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by
nontriviality
haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›
rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩
ext u; constructor
· rintro ⟨t, ht, ht... | Mathlib.Topology.Order.Basic.2497_0.Npdof1X5b8sCkZ2 | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop | Mathlib_Topology_Order_Basic |
case intro.intro.a.mpr
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
hb : s ⊆ Iio b
hs✝ : Set.Nonempty s → ∃ a < b, Ioo a b ⊆ s
✝ : Nontrivial (Filter { x // x ∈ s })
this : Nonempty ↑s
a : α
h : a < b
hs : Ioo a ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | intro hu | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by
nontriviality
haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›
rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩
ext u; constructor
· rintro ⟨t, ht, ht... | Mathlib.Topology.Order.Basic.2497_0.Npdof1X5b8sCkZ2 | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop | Mathlib_Topology_Order_Basic |
case intro.intro.a.mpr
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
hb : s ⊆ Iio b
hs✝ : Set.Nonempty s → ∃ a < b, Ioo a b ⊆ s
✝ : Nontrivial (Filter { x // x ∈ s })
this : Nonempty ↑s
a : α
h : a < b
hs : Ioo a ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by
nontriviality
haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›
rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩
ext u; constructor
· rintro ⟨t, ht, ht... | Mathlib.Topology.Order.Basic.2497_0.Npdof1X5b8sCkZ2 | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop | Mathlib_Topology_Order_Basic |
case intro.intro.a.mpr.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
hb : s ⊆ Iio b
hs✝ : Set.Nonempty s → ∃ a < b, Ioo a b ⊆ s
✝ : Nontrivial (Filter { x // x ∈ s })
this : Nonempty ↑s
a : α
h : a < b
hs : ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact ⟨Ioo x b, Ioo_mem_nhdsWithin_Iio' (hb x.2), fun z hz => hx _ hz.1.le⟩ | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by
nontriviality
haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›
rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩
ext u; constructor
· rintro ⟨t, ht, ht... | Mathlib.Topology.Order.Basic.2497_0.Npdof1X5b8sCkZ2 | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
ha : s ⊆ Ioi a
hs : Set.Nonempty s → ∃ b > a, Ioo a b ⊆ s
h : Set.Nonempty (⇑ofDual ⁻¹' s)
⊢ ∃ a_1 < toDual a, Ioo a_1 (toDual a) ⊆ ⇑ofDual ⁻¹' s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simpa only [OrderDual.exists, dual_Ioo] using hs h | theorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)
(hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot :=
comap_coe_nhdsWithin_Iio_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun h => by
| Mathlib.Topology.Order.Basic.2515_0.Npdof1X5b8sCkZ2 | theorem comap_coe_nhdsWithin_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)
(hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
hb : s ⊆ Iio b
hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s
⊢ map Subtype.val atTop = 𝓝[<] b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩) | theorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :
map ((↑) : s → α) atTop = 𝓝[<] b := by
| Mathlib.Topology.Order.Basic.2521_0.Npdof1X5b8sCkZ2 | theorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :
map ((↑) : s → α) atTop = 𝓝[<] b | Mathlib_Topology_Order_Basic |
case inl
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
hb : s ⊆ Iio b
hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s
hb' : Iio b = ∅
⊢ map Subtype.val atTop = 𝓝[<] b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩ | theorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :
map ((↑) : s → α) atTop = 𝓝[<] b := by
rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)
· | Mathlib.Topology.Order.Basic.2521_0.Npdof1X5b8sCkZ2 | theorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :
map ((↑) : s → α) atTop = 𝓝[<] b | Mathlib_Topology_Order_Basic |
case inl
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
hb : s ⊆ Iio b
hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s
hb' : Iio b = ∅
this : IsEmpty ↑s
⊢ map Subtype.val atTop = 𝓝[<] b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty] | theorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :
map ((↑) : s → α) atTop = 𝓝[<] b := by
rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)
· have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩
| Mathlib.Topology.Order.Basic.2521_0.Npdof1X5b8sCkZ2 | theorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :
map ((↑) : s → α) atTop = 𝓝[<] b | Mathlib_Topology_Order_Basic |
case inr.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
hb : s ⊆ Iio b
hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s
a : α
ha : a ∈ Iio b
⊢ map Subtype.val atTop = 𝓝[<] b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem] | theorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :
map ((↑) : s → α) atTop = 𝓝[<] b := by
rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)
· have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩
rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWith... | Mathlib.Topology.Order.Basic.2521_0.Npdof1X5b8sCkZ2 | theorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :
map ((↑) : s → α) atTop = 𝓝[<] b | Mathlib_Topology_Order_Basic |
case inr.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
hb : s ⊆ Iio b
hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s
a : α
ha : a ∈ Iio b
⊢ range Subtype.val ∈ 𝓝[<] b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [Subtype.range_val] | theorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :
map ((↑) : s → α) atTop = 𝓝[<] b := by
rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)
· have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩
rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWith... | Mathlib.Topology.Order.Basic.2521_0.Npdof1X5b8sCkZ2 | theorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :
map ((↑) : s → α) atTop = 𝓝[<] b | Mathlib_Topology_Order_Basic |
case inr.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
hb : s ⊆ Iio b
hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s
a : α
ha : a ∈ Iio b
⊢ s ∈ 𝓝[<] b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact (mem_nhdsWithin_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha) | theorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :
map ((↑) : s → α) atTop = 𝓝[<] b := by
rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)
· have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩
rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWith... | Mathlib.Topology.Order.Basic.2521_0.Npdof1X5b8sCkZ2 | theorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :
map ((↑) : s → α) atTop = 𝓝[<] b | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
ha : s ⊆ Ioi a
hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s
⊢ map Subtype.val atBot = 𝓝[>] a | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)
fun b' hb' => _ : _) | theorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :
map ((↑) : s → α) atBot = 𝓝[>] a := by
-- the elaborator gets stuck without `(... : _)`
| Mathlib.Topology.Order.Basic.2531_0.Npdof1X5b8sCkZ2 | theorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :
map ((↑) : s → α) atBot = 𝓝[>] a | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
ha : s ⊆ Ioi a
hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s
b' : αᵒᵈ
hb' : b' < toDual a
⊢ ∃ a_1 < toDual a, Ioo a_1 (toDual a) ⊆ ⇑ofDual ⁻¹' s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simpa only [OrderDual.exists, dual_Ioo] using hs b' hb' | theorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :
map ((↑) : s → α) atBot = 𝓝[>] a := by
-- the elaborator gets stuck without `(... : _)`
refine' (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)
fun b' hb' => _ : _)
| Mathlib.Topology.Order.Basic.2531_0.Npdof1X5b8sCkZ2 | theorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :
map ((↑) : s → α) atBot = 𝓝[>] a | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f : α → β
h : a < b
⊢ Tendsto (fun x => f ↑x) atTop l ↔ Tendsto f (𝓝[<] b) l | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [← map_coe_Ioo_atTop h, tendsto_map'_iff] | @[simp]
theorem tendsto_comp_coe_Ioo_atTop (h : a < b) :
Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by
| Mathlib.Topology.Order.Basic.2583_0.Npdof1X5b8sCkZ2 | @[simp]
theorem tendsto_comp_coe_Ioo_atTop (h : a < b) :
Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f : α → β
h : a < b
⊢ Tendsto (fun x => f ↑x) atTop l ↔ Tendsto (f ∘ Subtype.val) atTop l | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rfl | @[simp]
theorem tendsto_comp_coe_Ioo_atTop (h : a < b) :
Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by
rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; | Mathlib.Topology.Order.Basic.2583_0.Npdof1X5b8sCkZ2 | @[simp]
theorem tendsto_comp_coe_Ioo_atTop (h : a < b) :
Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f : α → β
h : a < b
⊢ Tendsto (fun x => f ↑x) atBot l ↔ Tendsto f (𝓝[>] a) l | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [← map_coe_Ioo_atBot h, tendsto_map'_iff] | @[simp]
theorem tendsto_comp_coe_Ioo_atBot (h : a < b) :
Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by
| Mathlib.Topology.Order.Basic.2589_0.Npdof1X5b8sCkZ2 | @[simp]
theorem tendsto_comp_coe_Ioo_atBot (h : a < b) :
Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f : α → β
h : a < b
⊢ Tendsto (fun x => f ↑x) atBot l ↔ Tendsto (f ∘ Subtype.val) atBot l | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rfl | @[simp]
theorem tendsto_comp_coe_Ioo_atBot (h : a < b) :
Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by
rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; | Mathlib.Topology.Order.Basic.2589_0.Npdof1X5b8sCkZ2 | @[simp]
theorem tendsto_comp_coe_Ioo_atBot (h : a < b) :
Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f : α → β
⊢ Tendsto (fun x => f ↑x) atBot l ↔ Tendsto f (𝓝[>] a) l | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [← map_coe_Ioi_atBot, tendsto_map'_iff] | @[simp, nolint simpNF]
theorem tendsto_comp_coe_Ioi_atBot :
Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by
| Mathlib.Topology.Order.Basic.2596_0.Npdof1X5b8sCkZ2 | @[simp, nolint simpNF]
theorem tendsto_comp_coe_Ioi_atBot :
Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f : α → β
⊢ Tendsto (fun x => f ↑x) atBot l ↔ Tendsto (f ∘ Subtype.val) atBot l | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rfl | @[simp, nolint simpNF]
theorem tendsto_comp_coe_Ioi_atBot :
Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by
rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; | Mathlib.Topology.Order.Basic.2596_0.Npdof1X5b8sCkZ2 | @[simp, nolint simpNF]
theorem tendsto_comp_coe_Ioi_atBot :
Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f : α → β
⊢ Tendsto (fun x => f ↑x) atTop l ↔ Tendsto f (𝓝[<] a) l | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [← map_coe_Iio_atTop, tendsto_map'_iff] | @[simp, nolint simpNF]
theorem tendsto_comp_coe_Iio_atTop :
Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by
| Mathlib.Topology.Order.Basic.2603_0.Npdof1X5b8sCkZ2 | @[simp, nolint simpNF]
theorem tendsto_comp_coe_Iio_atTop :
Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f : α → β
⊢ Tendsto (fun x => f ↑x) atTop l ↔ Tendsto (f ∘ Subtype.val) atTop l | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rfl | @[simp, nolint simpNF]
theorem tendsto_comp_coe_Iio_atTop :
Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by
rw [← map_coe_Iio_atTop, tendsto_map'_iff]; | Mathlib.Topology.Order.Basic.2603_0.Npdof1X5b8sCkZ2 | @[simp, nolint simpNF]
theorem tendsto_comp_coe_Iio_atTop :
Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f✝ : α → β
f : β → ↑(Ioo a b)
⊢ Tendsto f l atTop ↔ Tendsto (fun x => ↑(f x)) l (𝓝[<] b) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff] | @[simp]
theorem tendsto_Ioo_atTop {f : β → Ioo a b} :
Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by
| Mathlib.Topology.Order.Basic.2609_0.Npdof1X5b8sCkZ2 | @[simp]
theorem tendsto_Ioo_atTop {f : β → Ioo a b} :
Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f✝ : α → β
f : β → ↑(Ioo a b)
⊢ Tendsto (Subtype.val ∘ f) l (𝓝[<] b) ↔ Tendsto (fun x => ↑(f x)) l (𝓝[<] b) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rfl | @[simp]
theorem tendsto_Ioo_atTop {f : β → Ioo a b} :
Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by
rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; | Mathlib.Topology.Order.Basic.2609_0.Npdof1X5b8sCkZ2 | @[simp]
theorem tendsto_Ioo_atTop {f : β → Ioo a b} :
Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f✝ : α → β
f : β → ↑(Ioo a b)
⊢ Tendsto f l atBot ↔ Tendsto (fun x => ↑(f x)) l (𝓝[>] a) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff] | @[simp]
theorem tendsto_Ioo_atBot {f : β → Ioo a b} :
Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by
| Mathlib.Topology.Order.Basic.2615_0.Npdof1X5b8sCkZ2 | @[simp]
theorem tendsto_Ioo_atBot {f : β → Ioo a b} :
Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f✝ : α → β
f : β → ↑(Ioo a b)
⊢ Tendsto (Subtype.val ∘ f) l (𝓝[>] a) ↔ Tendsto (fun x => ↑(f x)) l (𝓝[>] a) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rfl | @[simp]
theorem tendsto_Ioo_atBot {f : β → Ioo a b} :
Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by
rw [← comap_coe_Ioo_nhdsWithin_Ioi, tendsto_comap_iff]; | Mathlib.Topology.Order.Basic.2615_0.Npdof1X5b8sCkZ2 | @[simp]
theorem tendsto_Ioo_atBot {f : β → Ioo a b} :
Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f✝ : α → β
f : β → ↑(Ioi a)
⊢ Tendsto f l atBot ↔ Tendsto (fun x => ↑(f x)) l (𝓝[>] a) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [← comap_coe_Ioi_nhdsWithin_Ioi, tendsto_comap_iff] | @[simp]
theorem tendsto_Ioi_atBot {f : β → Ioi a} :
Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by
| Mathlib.Topology.Order.Basic.2621_0.Npdof1X5b8sCkZ2 | @[simp]
theorem tendsto_Ioi_atBot {f : β → Ioi a} :
Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f✝ : α → β
f : β → ↑(Ioi a)
⊢ Tendsto (Subtype.val ∘ f) l (𝓝[>] a) ↔ Tendsto (fun x => ↑(f x)) l (𝓝[>] a) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rfl | @[simp]
theorem tendsto_Ioi_atBot {f : β → Ioi a} :
Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by
rw [← comap_coe_Ioi_nhdsWithin_Ioi, tendsto_comap_iff]; | Mathlib.Topology.Order.Basic.2621_0.Npdof1X5b8sCkZ2 | @[simp]
theorem tendsto_Ioi_atBot {f : β → Ioi a} :
Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f✝ : α → β
f : β → ↑(Iio a)
⊢ Tendsto f l atTop ↔ Tendsto (fun x => ↑(f x)) l (𝓝[<] a) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [← comap_coe_Iio_nhdsWithin_Iio, tendsto_comap_iff] | @[simp]
theorem tendsto_Iio_atTop {f : β → Iio a} :
Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by
| Mathlib.Topology.Order.Basic.2627_0.Npdof1X5b8sCkZ2 | @[simp]
theorem tendsto_Iio_atTop {f : β → Iio a} :
Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f✝ : α → β
f : β → ↑(Iio a)
⊢ Tendsto (Subtype.val ∘ f) l (𝓝[<] a) ↔ Tendsto (fun x => ↑(f x)) l (𝓝[<] a) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rfl | @[simp]
theorem tendsto_Iio_atTop {f : β → Iio a} :
Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by
rw [← comap_coe_Iio_nhdsWithin_Iio, tendsto_comap_iff]; | Mathlib.Topology.Order.Basic.2627_0.Npdof1X5b8sCkZ2 | @[simp]
theorem tendsto_Iio_atTop {f : β → Iio a} :
Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f : α → β
x : α
inst✝ : Nontrivial α
⊢ NeBot (𝓝[≠] x) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_ | instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by
| Mathlib.Topology.Order.Basic.2633_0.Npdof1X5b8sCkZ2 | instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : DenselyOrdered α
a b : α
s✝ : Set α
l : Filter β
f : α → β
x : α
inst✝ : Nontrivial α
s : Set α
hs : s ∈ 𝓝[≠] x
⊢ Set.Nonempty s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs | instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by
refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_
| Mathlib.Topology.Order.Basic.2633_0.Npdof1X5b8sCkZ2 | instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) | Mathlib_Topology_Order_Basic |
case intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : DenselyOrdered α
a b : α
s✝ : Set α
l : Filter β
f : α → β
x : α
inst✝ : Nontrivial α
s : Set α
hs : s ∈ 𝓝[≠] x
u : Set α
u_open : IsOpen u
xu : x ∈ u
us : u ∩ {x}ᶜ ⊆ s
⊢ Set.Non... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩ | instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by
refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_
obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs
| Mathlib.Topology.Order.Basic.2633_0.Npdof1X5b8sCkZ2 | instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) | Mathlib_Topology_Order_Basic |
case intro.intro.intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : DenselyOrdered α
a✝ b✝ : α
s✝ : Set α
l : Filter β
f : α → β
x : α
inst✝ : Nontrivial α
s : Set α
hs : s ∈ 𝓝[≠] x
u : Set α
u_open : IsOpen u
xu : x ∈ u
us : u ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | obtain ⟨y, hy⟩ : ∃ y, a < y ∧ y < b := exists_between a_lt_b | instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by
refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_
obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs
obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩
| Mathlib.Topology.Order.Basic.2633_0.Npdof1X5b8sCkZ2 | instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) | Mathlib_Topology_Order_Basic |
case intro.intro.intro.intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : DenselyOrdered α
a✝ b✝ : α
s✝ : Set α
l : Filter β
f : α → β
x : α
inst✝ : Nontrivial α
s : Set α
hs : s ∈ 𝓝[≠] x
u : Set α
u_open : IsOpen u
xu : x ∈ u
u... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases ne_or_eq x y with (xy | rfl) | instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by
refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_
obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs
obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩
obtain ⟨y... | Mathlib.Topology.Order.Basic.2633_0.Npdof1X5b8sCkZ2 | instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) | Mathlib_Topology_Order_Basic |
case intro.intro.intro.intro.intro.intro.intro.inl
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : DenselyOrdered α
a✝ b✝ : α
s✝ : Set α
l : Filter β
f : α → β
x : α
inst✝ : Nontrivial α
s : Set α
hs : s ∈ 𝓝[≠] x
u : Set α
u_open : IsOpen u
xu : x ∈... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact ⟨y, us ⟨hab hy, xy.symm⟩⟩ | instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by
refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_
obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs
obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩
obtain ⟨y... | Mathlib.Topology.Order.Basic.2633_0.Npdof1X5b8sCkZ2 | instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) | Mathlib_Topology_Order_Basic |
case intro.intro.intro.intro.intro.intro.intro.inr
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : DenselyOrdered α
a✝ b✝ : α
s✝ : Set α
l : Filter β
f : α → β
x : α
inst✝ : Nontrivial α
s : Set α
hs : s ∈ 𝓝[≠] x
u : Set α
u_open : IsOpen u
xu : x ∈... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | obtain ⟨z, hz⟩ : ∃ z, a < z ∧ z < x := exists_between hy.1 | instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by
refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_
obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs
obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩
obtain ⟨y... | Mathlib.Topology.Order.Basic.2633_0.Npdof1X5b8sCkZ2 | instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) | Mathlib_Topology_Order_Basic |
case intro.intro.intro.intro.intro.intro.intro.inr.intro
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : DenselyOrdered α
a✝ b✝ : α
s✝ : Set α
l : Filter β
f : α → β
x : α
inst✝ : Nontrivial α
s : Set α
hs : s ∈ 𝓝[≠] x
u : Set α
u_open : IsOpen u
xu... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact ⟨z, us ⟨hab ⟨hz.1, hz.2.trans hy.2⟩, hz.2.ne⟩⟩ | instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by
refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_
obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs
obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩
obtain ⟨y... | Mathlib.Topology.Order.Basic.2633_0.Npdof1X5b8sCkZ2 | instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
a b : α
s✝ : Set α
l : Filter β
f : α → β
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
⊢ ∃ t ⊆ s, Set.Countable t ∧ Dense t ∧ (∀ (x : α), IsBot x → x ∉ t) ∧ ∀ ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, htd⟩ | /-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a
separable space (e.g., if `α` has a second countable topology), then there exists a countable
dense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/
theorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivi... | Mathlib.Topology.Order.Basic.2643_0.Npdof1X5b8sCkZ2 | /-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a
separable space (e.g., if `α` has a second countable topology), then there exists a countable
dense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/
theorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivi... | Mathlib_Topology_Order_Basic |
case intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
a b : α
s✝ : Set α
l : Filter β
f : α → β
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
t : Set α
hts : t ⊆ s
htc : Set.Countable t
htd :... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | refine' ⟨t \ ({ x | IsBot x } ∪ { x | IsTop x }), _, _, _, fun x hx => _, fun x hx => _⟩ | /-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a
separable space (e.g., if `α` has a second countable topology), then there exists a countable
dense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/
theorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivi... | Mathlib.Topology.Order.Basic.2643_0.Npdof1X5b8sCkZ2 | /-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a
separable space (e.g., if `α` has a second countable topology), then there exists a countable
dense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/
theorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivi... | Mathlib_Topology_Order_Basic |
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