fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
Filter.Tendsto.eventually_le_const {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)
(h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=
h.eventually <| eventually_le_nhds hv | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Filter.Tendsto.eventually_le_const | null |
protected Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :
∃ y ∈ s, y < x :=
hs.orderDual.exists_gt x | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Dense.exists_lt | null |
protected Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :
∃ y ∈ s, y ≤ x :=
hs.orderDual.exists_ge x | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Dense.exists_le | null |
Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) :
∃ y ∈ s, y ≤ x :=
hs.orderDual.exists_ge' hbot x
/-! | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Dense.exists_le' | null |
Ioo_mem_nhdsGT_of_mem (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b :=
mem_nhdsWithin.2
⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ioo_mem_nhdsGT_of_mem | null |
Ioo_mem_nhdsGT (H : a < b) : Ioo a b ∈ 𝓝[>] a := Ioo_mem_nhdsGT_of_mem ⟨le_rfl, H⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ioo_mem_nhdsGT | null |
protected CovBy.nhdsGT (h : a ⋖ b) : 𝓝[>] a = ⊥ := h.toDual.nhdsLT | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | CovBy.nhdsGT | null |
protected SuccOrder.nhdsGT [SuccOrder α] : 𝓝[>] a = ⊥ := PredOrder.nhdsLT (α := αᵒᵈ) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | SuccOrder.nhdsGT | null |
SuccOrder.nhdsLT_eq_nhdsNE [SuccOrder α] (a : α) : 𝓝[<] a = 𝓝[≠] a :=
PredOrder.nhdsGT_eq_nhdsNE (α := αᵒᵈ) a | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | SuccOrder.nhdsLT_eq_nhdsNE | null |
SuccOrder.nhdsLE_eq_nhds [SuccOrder α] (a : α) : 𝓝[≤] a = 𝓝 a :=
PredOrder.nhdsGE_eq_nhds (α := αᵒᵈ) a | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | SuccOrder.nhdsLE_eq_nhds | null |
Ioc_mem_nhdsGT_of_mem (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b :=
mem_of_superset (Ioo_mem_nhdsGT_of_mem H) Ioo_subset_Ioc_self | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ioc_mem_nhdsGT_of_mem | null |
Ioc_mem_nhdsGT (H : a < b) : Ioc a b ∈ 𝓝[>] a := Ioc_mem_nhdsGT_of_mem ⟨le_rfl, H⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ioc_mem_nhdsGT | null |
Ico_mem_nhdsGT_of_mem (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b :=
mem_of_superset (Ioo_mem_nhdsGT_of_mem H) Ioo_subset_Ico_self | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ico_mem_nhdsGT_of_mem | null |
Ico_mem_nhdsGT (H : a < b) : Ico a b ∈ 𝓝[>] a := Ico_mem_nhdsGT_of_mem ⟨le_rfl, H⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ico_mem_nhdsGT | null |
Icc_mem_nhdsGT_of_mem (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b :=
mem_of_superset (Ioo_mem_nhdsGT_of_mem H) Ioo_subset_Icc_self | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Icc_mem_nhdsGT_of_mem | null |
Icc_mem_nhdsGT (H : a < b) : Icc a b ∈ 𝓝[>] a := Icc_mem_nhdsGT_of_mem ⟨le_rfl, H⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Icc_mem_nhdsGT | null |
nhdsWithin_Ioc_eq_nhdsGT (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a :=
nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | nhdsWithin_Ioc_eq_nhdsGT | null |
nhdsWithin_Ioo_eq_nhdsGT (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a :=
nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | nhdsWithin_Ioo_eq_nhdsGT | null |
continuousWithinAt_Ioc_iff_Ioi (h : a < b) :
ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by
simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsGT h]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | continuousWithinAt_Ioc_iff_Ioi | null |
continuousWithinAt_Ioo_iff_Ioi (h : a < b) :
ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by
simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsGT h]
/-! | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | continuousWithinAt_Ioo_iff_Ioi | null |
protected CovBy.nhdsGE (H : a ⋖ b) : 𝓝[≥] a = pure a := H.toDual.nhdsLE | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | CovBy.nhdsGE | null |
protected SuccOrder.nhdsGE [SuccOrder α] : 𝓝[≥] a = pure a :=
PredOrder.nhdsLE (α := αᵒᵈ) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | SuccOrder.nhdsGE | null |
Ico_mem_nhdsGE (H : a < b) : Ico a b ∈ 𝓝[≥] a :=
inter_mem_nhdsWithin _ <| Iio_mem_nhds H | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ico_mem_nhdsGE | null |
Ico_mem_nhdsGE_of_mem (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b :=
mem_of_superset (Ico_mem_nhdsGE H.2) <| Ico_subset_Ico_left H.1 | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ico_mem_nhdsGE_of_mem | null |
Ioo_mem_nhdsGE_of_mem (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b :=
mem_of_superset (Ico_mem_nhdsGE H.2) <| Ico_subset_Ioo_left H.1 | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ioo_mem_nhdsGE_of_mem | null |
Ioc_mem_nhdsGE_of_mem (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b :=
mem_of_superset (Ioo_mem_nhdsGE_of_mem H) Ioo_subset_Ioc_self | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ioc_mem_nhdsGE_of_mem | null |
Icc_mem_nhdsGE_of_mem (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b :=
mem_of_superset (Ico_mem_nhdsGE_of_mem H) Ico_subset_Icc_self | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Icc_mem_nhdsGE_of_mem | null |
Icc_mem_nhdsGE (H : a < b) : Icc a b ∈ 𝓝[≥] a := Icc_mem_nhdsGE_of_mem ⟨le_rfl, H⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Icc_mem_nhdsGE | null |
nhdsWithin_Icc_eq_nhdsGE (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a :=
nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iic_mem_nhds h
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | nhdsWithin_Icc_eq_nhdsGE | null |
nhdsWithin_Ico_eq_nhdsGE (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a :=
nhdsWithin_inter_of_mem' <| nhdsWithin_le_nhds <| Iio_mem_nhds h
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | nhdsWithin_Ico_eq_nhdsGE | null |
continuousWithinAt_Icc_iff_Ici (h : a < b) :
ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by
simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsGE h]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | continuousWithinAt_Icc_iff_Ici | null |
continuousWithinAt_Ico_iff_Ici (h : a < b) :
ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by
simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsGE h] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | continuousWithinAt_Ico_iff_Ici | null |
isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } :=
t.isClosed_le' | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | isClosed_le_prod | null |
isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :
IsClosed { b | f b ≤ g b } :=
continuous_iff_isClosed.mp (hf.prodMk hg) _ isClosed_le_prod | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | isClosed_le | null |
isClosed_Icc {a b : α} : IsClosed (Icc a b) :=
IsClosed.inter isClosed_Ici isClosed_Iic
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | isClosed_Icc | null |
closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=
isClosed_Icc.closure_eq | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | closure_Icc | null |
le_of_tendsto_of_tendsto_of_frequently {f g : β → α} {b : Filter β} {a₁ a₂ : α}
(hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∃ᶠ x in b, f x ≤ g x) : a₁ ≤ a₂ :=
t.isClosed_le'.mem_of_frequently_of_tendsto h (hf.prodMk_nhds hg) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | le_of_tendsto_of_tendsto_of_frequently | null |
le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]
(hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ :=
le_of_tendsto_of_tendsto_of_frequently hf hg <| Eventually.frequently h
alias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | le_of_tendsto_of_tendsto | null |
le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]
(hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ :=
le_of_tendsto_of_tendsto hf hg (Eventually.of_forall h)
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | le_of_tendsto_of_tendsto' | null |
closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :
closure { b | f b ≤ g b } = { b | f b ≤ g b } :=
(isClosed_le hf hg).closure_eq | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | closure_le_eq | null |
closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)
(hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } :=
(closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | closure_lt_subset_le | null |
ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}
(hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)
(h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=
show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from
OrderClosedTopology.isClosed_le'.closure_subset ((hf.prodMk hg).mem_closure hx h) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | ContinuousWithinAt.closure_le | null |
IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)
(hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) :=
(hf.prodMk hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le' | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | IsClosed.isClosed_le | If `s` is a closed set and two functions `f` and `g` are continuous on `s`,
then the set `{x ∈ s | f x ≤ g x}` is a closed set. |
le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)
(hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :
f x ≤ g x :=
have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩
(closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2 | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | le_on_closure | null |
IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)
(hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } :=
(hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | IsClosed.epigraph | null |
IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)
(hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ p.2 ≤ f p.1 } :=
(hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | IsClosed.hypograph | null |
isClosed_monotoneOn [Preorder β] {s : Set β} : IsClosed {f : β → α | MonotoneOn f s} := by
simp only [isClosed_iff_clusterPt, clusterPt_principal_iff_frequently]
intro g hg a ha b hb hab
have hmain (x) : Tendsto (fun f' ↦ f' x) (𝓝 g) (𝓝 (g x)) := continuousAt_apply x _
exact le_of_tendsto_of_tendsto_of_frequently (hmain a) (hmain b) (hg.mono fun g h ↦ h ha hb hab) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | isClosed_monotoneOn | The set of monotone functions on a set is closed. |
isClosed_monotone [Preorder β] : IsClosed {f : β → α | Monotone f} := by
simp_rw [← monotoneOn_univ]
exact isClosed_monotoneOn | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | isClosed_monotone | The set of monotone functions is closed. |
isClosed_antitoneOn [Preorder β] {s : Set β} : IsClosed {f : β → α | AntitoneOn f s} :=
isClosed_monotoneOn (α := αᵒᵈ) (β := β) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | isClosed_antitoneOn | The set of antitone functions on a set is closed. |
isClosed_antitone [Preorder β] : IsClosed {f : β → α | Antitone f} :=
isClosed_monotone (α := αᵒᵈ) (β := β) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | isClosed_antitone | The set of antitone functions is closed. |
isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :
IsOpen { b | f b < g b } := by
simpa only [lt_iff_not_ge] using (isClosed_le hg hf).isOpen_compl | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | isOpen_lt | null |
isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } :=
isOpen_lt continuous_fst continuous_snd
variable {a b : α} | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | isOpen_lt_prod | null |
isOpen_Ioo : IsOpen (Ioo a b) :=
IsOpen.inter isOpen_Ioi isOpen_Iio
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | isOpen_Ioo | null |
interior_Ioo : interior (Ioo a b) = Ioo a b :=
isOpen_Ioo.interior_eq | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | interior_Ioo | null |
Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by
simp only [interior_Ioo, subset_closure] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ioo_subset_closure_interior | null |
Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=
IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ioo_mem_nhds | null |
Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=
mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ioc_mem_nhds | null |
Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=
mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ico_mem_nhds | null |
Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=
mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Icc_mem_nhds | null |
DiscreteTopology.of_predOrder_succOrder [PredOrder α] [SuccOrder α] :
DiscreteTopology α := by
refine discreteTopology_iff_nhds.mpr fun a ↦ ?_
rw [← nhdsWithin_univ, ← Iic_union_Ioi, nhdsWithin_union, PredOrder.nhdsLE, SuccOrder.nhdsGT,
sup_bot_eq] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | DiscreteTopology.of_predOrder_succOrder | The only order closed topology on a linear order which is a `PredOrder` and a `SuccOrder`
is the discrete topology.
This theorem is not an instance,
because it causes searches for `PredOrder` and `SuccOrder` with their `Preorder` arguments
and very rarely matches. |
lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) :
{ b | f b < g b } ⊆ interior { b | f b ≤ g b } :=
(interior_maximal fun _ => le_of_lt) <| isOpen_lt hf hg | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | lt_subset_interior_le | null |
frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) :
frontier { b | f b ≤ g b } ⊆ { b | f b = g b } := by
rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg]
rintro b ⟨hb₁, hb₂⟩
refine le_antisymm hb₁ (closure_lt_subset_le hg hf ?_)
convert hb₂ using 2; simp only [not_le.symm]; rfl | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | frontier_le_subset_eq | null |
frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} :=
frontier_le_subset_eq (@continuous_id α _) continuous_const | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | frontier_Iic_subset | null |
frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} :=
frontier_Iic_subset (α := αᵒᵈ) _ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | frontier_Ici_subset | null |
frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) :
frontier { b | f b < g b } ⊆ { b | f b = g b } := by
simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | frontier_lt_subset_eq | null |
continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}
(hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x | f x ≤ g x })
(hg' : ContinuousOn g' { x | g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) :
Continuous fun x => if f x ≤ g x then f' x else g' x := by
refine continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) ?_ (hg'.mono ?_)
· rwa [(isClosed_le hf hg).closure_eq]
· simp only [not_le]
exact closure_lt_subset_le hg hf | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | continuous_if_le | null |
Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ}
(hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g)
(hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x :=
continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Continuous.if_le | null |
Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))
(hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=
let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi
(hg.eventually (Ioi_mem_nhds hb)).mp <| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ =>
h _ h₁ _ h₂
nonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀)
(hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x :=
hf.eventually_lt hg hfg
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Filter.Tendsto.eventually_lt | null |
protected Continuous.min (hf : Continuous f) (hg : Continuous g) :
Continuous fun b => min (f b) (g b) := by
simp only [min_def]
exact hf.if_le hg hf hg fun x => id
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Continuous.min | null |
protected Continuous.max (hf : Continuous f) (hg : Continuous g) :
Continuous fun b => max (f b) (g b) :=
Continuous.min (α := αᵒᵈ) hf hg | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Continuous.max | null |
continuous_min : Continuous fun p : α × α => min p.1 p.2 :=
continuous_fst.min continuous_snd | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | continuous_min | null |
continuous_max : Continuous fun p : α × α => max p.1 p.2 :=
continuous_fst.max continuous_snd | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | continuous_max | null |
protected Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))
(hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=
(continuous_max.tendsto (a₁, a₂)).comp (hf.prodMk_nhds hg) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Filter.Tendsto.max | null |
protected Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁))
(hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=
(continuous_min.tendsto (a₁, a₂)).comp (hf.prodMk_nhds hg) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Filter.Tendsto.min | null |
protected Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :
Tendsto (fun i => max a (f i)) l (𝓝 a) := by
simpa only [sup_idem] using (tendsto_const_nhds (x := a)).max h | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Filter.Tendsto.max_right | null |
protected Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :
Tendsto (fun i => max (f i) a) l (𝓝 a) := by
simp_rw [max_comm _ a]
exact h.max_right | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Filter.Tendsto.max_left | null |
Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :
Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by
obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h
exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Filter.tendsto_nhds_max_right | null |
Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :
Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by
simp_rw [max_comm _ a]
exact Filter.tendsto_nhds_max_right h | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Filter.tendsto_nhds_max_left | null |
Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :
Tendsto (fun i => min a (f i)) l (𝓝 a) :=
Filter.Tendsto.max_right (α := αᵒᵈ) h | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Filter.Tendsto.min_right | null |
Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) :
Tendsto (fun i => min (f i) a) l (𝓝 a) :=
Filter.Tendsto.max_left (α := αᵒᵈ) h | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Filter.Tendsto.min_left | null |
Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :
Tendsto (fun i => min a (f i)) l (𝓝[<] a) :=
Filter.tendsto_nhds_max_right (α := αᵒᵈ) h | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Filter.tendsto_nhds_min_right | null |
Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) :
Tendsto (fun i => min (f i) a) l (𝓝[<] a) :=
Filter.tendsto_nhds_max_left (α := αᵒᵈ) h | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Filter.tendsto_nhds_min_left | null |
Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) :
∃ z ∈ s, z ∈ Ioo x y :=
hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Dense.exists_between | null |
Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :
Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by
refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2))
rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩
exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Dense.Ioi_eq_biUnion | null |
Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) :
Iio x = ⋃ y ∈ s ∩ Iio x, Iio y :=
Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Dense.Iio_eq_biUnion | null |
Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] :
OrderClosedTopology (α → β) :=
inferInstance | instance | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Pi.orderClosedTopology' | null |
isLocalMax_of_mono_anti
{α : Type*} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]
{β : Type*} [Preorder β]
{a b c : α} (g₀ : a < b) (g₁ : b < c) {f : α → β}
(h₀ : MonotoneOn f (Ioc a b))
(h₁ : AntitoneOn f (Ico b c)) : IsLocalMax f b :=
isLocalMax_of_mono_anti' (Ioc_mem_nhdsLE g₀) (Ico_mem_nhdsGE g₁) h₀ h₁ | lemma | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/OrderClosedExtr.lean | isLocalMax_of_mono_anti | If `f` is monotone on `(a,b]` and antitone on `[b,c)` then `f` has
a local maximum at `b`. |
isLocalMin_of_anti_mono
{α : Type*} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α]
{β : Type*} [Preorder β] {a b c : α} (g₀ : a < b) (g₁ : b < c) {f : α → β}
(h₀ : AntitoneOn f (Ioc a b)) (h₁ : MonotoneOn f (Ico b c)) : IsLocalMin f b :=
mem_of_superset (Ioo_mem_nhds g₀ g₁) (fun x hx => by rcases le_total x b <;> aesop) | lemma | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/OrderClosedExtr.lean | isLocalMin_of_anti_mono | If `f` is antitone on `(a,b]` and monotone on `[b,c)` then `f` has
a local minimum at `b`. |
protected partialSups (hf : ∀ k ≤ n, Tendsto (f k) l (𝓝 (g k))) :
Tendsto (partialSups f n) l (𝓝 (partialSups g n)) := by
simp only [partialSups_eq_sup'_range]
refine finset_sup'_nhds _ ?_
simpa [Nat.lt_succ_iff] | lemma | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Order.PartialSups"
] | Mathlib/Topology/Order/PartialSups.lean | partialSups | null |
protected partialSups_apply (hf : ∀ k ≤ n, Tendsto (f k) l (𝓝 (g k))) :
Tendsto (fun a ↦ partialSups (f · a) n) l (𝓝 (partialSups g n)) := by
simpa only [← Pi.partialSups_apply] using Tendsto.partialSups hf | lemma | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Order.PartialSups"
] | Mathlib/Topology/Order/PartialSups.lean | partialSups_apply | null |
protected ContinuousAt.partialSups_apply (hf : ∀ k ≤ n, ContinuousAt (f k) x) :
ContinuousAt (fun a ↦ partialSups (f · a) n) x :=
Tendsto.partialSups_apply hf | lemma | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Order.PartialSups"
] | Mathlib/Topology/Order/PartialSups.lean | ContinuousAt.partialSups_apply | null |
protected ContinuousAt.partialSups (hf : ∀ k ≤ n, ContinuousAt (f k) x) :
ContinuousAt (partialSups f n) x := by
simpa only [← Pi.partialSups_apply] using ContinuousAt.partialSups_apply hf | lemma | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Order.PartialSups"
] | Mathlib/Topology/Order/PartialSups.lean | ContinuousAt.partialSups | null |
protected ContinuousWithinAt.partialSups_apply (hf : ∀ k ≤ n, ContinuousWithinAt (f k) s x) :
ContinuousWithinAt (fun a ↦ partialSups (f · a) n) s x :=
Tendsto.partialSups_apply hf | lemma | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Order.PartialSups"
] | Mathlib/Topology/Order/PartialSups.lean | ContinuousWithinAt.partialSups_apply | null |
protected ContinuousWithinAt.partialSups (hf : ∀ k ≤ n, ContinuousWithinAt (f k) s x) :
ContinuousWithinAt (partialSups f n) s x := by
simpa only [← Pi.partialSups_apply] using ContinuousWithinAt.partialSups_apply hf | lemma | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Order.PartialSups"
] | Mathlib/Topology/Order/PartialSups.lean | ContinuousWithinAt.partialSups | null |
protected ContinuousOn.partialSups_apply (hf : ∀ k ≤ n, ContinuousOn (f k) s) :
ContinuousOn (fun a ↦ partialSups (f · a) n) s := fun x hx ↦
ContinuousWithinAt.partialSups_apply fun k hk ↦ hf k hk x hx | lemma | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Order.PartialSups"
] | Mathlib/Topology/Order/PartialSups.lean | ContinuousOn.partialSups_apply | null |
protected ContinuousOn.partialSups (hf : ∀ k ≤ n, ContinuousOn (f k) s) :
ContinuousOn (partialSups f n) s := fun x hx ↦
ContinuousWithinAt.partialSups fun k hk ↦ hf k hk x hx | lemma | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Order.PartialSups"
] | Mathlib/Topology/Order/PartialSups.lean | ContinuousOn.partialSups | null |
protected Continuous.partialSups_apply (hf : ∀ k ≤ n, Continuous (f k)) :
Continuous (fun a ↦ partialSups (f · a) n) :=
continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.partialSups_apply fun k hk ↦
(hf k hk).continuousAt | lemma | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Order.PartialSups"
] | Mathlib/Topology/Order/PartialSups.lean | Continuous.partialSups_apply | null |
protected Continuous.partialSups (hf : ∀ k ≤ n, Continuous (f k)) :
Continuous (partialSups f n) :=
continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.partialSups fun k hk ↦ (hf k hk).continuousAt | lemma | Topology | [
"Mathlib.Topology.Order.Lattice",
"Mathlib.Order.PartialSups"
] | Mathlib/Topology/Order/PartialSups.lean | Continuous.partialSups | null |
PriestleySpace (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
priestley {x y : α} : ¬x ≤ y → ∃ U : Set α, IsClopen U ∧ IsUpperSet U ∧ x ∈ U ∧ y ∉ U
variable [TopologicalSpace α] | class | Topology | [
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/Priestley.lean | PriestleySpace | A Priestley space is an ordered topological space such that any two distinct points can be
separated by a clopen upper set. Compactness is often assumed, but we do not include it here. |
exists_isClopen_upper_of_not_le :
¬x ≤ y → ∃ U : Set α, IsClopen U ∧ IsUpperSet U ∧ x ∈ U ∧ y ∉ U :=
PriestleySpace.priestley | theorem | Topology | [
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/Priestley.lean | exists_isClopen_upper_of_not_le | null |
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