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 ⌀ |
|---|---|---|---|---|---|---|
comap_coe_nhdsGT_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) :
comap ((↑) : s → α) (𝓝[>] a) = atBot := by
apply comap_coe_nhdsLT_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)
simp only [OrderDual.exists, Ioo_toDual]
exact hs | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | comap_coe_nhdsGT_of_Ioo_subset | null |
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', nhdsWithin_empty... | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | map_coe_atTop_of_Ioo_subset | null |
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
refine (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)
fun b' hb' => ?_ :)
simpa using hs b' hb' | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | map_coe_atBot_of_Ioo_subset | null |
comap_coe_Ioo_nhdsLT (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=
comap_coe_nhdsLT_of_Ioo_subset Ioo_subset_Iio_self fun h => ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩ | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | comap_coe_Ioo_nhdsLT | The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at
the right endpoint in the ambient order. |
comap_coe_Ioo_nhdsGT (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=
comap_coe_nhdsGT_of_Ioo_subset Ioo_subset_Ioi_self fun h => ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩ | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | comap_coe_Ioo_nhdsGT | The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at
the left endpoint in the ambient order. |
comap_coe_Ioi_nhdsGT (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=
comap_coe_nhdsGT_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩ | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | comap_coe_Ioi_nhdsGT | null |
comap_coe_Iio_nhdsLT (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=
comap_coe_Ioi_nhdsGT (α := αᵒᵈ) a
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | comap_coe_Iio_nhdsLT | null |
map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=
map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | map_coe_Ioo_atTop | null |
map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=
map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | map_coe_Ioo_atBot | null |
map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=
map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | map_coe_Ioi_atBot | null |
map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=
map_coe_Ioi_atBot (α := αᵒᵈ) _
variable {l : Filter β} {f : α → β}
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | map_coe_Iio_atTop | null |
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]; rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | tendsto_comp_coe_Ioo_atTop | null |
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]; rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | tendsto_comp_coe_Ioo_atBot | null |
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]; rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | tendsto_comp_coe_Ioi_atBot | null |
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]; rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | tendsto_comp_coe_Iio_atTop | null |
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_nhdsLT, tendsto_comap_iff]; rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | tendsto_Ioo_atTop | null |
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_nhdsGT, tendsto_comap_iff]; rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | tendsto_Ioo_atBot | null |
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_nhdsGT, tendsto_comap_iff]; rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | tendsto_Ioi_atBot | null |
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_nhdsLT, tendsto_comap_iff]; rfl | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | tendsto_Iio_atTop | null |
Dense.exists_countable_dense_subset_no_bot_top [Nontrivial α] {s : Set α} [SeparableSpace s]
(hs : Dense s) :
∃ t, t ⊆ s ∧ t.Countable ∧ Dense t ∧ (∀ x, IsBot x → x ∉ t) ∧ ∀ x, IsTop x → x ∉ t := by
rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, htd⟩
refine ⟨t \ ({ x | IsBot x } ∪ { x | IsTop x ... | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | Dense.exists_countable_dense_subset_no_bot_top | 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`. |
exists_countable_dense_no_bot_top [SeparableSpace α] [Nontrivial α] :
∃ s : Set α, s.Countable ∧ Dense s ∧ (∀ x, IsBot x → x ∉ s) ∧ ∀ x, IsTop x → x ∉ s := by
simpa using dense_univ.exists_countable_dense_subset_no_bot_top | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | exists_countable_dense_no_bot_top | If `α` is a nontrivial separable dense linear order, then there exists a
countable dense set `s : Set α` that contains neither top nor bottom elements of `α`.
For a dense set containing both bot and top elements, see
`exists_countable_dense_bot_top`. |
@[simp]
isClosed_Ico_iff {a b : α} : IsClosed (Set.Ico a b) ↔ b ≤ a := by
refine ⟨fun h => le_of_not_gt fun hab => ?_, by simp_all⟩
have := h.closure_eq
rw [closure_Ico hab.ne, Icc_eq_Ico_same_iff] at this
exact this hab.le | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | isClosed_Ico_iff | `Set.Ico a b` is only closed if it is empty. |
@[simp]
isClosed_Ioc_iff {a b : α} : IsClosed (Set.Ioc a b) ↔ b ≤ a := by
refine ⟨fun h => le_of_not_gt fun hab => ?_, by simp_all⟩
have := h.closure_eq
rw [closure_Ioc hab.ne, Icc_eq_Ioc_same_iff] at this
exact this hab.le | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | isClosed_Ioc_iff | `Set.Ioc a b` is only closed if it is empty. |
@[simp]
isClosed_Ioo_iff {a b : α} : IsClosed (Set.Ioo a b) ↔ b ≤ a := by
refine ⟨fun h => le_of_not_gt fun hab => ?_, by simp_all⟩
have := h.closure_eq
rw [closure_Ioo hab.ne, Icc_eq_Ioo_same_iff] at this
exact this hab.le | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | isClosed_Ioo_iff | `Set.Ioo a b` is only closed if it is empty. |
continuousOn_Icc_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {la lb : β}
(hab : a ≠ b) (hf : ContinuousOn f (Ioo a b)) (ha : Tendsto f (𝓝[>] a) (𝓝 la))
(hb : Tendsto f (𝓝[<] b) (𝓝 lb)) : ContinuousOn... | theorem | Topology | [
"Mathlib.Topology.ExtendFrom",
"Mathlib.Topology.Order.DenselyOrdered"
] | Mathlib/Topology/Order/ExtendFrom.lean | continuousOn_Icc_extendFrom_Ioo | null |
eq_lim_at_left_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [T2Space β] {f : α → β} {a b : α} {la : β} (hab : a < b)
(ha : Tendsto f (𝓝[>] a) (𝓝 la)) : extendFrom (Ioo a b) f a = la := by
apply extendFrom_eq
· rw [closure_Ioo hab.ne]
sim... | theorem | Topology | [
"Mathlib.Topology.ExtendFrom",
"Mathlib.Topology.Order.DenselyOrdered"
] | Mathlib/Topology/Order/ExtendFrom.lean | eq_lim_at_left_extendFrom_Ioo | null |
eq_lim_at_right_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [T2Space β] {f : α → β} {a b : α} {lb : β} (hab : a < b)
(hb : Tendsto f (𝓝[<] b) (𝓝 lb)) : extendFrom (Ioo a b) f b = lb := by
apply extendFrom_eq
· rw [closure_Ioo hab.ne]
si... | theorem | Topology | [
"Mathlib.Topology.ExtendFrom",
"Mathlib.Topology.Order.DenselyOrdered"
] | Mathlib/Topology/Order/ExtendFrom.lean | eq_lim_at_right_extendFrom_Ioo | null |
continuousOn_Ico_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {la : β}
(hab : a < b) (hf : ContinuousOn f (Ioo a b)) (ha : Tendsto f (𝓝[>] a) (𝓝 la)) :
ContinuousOn (extendFrom (Ioo a b) f) (Ico a b) :=... | theorem | Topology | [
"Mathlib.Topology.ExtendFrom",
"Mathlib.Topology.Order.DenselyOrdered"
] | Mathlib/Topology/Order/ExtendFrom.lean | continuousOn_Ico_extendFrom_Ioo | null |
continuousOn_Ioc_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {lb : β}
(hab : a < b) (hf : ContinuousOn f (Ioo a b)) (hb : Tendsto f (𝓝[<] b) (𝓝 lb)) :
ContinuousOn (extendFrom (Ioo a b) f) (Ioc a b) :=... | theorem | Topology | [
"Mathlib.Topology.ExtendFrom",
"Mathlib.Topology.Order.DenselyOrdered"
] | Mathlib/Topology/Order/ExtendFrom.lean | continuousOn_Ioc_extendFrom_Ioo | null |
protected IsMaxOn.closure (h : IsMaxOn f s a) (hc : ContinuousOn f (closure s)) :
IsMaxOn f (closure s) a := fun x hx =>
ContinuousWithinAt.closure_le hx ((hc x hx).mono subset_closure) continuousWithinAt_const h | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/ExtrClosure.lean | IsMaxOn.closure | null |
protected IsMinOn.closure (h : IsMinOn f s a) (hc : ContinuousOn f (closure s)) :
IsMinOn f (closure s) a :=
h.dual.closure hc | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/ExtrClosure.lean | IsMinOn.closure | null |
protected IsExtrOn.closure (h : IsExtrOn f s a) (hc : ContinuousOn f (closure s)) :
IsExtrOn f (closure s) a :=
h.elim (fun h => Or.inl <| h.closure hc) fun h => Or.inr <| h.closure hc | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/ExtrClosure.lean | IsExtrOn.closure | null |
protected IsLocalMaxOn.closure (h : IsLocalMaxOn f s a) (hc : ContinuousOn f (closure s)) :
IsLocalMaxOn f (closure s) a := by
rcases mem_nhdsWithin.1 h with ⟨U, Uo, aU, hU⟩
refine mem_nhdsWithin.2 ⟨U, Uo, aU, ?_⟩
rintro x ⟨hxU, hxs⟩
refine ContinuousWithinAt.closure_le ?_ ?_ continuousWithinAt_const hU
·... | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/ExtrClosure.lean | IsLocalMaxOn.closure | null |
protected IsLocalMinOn.closure (h : IsLocalMinOn f s a) (hc : ContinuousOn f (closure s)) :
IsLocalMinOn f (closure s) a :=
IsLocalMaxOn.closure h.dual hc | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/ExtrClosure.lean | IsLocalMinOn.closure | null |
protected IsLocalExtrOn.closure (h : IsLocalExtrOn f s a)
(hc : ContinuousOn f (closure s)) : IsLocalExtrOn f (closure s) a :=
h.elim (fun h => Or.inl <| h.closure hc) fun h => Or.inr <| h.closure hc | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/ExtrClosure.lean | IsLocalExtrOn.closure | null |
protected tendsto_nhds_atTop [NoMaxOrder X] : Tendsto 𝓝 (atTop : Filter X) (𝓝 atTop) :=
Filter.tendsto_nhds_atTop_iff.2 fun x => (eventually_gt_atTop x).mono fun _ => le_mem_nhds | theorem | Topology | [
"Mathlib.Topology.Filter",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/Filter.lean | tendsto_nhds_atTop | null |
protected tendsto_nhds_atBot [NoMinOrder X] : Tendsto 𝓝 (atBot : Filter X) (𝓝 atBot) :=
@Filter.tendsto_nhds_atTop Xᵒᵈ _ _ _ _ | theorem | Topology | [
"Mathlib.Topology.Filter",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/Filter.lean | tendsto_nhds_atBot | null |
Tendsto.nhds_atTop [NoMaxOrder X] {f : α → X} {l : Filter α} (h : Tendsto f l atTop) :
Tendsto (𝓝 ∘ f) l (𝓝 atTop) :=
Filter.tendsto_nhds_atTop.comp h | theorem | Topology | [
"Mathlib.Topology.Filter",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/Filter.lean | Tendsto.nhds_atTop | null |
Tendsto.nhds_atBot [NoMinOrder X] {f : α → X} {l : Filter α} (h : Tendsto f l atBot) :
Tendsto (𝓝 ∘ f) l (𝓝 atBot) :=
@Tendsto.nhds_atTop α Xᵒᵈ _ _ _ _ _ _ h | theorem | Topology | [
"Mathlib.Topology.Filter",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/Filter.lean | Tendsto.nhds_atBot | null |
hull (T : Set α) (a : α) := T ↓∩ Ici a
variable {T : Set α}
/- The set of relative-closed sets of the form `hull T a` for some `a` in `α` is closed under
pairwise union. -/ | abbrev | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | hull | For `a` of type `α` the set of element of `T` which dominate `a` is the `hull` of `a` in `T`. |
hull_inf (hT : ∀ p ∈ T, InfPrime p) (a b : α) :
hull T (a ⊓ b) = hull T a ∪ hull T b := by
ext p
constructor <;> intro h
· exact (hT p p.2).2 h
· rcases h with (h1 | h3)
· exact inf_le_of_left_le h1
· exact inf_le_of_right_le h3
variable [OrderTop α]
/- Every relative-closed set of the form `T ↓∩ (↑... | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | hull_inf | null |
hull_finsetInf (hT : ∀ p ∈ T, InfPrime p) (F : Finset α) :
hull T (inf F id) = T ↓∩ upperClosure F.toSet := by
rw [coe_upperClosure]
induction F using Finset.cons_induction with
| empty =>
simp only [coe_empty, mem_empty_iff_false, iUnion_of_empty, iUnion_empty, Set.preimage_empty,
inf_empty]
by... | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | hull_finsetInf | null |
preimage_upperClosure_compl_finset (hT : ∀ p ∈ T, InfPrime p) (F : Finset α) :
T ↓∩ (upperClosure F.toSet)ᶜ = (hull T (inf F id))ᶜ := by
rw [Set.preimage_compl, (hull_finsetInf hT)]
variable [TopologicalSpace α] [IsLower α]
/-
The relative-open sets of the form `(hull T a)ᶜ` for `a` in `α` form a basis for the re... | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | preimage_upperClosure_compl_finset | null |
isTopologicalBasis_relativeLower (hT : ∀ p ∈ T, InfPrime p) :
IsTopologicalBasis { S : Set T | ∃ (a : α), (hull T a)ᶜ = S } := by
convert isTopologicalBasis_subtype Topology.IsLower.isTopologicalBasis T
ext R
simp only [preimage_compl, mem_setOf_eq, IsLower.lowerBasis, mem_image, exists_exists_and_eq_and]
c... | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | isTopologicalBasis_relativeLower | null |
hull_iSup {ι : Sort v} (s : ι → α) : hull T (iSup s) = ⋂ i, hull T (s i) := by aesop | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | hull_iSup | null |
hull_sSup (S : Set α) : hull T (sSup S) = ⋂₀ { hull T a | a ∈ S } := by aesop
/- When `α` is complete, a set is Lower topology relative-open if and only if it is of the form
`(hull T a)ᶜ` for some `a` in `α`.-/ | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | hull_sSup | null |
isOpen_iff [TopologicalSpace α] [IsLower α] (hT : ∀ p ∈ T, InfPrime p)
(S : Set T) : IsOpen S ↔ ∃ (a : α), S = (hull T a)ᶜ := by
constructor <;> intro h
· let R := {a : α | (hull T a)ᶜ ⊆ S}
use sSup R
rw [IsTopologicalBasis.open_eq_sUnion' (isTopologicalBasis_relativeLower hT) h]
aesop
· obtain ⟨a... | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | isOpen_iff | null |
isClosed_iff [TopologicalSpace α] [IsLower α] (hT : ∀ p ∈ T, InfPrime p)
{S : Set T} : IsClosed S ↔ ∃ (a : α), S = hull T a := by
simp only [← isOpen_compl_iff, isOpen_iff hT, compl_inj_iff] | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | isClosed_iff | null |
kernel (S : Set T) := sInf (Subtype.val '' S)
/- The pair of maps `kernel` and `hull` form an antitone Galois connection between the
subsets of `T` and `α`. -/
open OrderDual in | abbrev | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | kernel | For a subset `S` of `T`, `kernel S` is the infimum of `S` (considered as a set of `α`) |
gc : GaloisConnection (α := Set T) (β := αᵒᵈ)
(fun S => toDual (kernel S)) (fun a => hull T (ofDual a)) := fun S a => by
simp only [toDual_sInf, sSup_le_iff, mem_preimage, mem_image, Subtype.exists, exists_and_right,
exists_eq_right, ← ofDual_le_ofDual, forall_exists_index, OrderDual.forall, ofDual_toDual]
... | theorem | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | gc | null |
gc_closureOperator (S : Set T) : gc.closureOperator S = hull T (kernel S) := by
simp only [toDual_sInf, GaloisConnection.closureOperator_apply, ofDual_sSup]
rw [← preimage_comp, ← OrderDual.toDual_symm_eq, Equiv.symm_comp_self, preimage_id_eq, id_eq]
variable (T) | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | gc_closureOperator | null |
OrderGenerates := ∀ (a : α), ∃ (S : Set T), a = kernel S
variable {T} | def | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | OrderGenerates | `T` order generates `α` if, for every `a` in `α`, there exists a subset of `T` such that `a` is
the `kernel` of `S`. |
gi (hG : OrderGenerates T) : GaloisInsertion (α := Set T) (β := αᵒᵈ)
(OrderDual.toDual ∘ kernel)
(hull T ∘ OrderDual.ofDual) :=
gc.toGaloisInsertion fun a ↦ by
rw [OrderDual.le_toDual]
obtain ⟨S, hS⟩ := hG a
exact le_of_le_of_eq (sInf_le_sInf (image_val_mono (fun c hcS => mem_preimage.mpr (mem_Ici... | def | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | gi | When `T` is order generating, the `kernel` and the `hull` form a Galois insertion |
kernel_hull (hG : OrderGenerates T) (a : α) : kernel (hull T a) = a := by
conv_rhs => rw [← OrderDual.ofDual_toDual a, ← (gi hG).l_u_eq a]
rfl | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | kernel_hull | null |
hull_kernel_of_isClosed [TopologicalSpace α] [IsLower α]
(hT : ∀ p ∈ T, InfPrime p) (hG : OrderGenerates T) {C : Set T} (h : IsClosed C) :
hull T (kernel C) = C := by
obtain ⟨a, ha⟩ := (isClosed_iff hT).mp h
rw [ha, kernel_hull hG] | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | hull_kernel_of_isClosed | null |
closedsGC_closureOperator [TopologicalSpace α] [IsLower α]
(hT : ∀ p ∈ T, InfPrime p) (hG : OrderGenerates T) (S : Set T) :
(TopologicalSpace.Closeds.gc (α := T)).closureOperator S = hull T (kernel S) := by
simp only [GaloisConnection.closureOperator_apply, Closeds.coe_closure, closure, le_antisymm_iff]
con... | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | closedsGC_closureOperator | null |
intermediate_value_univ₂ [PreconnectedSpace X] {a b : X} {f g : X → α} (hf : Continuous f)
(hg : Continuous g) (ha : f a ≤ g a) (hb : g b ≤ f b) : ∃ x, f x = g x := by
obtain ⟨x, _, hfg, hgf⟩ : (univ ∩ { x | f x ≤ g x ∧ g x ≤ f x }).Nonempty :=
isPreconnected_closed_iff.1 PreconnectedSpace.isPreconnected_univ... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | intermediate_value_univ₂ | Intermediate value theorem for two functions: if `f` and `g` are two continuous functions
on a preconnected space and `f a ≤ g a` and `g b ≤ f b`, then for some `x` we have `f x = g x`. |
intermediate_value_univ₂_eventually₁ [PreconnectedSpace X] {a : X} {l : Filter X} [NeBot l]
{f g : X → α} (hf : Continuous f) (hg : Continuous g) (ha : f a ≤ g a) (he : g ≤ᶠ[l] f) :
∃ x, f x = g x :=
let ⟨_, h⟩ := he.exists; intermediate_value_univ₂ hf hg ha h | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | intermediate_value_univ₂_eventually₁ | null |
intermediate_value_univ₂_eventually₂ [PreconnectedSpace X] {l₁ l₂ : Filter X} [NeBot l₁]
[NeBot l₂] {f g : X → α} (hf : Continuous f) (hg : Continuous g) (he₁ : f ≤ᶠ[l₁] g)
(he₂ : g ≤ᶠ[l₂] f) : ∃ x, f x = g x :=
let ⟨_, h₁⟩ := he₁.exists
let ⟨_, h₂⟩ := he₂.exists
intermediate_value_univ₂ hf hg h₁ h₂ | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | intermediate_value_univ₂_eventually₂ | null |
IsPreconnected.intermediate_value₂ {s : Set X} (hs : IsPreconnected s) {a b : X}
(ha : a ∈ s) (hb : b ∈ s) {f g : X → α} (hf : ContinuousOn f s) (hg : ContinuousOn g s)
(ha' : f a ≤ g a) (hb' : g b ≤ f b) : ∃ x ∈ s, f x = g x :=
let ⟨x, hx⟩ :=
@intermediate_value_univ₂ s α _ _ _ _ (Subtype.preconnectedSpa... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsPreconnected.intermediate_value₂ | Intermediate value theorem for two functions: if `f` and `g` are two functions continuous
on a preconnected set `s` and for some `a b ∈ s` we have `f a ≤ g a` and `g b ≤ f b`,
then for some `x ∈ s` we have `f x = g x`. |
IsPreconnected.intermediate_value₂_eventually₁ {s : Set X} (hs : IsPreconnected s) {a : X}
{l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f g : X → α} (hf : ContinuousOn f s)
(hg : ContinuousOn g s) (ha' : f a ≤ g a) (he : g ≤ᶠ[l] f) : ∃ x ∈ s, f x = g x := by
rw [continuousOn_iff_continuous_restrict]... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsPreconnected.intermediate_value₂_eventually₁ | null |
IsPreconnected.intermediate_value₂_eventually₂ {s : Set X} (hs : IsPreconnected s)
{l₁ l₂ : Filter X} [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f g : X → α}
(hf : ContinuousOn f s) (hg : ContinuousOn g s) (he₁ : f ≤ᶠ[l₁] g) (he₂ : g ≤ᶠ[l₂] f) :
∃ x ∈ s, f x = g x := by
rw [continuousOn_if... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsPreconnected.intermediate_value₂_eventually₂ | null |
IsPreconnected.intermediate_value {s : Set X} (hs : IsPreconnected s) {a b : X} (ha : a ∈ s)
(hb : b ∈ s) {f : X → α} (hf : ContinuousOn f s) : Icc (f a) (f b) ⊆ f '' s := fun _x hx =>
hs.intermediate_value₂ ha hb hf continuousOn_const hx.1 hx.2 | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsPreconnected.intermediate_value | **Intermediate Value Theorem** for continuous functions on connected sets. |
IsPreconnected.intermediate_value_Ico {s : Set X} (hs : IsPreconnected s) {a : X}
{l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α}
(ht : Tendsto f l (𝓝 v)) : Ico (f a) v ⊆ f '' s := fun _ h =>
hs.intermediate_value₂_eventually₁ ha hl hf continuousOn_const h.1 ... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsPreconnected.intermediate_value_Ico | null |
IsPreconnected.intermediate_value_Ioc {s : Set X} (hs : IsPreconnected s) {a : X}
{l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α}
(ht : Tendsto f l (𝓝 v)) : Ioc v (f a) ⊆ f '' s := fun _ h =>
(hs.intermediate_value₂_eventually₁ ha hl continuousOn_const hf h.2... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsPreconnected.intermediate_value_Ioc | null |
IsPreconnected.intermediate_value_Ioo {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
[NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
{v₁ v₂ : α} (ht₁ : Tendsto f l₁ (𝓝 v₁)) (ht₂ : Tendsto f l₂ (𝓝 v₂)) :
Ioo v₁ v₂ ⊆ f '' s := fun _ h =>
hs.intermediat... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsPreconnected.intermediate_value_Ioo | null |
IsPreconnected.intermediate_value_Ici {s : Set X} (hs : IsPreconnected s) {a : X}
{l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
(ht : Tendsto f l atTop) : Ici (f a) ⊆ f '' s := fun y h =>
hs.intermediate_value₂_eventually₁ ha hl hf continuousOn_const h (tendsto_atTo... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsPreconnected.intermediate_value_Ici | null |
IsPreconnected.intermediate_value_Iic {s : Set X} (hs : IsPreconnected s) {a : X}
{l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
(ht : Tendsto f l atBot) : Iic (f a) ⊆ f '' s := fun y h =>
(hs.intermediate_value₂_eventually₁ ha hl continuousOn_const hf h (tendsto_atB... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsPreconnected.intermediate_value_Iic | null |
IsPreconnected.intermediate_value_Ioi {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
[NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
{v : α} (ht₁ : Tendsto f l₁ (𝓝 v)) (ht₂ : Tendsto f l₂ atTop) : Ioi v ⊆ f '' s := fun y h =>
hs.intermediate_value₂_eventu... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsPreconnected.intermediate_value_Ioi | null |
IsPreconnected.intermediate_value_Iio {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
[NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
{v : α} (ht₁ : Tendsto f l₁ atBot) (ht₂ : Tendsto f l₂ (𝓝 v)) : Iio v ⊆ f '' s := fun y h =>
hs.intermediate_value₂_eventu... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsPreconnected.intermediate_value_Iio | null |
IsPreconnected.intermediate_value_Iii {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
[NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
(ht₁ : Tendsto f l₁ atBot) (ht₂ : Tendsto f l₂ atTop) : univ ⊆ f '' s := fun y _ =>
hs.intermediate_value₂_eventually₂ hl₁ ... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsPreconnected.intermediate_value_Iii | null |
intermediate_value_univ [PreconnectedSpace X] (a b : X) {f : X → α} (hf : Continuous f) :
Icc (f a) (f b) ⊆ range f := fun _ hx => intermediate_value_univ₂ hf continuous_const hx.1 hx.2 | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | intermediate_value_univ | **Intermediate Value Theorem** for continuous functions on connected spaces. |
mem_range_of_exists_le_of_exists_ge [PreconnectedSpace X] {c : α} {f : X → α}
(hf : Continuous f) (h₁ : ∃ a, f a ≤ c) (h₂ : ∃ b, c ≤ f b) : c ∈ range f :=
let ⟨a, ha⟩ := h₁; let ⟨b, hb⟩ := h₂; intermediate_value_univ a b hf ⟨ha, hb⟩
/-! | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | mem_range_of_exists_le_of_exists_ge | **Intermediate Value Theorem** for continuous functions on connected spaces. |
IsPreconnected.Icc_subset {s : Set α} (hs : IsPreconnected s) {a b : α} (ha : a ∈ s)
(hb : b ∈ s) : Icc a b ⊆ s := by
simpa only [image_id] using hs.intermediate_value ha hb continuousOn_id | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsPreconnected.Icc_subset | If a preconnected set contains endpoints of an interval, then it includes the whole interval. |
IsPreconnected.ordConnected {s : Set α} (h : IsPreconnected s) : OrdConnected s :=
⟨fun _ hx _ hy => h.Icc_subset hx hy⟩ | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsPreconnected.ordConnected | null |
IsConnected.Icc_subset {s : Set α} (hs : IsConnected s) {a b : α} (ha : a ∈ s)
(hb : b ∈ s) : Icc a b ⊆ s :=
hs.2.Icc_subset ha hb | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsConnected.Icc_subset | If a preconnected set contains endpoints of an interval, then it includes the whole interval. |
IsPreconnected.eq_univ_of_unbounded {s : Set α} (hs : IsPreconnected s) (hb : ¬BddBelow s)
(ha : ¬BddAbove s) : s = univ := by
refine eq_univ_of_forall fun x => ?_
obtain ⟨y, ys, hy⟩ : ∃ y ∈ s, y < x := not_bddBelow_iff.1 hb x
obtain ⟨z, zs, hz⟩ : ∃ z ∈ s, x < z := not_bddAbove_iff.1 ha x
exact hs.Icc_subse... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsPreconnected.eq_univ_of_unbounded | If preconnected set in a linear order space is unbounded below and above, then it is the whole
space. |
IsConnected.Ioo_csInf_csSup_subset {s : Set α} (hs : IsConnected s) (hb : BddBelow s)
(ha : BddAbove s) : Ioo (sInf s) (sSup s) ⊆ s := fun _x hx =>
let ⟨_y, ys, hy⟩ := (isGLB_lt_iff (isGLB_csInf hs.nonempty hb)).1 hx.1
let ⟨_z, zs, hz⟩ := (lt_isLUB_iff (isLUB_csSup hs.nonempty ha)).1 hx.2
hs.Icc_subset ys zs ... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsConnected.Ioo_csInf_csSup_subset | A bounded connected subset of a conditionally complete linear order includes the open interval
`(Inf s, Sup s)`. |
eq_Icc_csInf_csSup_of_connected_bdd_closed {s : Set α} (hc : IsConnected s)
(hb : BddBelow s) (ha : BddAbove s) (hcl : IsClosed s) : s = Icc (sInf s) (sSup s) :=
(subset_Icc_csInf_csSup hb ha).antisymm <|
hc.Icc_subset (hcl.csInf_mem hc.nonempty hb) (hcl.csSup_mem hc.nonempty ha) | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | eq_Icc_csInf_csSup_of_connected_bdd_closed | null |
IsPreconnected.Ioi_csInf_subset {s : Set α} (hs : IsPreconnected s) (hb : BddBelow s)
(ha : ¬BddAbove s) : Ioi (sInf s) ⊆ s := fun x hx =>
have sne : s.Nonempty := nonempty_of_not_bddAbove ha
let ⟨_y, ys, hy⟩ : ∃ y ∈ s, y < x := (isGLB_lt_iff (isGLB_csInf sne hb)).1 hx
let ⟨_z, zs, hz⟩ : ∃ z ∈ s, x < z := not... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsPreconnected.Ioi_csInf_subset | null |
IsPreconnected.Iio_csSup_subset {s : Set α} (hs : IsPreconnected s) (hb : ¬BddBelow s)
(ha : BddAbove s) : Iio (sSup s) ⊆ s :=
IsPreconnected.Ioi_csInf_subset (α := αᵒᵈ) hs ha hb | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsPreconnected.Iio_csSup_subset | null |
IsPreconnected.mem_intervals {s : Set α} (hs : IsPreconnected s) :
s ∈
({Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s),
Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅} : Set (Set α)) := by
rcases s.eq_empty_or_nonempty with (rfl | hne)
... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsPreconnected.mem_intervals | A preconnected set in a conditionally complete linear order is either one of the intervals
`[Inf s, Sup s]`, `[Inf s, Sup s)`, `(Inf s, Sup s]`, `(Inf s, Sup s)`, `[Inf s, +∞)`,
`(Inf s, +∞)`, `(-∞, Sup s]`, `(-∞, Sup s)`, `(-∞, +∞)`, or `∅`. The converse statement requires
`α` to be densely ordered. |
setOf_isPreconnected_subset_of_ordered :
{ s : Set α | IsPreconnected s } ⊆
(range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo)) ∪
(range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) := by
intro s hs
rcases hs.mem_intervals with (hs | hs | hs | hs | hs | hs... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | setOf_isPreconnected_subset_of_ordered | A preconnected set is either one of the intervals `Icc`, `Ico`, `Ioc`, `Ioo`, `Ici`, `Ioi`,
`Iic`, `Iio`, or `univ`, or `∅`. The converse statement requires `α` to be densely ordered. Though
one can represent `∅` as `(Inf ∅, Inf ∅)`, we include it into the list of possible cases to improve
readability. |
IsClosed.mem_of_ge_of_forall_exists_gt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b))
(ha : a ∈ s) (hab : a ≤ b) (hgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).Nonempty) : b ∈ s := by
let S := s ∩ Icc a b
replace ha : a ∈ S := ⟨ha, left_mem_Icc.2 hab⟩
have Sbd : BddAbove S := ⟨b, fun z hz => hz.2.2⟩
let c := ... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsClosed.mem_of_ge_of_forall_exists_gt | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and the set `s ∩ [a, b)` has no maximal point, then `b ∈ s`. |
IsClosed.Icc_subset_of_forall_exists_gt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b))
(ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Ico a b, ∀ y ∈ Ioi x, (s ∩ Ioc x y).Nonempty) : Icc a b ⊆ s := by
intro y hy
have : IsClosed (s ∩ Icc a y) := by
suffices s ∩ Icc a y = s ∩ Icc a b ∩ Icc a y from this ▸ hs.inter isClose... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsClosed.Icc_subset_of_forall_exists_gt | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and for any `a ≤ x < y ≤ b`, `x ∈ s`, the set `s ∩ (x, y]`
is not empty, then `[a, b] ⊆ s`. |
IsClosed.mem_of_ge_of_forall_exists_lt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b))
(hb : b ∈ s) (hab : a ≤ b) (hgt : ∀ x ∈ s ∩ Ioc a b, (s ∩ Ico a x).Nonempty) : a ∈ s := by
suffices OrderDual.toDual a ∈ ofDual ⁻¹' s by aesop
have : IsClosed (OrderDual.ofDual ⁻¹' (s ∩ Icc a b)) := hs
rw [preimage_inte... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsClosed.mem_of_ge_of_forall_exists_lt | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `b`, and the set `s ∩ (a, b]` has no minimal point, then `a ∈ s`. |
IsClosed.Icc_subset_of_forall_exists_lt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b))
(hb : b ∈ s) (hgt : ∀ x ∈ s ∩ Ioc a b, ∀ y ∈ Iio x, (s ∩ Ico y x).Nonempty) : Icc a b ⊆ s := by
intro y hy
have : IsClosed (s ∩ Icc y b) := by
suffices s ∩ Icc y b = s ∩ Icc a b ∩ Icc y b from this ▸ hs.inter isClose... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsClosed.Icc_subset_of_forall_exists_lt | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `b`, and for any `a ≤ y < x ≤ b`, `x ∈ s`, the set `s ∩ [y, x)`
is not empty, then `[a, b] ⊆ s`. |
IsClosed.Icc_subset_of_forall_mem_nhdsGT_of_Icc_subset {a b : α} {s : Set α}
(hs : IsClosed (s ∩ Icc a b)) (ha : a ∈ s)
(h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t) :
Icc a b ⊆ s := by
rcases lt_or_ge b a with hab | hab
· simp_all
set A := {t ∈ Icc a b | Icc a t ⊆ s}
have a_mem : a ∈ A := ⟨left_me... | lemma | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsClosed.Icc_subset_of_forall_mem_nhdsGT_of_Icc_subset | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and for any `x ∈ [a, b)` such that `[a, x]` is included in `s`,
the set `s` includes some open neighborhood of `x` within `(x, +∞)`, then `[a, b] ⊆ s`. |
IsClosed.Icc_subset_of_forall_mem_nhdsWithin {a b : α} {s : Set α}
(hs : IsClosed (s ∩ Icc a b)) (ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[>] x) :
Icc a b ⊆ s :=
hs.Icc_subset_of_forall_mem_nhdsGT_of_Icc_subset ha
(fun _t ht h't ↦ hgt _ ⟨h't ⟨ht.1, le_rfl⟩, ht⟩) | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | IsClosed.Icc_subset_of_forall_mem_nhdsWithin | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and for any `x ∈ s ∩ [a, b)` the set `s` includes some open
neighborhood of `x` within `(x, +∞)`, then `[a, b] ⊆ s`. |
isPreconnected_Icc_aux (x y : α) (s t : Set α) (hxy : x ≤ y) (hs : IsClosed s)
(ht : IsClosed t) (hab : Icc a b ⊆ s ∪ t) (hx : x ∈ Icc a b ∩ s) (hy : y ∈ Icc a b ∩ t) :
(Icc a b ∩ (s ∩ t)).Nonempty := by
have xyab : Icc x y ⊆ Icc a b := Icc_subset_Icc hx.1.1 hy.1.2
by_contra hst
suffices Icc x y ⊆ s from
... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isPreconnected_Icc_aux | null |
isPreconnected_Icc : IsPreconnected (Icc a b) :=
isPreconnected_closed_iff.2
(by
rintro s t hs ht hab ⟨x, hx⟩ ⟨y, hy⟩
rcases le_total x y with h | h
· exact isPreconnected_Icc_aux x y s t h hs ht hab hx hy
· rw [inter_comm s t]
rw [union_comm s t] at hab
exact isPreconnecte... | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isPreconnected_Icc | A closed interval in a densely ordered conditionally complete linear order is preconnected. |
isPreconnected_uIcc : IsPreconnected ([[a, b]]) :=
isPreconnected_Icc | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isPreconnected_uIcc | null |
Set.OrdConnected.isPreconnected {s : Set α} (h : s.OrdConnected) : IsPreconnected s :=
isPreconnected_of_forall_pair fun x hx y hy =>
⟨[[x, y]], h.uIcc_subset hx hy, left_mem_uIcc, right_mem_uIcc, isPreconnected_uIcc⟩ | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | Set.OrdConnected.isPreconnected | null |
isPreconnected_iff_ordConnected {s : Set α} : IsPreconnected s ↔ OrdConnected s :=
⟨IsPreconnected.ordConnected, Set.OrdConnected.isPreconnected⟩ | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isPreconnected_iff_ordConnected | null |
isPreconnected_Ici : IsPreconnected (Ici a) :=
ordConnected_Ici.isPreconnected | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isPreconnected_Ici | null |
isPreconnected_Iic : IsPreconnected (Iic a) :=
ordConnected_Iic.isPreconnected | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isPreconnected_Iic | null |
isPreconnected_Iio : IsPreconnected (Iio a) :=
ordConnected_Iio.isPreconnected | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isPreconnected_Iio | null |
isPreconnected_Ioi : IsPreconnected (Ioi a) :=
ordConnected_Ioi.isPreconnected | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isPreconnected_Ioi | null |
isPreconnected_Ioo : IsPreconnected (Ioo a b) :=
ordConnected_Ioo.isPreconnected | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isPreconnected_Ioo | null |
isPreconnected_Ioc : IsPreconnected (Ioc a b) :=
ordConnected_Ioc.isPreconnected | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isPreconnected_Ioc | null |
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