statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
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t2_iff_nhds : T2Space X ↔ ∀ {x y : X}, NeBot (𝓝 x ⊓ 𝓝 y) → x = y | by
simp only [t2Space_iff_disjoint_nhds, disjoint_iff, neBot_iff, Ne, not_imp_comm, Pairwise] | theorem | t2_iff_nhds | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Pairwise",
"T2Space",
"disjoint_iff",
"not_imp_comm",
"t2Space_iff_disjoint_nhds"
] | A space is T₂ iff the neighbourhoods of distinct points generate the bottom filter. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
eq_of_nhds_neBot [T2Space X] {x y : X} (h : NeBot (𝓝 x ⊓ 𝓝 y)) : x = y | t2_iff_nhds.mp ‹_› h | theorem | eq_of_nhds_neBot | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t2Space_iff_nhds :
T2Space X ↔ Pairwise fun x y : X => ∃ U ∈ 𝓝 x, ∃ V ∈ 𝓝 y, Disjoint U V | by
simp only [t2Space_iff_disjoint_nhds, Filter.disjoint_iff, Pairwise] | theorem | t2Space_iff_nhds | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Disjoint",
"Filter.disjoint_iff",
"Pairwise",
"T2Space",
"t2Space_iff_disjoint_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t2_separation_nhds [T2Space X] {x y : X} (h : x ≠ y) :
∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ Disjoint u v | let ⟨u, v, open_u, open_v, x_in, y_in, huv⟩ := t2_separation h
⟨u, v, open_u.mem_nhds x_in, open_v.mem_nhds y_in, huv⟩ | theorem | t2_separation_nhds | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Disjoint",
"T2Space",
"t2_separation"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t2_separation_compact_nhds [LocallyCompactSpace X] [T2Space X] {x y : X} (h : x ≠ y) :
∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ IsCompact u ∧ IsCompact v ∧ Disjoint u v | by
simpa only [exists_prop, ← exists_and_left, and_comm, and_assoc, and_left_comm] using
((compact_basis_nhds x).disjoint_iff (compact_basis_nhds y)).1 (disjoint_nhds_nhds.2 h) | theorem | t2_separation_compact_nhds | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Disjoint",
"IsCompact",
"LocallyCompactSpace",
"T2Space",
"compact_basis_nhds",
"disjoint_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t2_iff_ultrafilter :
T2Space X ↔ ∀ {x y : X} (f : Ultrafilter X), ↑f ≤ 𝓝 x → ↑f ≤ 𝓝 y → x = y | t2_iff_nhds.trans <| by simp only [← exists_ultrafilter_iff, and_imp, le_inf_iff, exists_imp] | theorem | t2_iff_ultrafilter | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"T2Space",
"Ultrafilter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t2_iff_isClosed_diagonal : T2Space X ↔ IsClosed (diagonal X) | by
simp only [t2Space_iff_disjoint_nhds, ← isOpen_compl_iff, isOpen_iff_mem_nhds, Prod.forall,
nhds_prod_eq, compl_diagonal_mem_prod, mem_compl_iff, mem_diagonal_iff, Pairwise] | theorem | t2_iff_isClosed_diagonal | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"IsClosed",
"Pairwise",
"T2Space",
"isOpen_compl_iff",
"isOpen_iff_mem_nhds",
"nhds_prod_eq",
"t2Space_iff_disjoint_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_diagonal [T2Space X] : IsClosed (diagonal X) | t2_iff_isClosed_diagonal.mp ‹_› | theorem | isClosed_diagonal | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"IsClosed",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t2Space_iff_of_isOpenQuotientMap [TopologicalSpace Y] {π : X → Y}
(h : IsOpenQuotientMap π) : T2Space Y ↔ IsClosed {q : X × X | π q.1 = π q.2} | by
rw [t2_iff_isClosed_diagonal]
replace h := IsOpenQuotientMap.prodMap h h
refine ⟨fun H ↦ H.preimage h.continuous, fun H ↦ ?_⟩
simp_rw [← isOpen_compl_iff] at H ⊢
convert! h.isOpenMap _ H
exact (h.surjective.image_preimage _).symm | theorem | t2Space_iff_of_isOpenQuotientMap | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"IsClosed",
"IsOpenQuotientMap",
"IsOpenQuotientMap.prodMap",
"T2Space",
"TopologicalSpace",
"isOpen_compl_iff",
"symm",
"t2_iff_isClosed_diagonal"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_nhds_unique [T2Space X] {f : Y → X} {l : Filter Y} {a b : X} [NeBot l]
(ha : Tendsto f l (𝓝 a)) (hb : Tendsto f l (𝓝 b)) : a = b | (tendsto_nhds_unique_inseparable ha hb).eq | theorem | tendsto_nhds_unique | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Filter",
"T2Space",
"tendsto_nhds_unique_inseparable"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_nhds_unique' [T2Space X] {f : Y → X} {l : Filter Y} {a b : X} (_ : NeBot l)
(ha : Tendsto f l (𝓝 a)) (hb : Tendsto f l (𝓝 b)) : a = b | tendsto_nhds_unique ha hb | theorem | tendsto_nhds_unique' | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Filter",
"T2Space",
"tendsto_nhds_unique"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_nhds_unique_of_eventuallyEq [T2Space X] {f g : Y → X} {l : Filter Y} {a b : X}
[NeBot l] (ha : Tendsto f l (𝓝 a)) (hb : Tendsto g l (𝓝 b)) (hfg : f =ᶠ[l] g) : a = b | tendsto_nhds_unique (ha.congr' hfg) hb | theorem | tendsto_nhds_unique_of_eventuallyEq | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Filter",
"T2Space",
"tendsto_nhds_unique"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_nhds_unique_of_frequently_eq [T2Space X] {f g : Y → X} {l : Filter Y} {a b : X}
(ha : Tendsto f l (𝓝 a)) (hb : Tendsto g l (𝓝 b)) (hfg : ∃ᶠ x in l, f x = g x) : a = b | have : ∃ᶠ z : X × X in 𝓝 (a, b), z.1 = z.2 := (ha.prodMk_nhds hb).frequently hfg
not_not.1 fun hne => this (isClosed_diagonal.isOpen_compl.mem_nhds hne) | theorem | tendsto_nhds_unique_of_frequently_eq | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Filter",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.nhdsSet_inter_eq [T2Space X] {s t : Set X} (hs : IsCompact s) (ht : IsCompact t) :
𝓝ˢ (s ∩ t) = 𝓝ˢ s ⊓ 𝓝ˢ t | by
refine le_antisymm (nhdsSet_inter_le _ _) ?_
simp_rw [hs.nhdsSet_inf_eq_biSup, ht.inf_nhdsSet_eq_biSup, nhdsSet, sSup_image]
refine iSup₂_le fun x hxs ↦ iSup₂_le fun y hyt ↦ ?_
rcases eq_or_ne x y with (rfl | hne)
· exact le_iSup₂_of_le x ⟨hxs, hyt⟩ (inf_idem _).le
· exact (disjoint_nhds_nhds.mpr hne).eq... | theorem | IsCompact.nhdsSet_inter_eq | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"IsCompact",
"Set",
"T2Space",
"eq_or_ne",
"iSup₂_le",
"le_antisymm",
"le_iSup₂_of_le",
"nhdsSet",
"nhdsSet_inter_le",
"sSup_image"
] | If `s` and `t` are compact sets in a T₂ space, then the set neighborhoods filter of `s ∩ t`
is the infimum of set neighborhoods filters for `s` and `t`.
For general sets, only the `≤` inequality holds, see `nhdsSet_inter_le`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.separation_of_notMem {X : Type u_1} [TopologicalSpace X] [T2Space X] {x : X}
{t : Set X} (H1 : IsCompact t) (H2 : x ∉ t) :
∃ (U : Set X), ∃ (V : Set X), IsOpen U ∧ IsOpen V ∧ t ⊆ U ∧ x ∈ V ∧ Disjoint U V | by
simpa [SeparatedNhds] using SeparatedNhds.of_isCompact_isCompact_isClosed H1 isCompact_singleton
isClosed_singleton <| disjoint_singleton_right.mpr H2 | lemma | IsCompact.separation_of_notMem | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Disjoint",
"IsCompact",
"IsOpen",
"SeparatedNhds",
"SeparatedNhds.of_isCompact_isCompact_isClosed",
"Set",
"T2Space",
"TopologicalSpace",
"isClosed_singleton",
"isCompact_singleton"
] | In a `T2Space X`, for a compact set `t` and a point `x` outside `t`, there are open sets `U`,
`V` that separate `t` and `x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.disjoint_nhdsSet_nhds {X : Type u_1} [TopologicalSpace X] [T2Space X] {x : X}
{t : Set X} (H1 : IsCompact t) (H2 : x ∉ t) :
Disjoint (𝓝ˢ t) (𝓝 x) | by
simpa using SeparatedNhds.disjoint_nhdsSet <| .of_isCompact_isCompact_isClosed H1
isCompact_singleton isClosed_singleton <| disjoint_singleton_right.mpr H2 | lemma | IsCompact.disjoint_nhdsSet_nhds | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Disjoint",
"IsCompact",
"Set",
"T2Space",
"TopologicalSpace",
"isClosed_singleton",
"isCompact_singleton"
] | In a `T2Space X`, for a compact set `t` and a point `x` outside `t`, `𝓝ˢ t` and `𝓝 x` are
disjoint. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.InjOn.exists_mem_nhdsSet {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
[T2Space Y] {f : X → Y} {s : Set X} (inj : InjOn f s) (sc : IsCompact s)
(fc : ∀ x ∈ s, ContinuousAt f x) (loc : ∀ x ∈ s, ∃ u ∈ 𝓝 x, InjOn f u) :
∃ t ∈ 𝓝ˢ s, InjOn f t | by
have : ∀ x ∈ s ×ˢ s, ∀ᶠ y in 𝓝 x, f y.1 = f y.2 → y.1 = y.2 := fun (x, y) ⟨hx, hy⟩ ↦ by
rcases eq_or_ne x y with rfl | hne
· rcases loc x hx with ⟨u, hu, hf⟩
exact Filter.mem_of_superset (prod_mem_nhds hu hu) <| forall_prod_set.2 hf
· suffices ∀ᶠ z in 𝓝 (x, y), f z.1 ≠ f z.2 from this.mono fun ... | theorem | Set.InjOn.exists_mem_nhdsSet | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"ContinuousAt",
"Filter.mem_of_superset",
"IsCompact",
"Set",
"T2Space",
"TopologicalSpace",
"eq_or_ne",
"eventually_nhdsSet_iff_forall",
"prod_mem_nhds"
] | If a function `f` is
- injective on a compact set `s`;
- continuous at every point of this set;
- injective on a neighborhood of each point of this set,
then it is injective on a neighborhood of this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.InjOn.exists_isOpen_superset {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
[T2Space Y] {f : X → Y} {s : Set X} (inj : InjOn f s) (sc : IsCompact s)
(fc : ∀ x ∈ s, ContinuousAt f x) (loc : ∀ x ∈ s, ∃ u ∈ 𝓝 x, InjOn f u) :
∃ t, IsOpen t ∧ s ⊆ t ∧ InjOn f t | let ⟨_t, hst, ht⟩ := inj.exists_mem_nhdsSet sc fc loc
let ⟨u, huo, hsu, hut⟩ := mem_nhdsSet_iff_exists.1 hst
⟨u, huo, hsu, ht.mono hut⟩ | theorem | Set.InjOn.exists_isOpen_superset | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"ContinuousAt",
"IsCompact",
"IsOpen",
"Set",
"T2Space",
"TopologicalSpace"
] | If a function `f` is
- injective on a compact set `s`;
- continuous at every point of this set;
- injective on a neighborhood of each point of this set,
then it is injective on an open neighborhood of this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
lim_eq {x : X} [NeBot f] (h : f ≤ 𝓝 x) : @lim _ _ ⟨x⟩ f = x | tendsto_nhds_unique (le_nhds_lim ⟨x, h⟩) h | theorem | lim_eq | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"le_nhds_lim",
"tendsto_nhds_unique"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lim_eq_iff [NeBot f] (h : ∃ x : X, f ≤ 𝓝 x) {x} : @lim _ _ ⟨x⟩ f = x ↔ f ≤ 𝓝 x | ⟨fun c => c ▸ le_nhds_lim h, lim_eq⟩ | theorem | lim_eq_iff | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"le_nhds_lim"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ultrafilter.lim_eq_iff_le_nhds [CompactSpace X] {x : X} {F : Ultrafilter X} :
F.lim = x ↔ ↑F ≤ 𝓝 x | ⟨fun h => h ▸ F.le_nhds_lim, lim_eq⟩ | theorem | Ultrafilter.lim_eq_iff_le_nhds | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"CompactSpace",
"Ultrafilter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_iff_ultrafilter' [CompactSpace X] (U : Set X) :
IsOpen U ↔ ∀ F : Ultrafilter X, F.lim ∈ U → U ∈ F.1 | by
rw [isOpen_iff_ultrafilter]
refine ⟨fun h F hF => h F.lim hF F F.le_nhds_lim, ?_⟩
intro cond x hx f h
rw [← Ultrafilter.lim_eq_iff_le_nhds.2 h] at hx
exact cond _ hx | theorem | isOpen_iff_ultrafilter' | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"CompactSpace",
"IsOpen",
"Set",
"Ultrafilter",
"isOpen_iff_ultrafilter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.limUnder_eq {x : X} {f : Filter Y} [NeBot f] {g : Y → X}
(h : Tendsto g f (𝓝 x)) : @limUnder _ _ _ ⟨x⟩ f g = x | lim_eq h | theorem | Filter.Tendsto.limUnder_eq | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Filter",
"lim_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.limUnder_eq_iff {f : Filter Y} [NeBot f] {g : Y → X} (h : ∃ x, Tendsto g f (𝓝 x))
{x} : @limUnder _ _ _ ⟨x⟩ f g = x ↔ Tendsto g f (𝓝 x) | ⟨fun c => c ▸ tendsto_nhds_limUnder h, Filter.Tendsto.limUnder_eq⟩ | theorem | Filter.limUnder_eq_iff | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Filter",
"tendsto_nhds_limUnder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.limUnder_eq [TopologicalSpace Y] {f : Y → X} (h : Continuous f) (y : Y) :
@limUnder _ _ _ ⟨f y⟩ (𝓝 y) f = f y | (h.tendsto y).limUnder_eq | theorem | Continuous.limUnder_eq | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Continuous",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lim_nhds (x : X) : @lim _ _ ⟨x⟩ (𝓝 x) = x | lim_eq le_rfl | theorem | lim_nhds | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"le_rfl",
"lim_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
limUnder_nhds_id (x : X) : @limUnder _ _ _ ⟨x⟩ (𝓝 x) id = x | lim_nhds x | theorem | limUnder_nhds_id | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"lim_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lim_nhdsWithin {x : X} {s : Set X} (h : x ∈ closure s) : @lim _ _ ⟨x⟩ (𝓝[s] x) = x | haveI : NeBot (𝓝[s] x) := mem_closure_iff_clusterPt.1 h
lim_eq inf_le_left | theorem | lim_nhdsWithin | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Set",
"closure",
"lim_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
limUnder_nhdsWithin_id {x : X} {s : Set X} (h : x ∈ closure s) :
@limUnder _ _ _ ⟨x⟩ (𝓝[s] x) id = x | lim_nhdsWithin h | theorem | limUnder_nhdsWithin_id | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Set",
"closure",
"lim_nhdsWithin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
separated_by_continuous [TopologicalSpace Y] [T2Space Y]
{f : X → Y} (hf : Continuous f) {x y : X} (h : f x ≠ f y) :
∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v | let ⟨u, v, uo, vo, xu, yv, uv⟩ := t2_separation h
⟨f ⁻¹' u, f ⁻¹' v, uo.preimage hf, vo.preimage hf, xu, yv, uv.preimage _⟩ | theorem | separated_by_continuous | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Continuous",
"Disjoint",
"IsOpen",
"Set",
"T2Space",
"TopologicalSpace",
"t2_separation"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
separated_by_isOpenEmbedding [TopologicalSpace Y] [T2Space X]
{f : X → Y} (hf : IsOpenEmbedding f) {x y : X} (h : x ≠ y) :
∃ u v : Set Y, IsOpen u ∧ IsOpen v ∧ f x ∈ u ∧ f y ∈ v ∧ Disjoint u v | let ⟨u, v, uo, vo, xu, yv, uv⟩ := t2_separation h
⟨f '' u, f '' v, hf.isOpenMap _ uo, hf.isOpenMap _ vo, mem_image_of_mem _ xu,
mem_image_of_mem _ yv, disjoint_image_of_injective hf.injective uv⟩ | theorem | separated_by_isOpenEmbedding | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Disjoint",
"IsOpen",
"Set",
"T2Space",
"TopologicalSpace",
"t2_separation"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Prod.t2Space [T2Space X] [TopologicalSpace Y] [T2Space Y] : T2Space (X × Y) | inferInstance | instance | Prod.t2Space | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"T2Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
T2Space.of_injective_continuous [TopologicalSpace Y] [T2Space Y] {f : X → Y}
(hinj : Injective f) (hc : Continuous f) : T2Space X | ⟨fun _ _ h => separated_by_continuous hc (hinj.ne h)⟩ | theorem | T2Space.of_injective_continuous | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Continuous",
"T2Space",
"TopologicalSpace",
"separated_by_continuous"
] | If the codomain of an injective continuous function is a Hausdorff space, then so is its
domain. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Topology.IsEmbedding.t2Space [TopologicalSpace Y] [T2Space Y] {f : X → Y}
(hf : IsEmbedding f) : T2Space X | .of_injective_continuous hf.injective hf.continuous | theorem | Topology.IsEmbedding.t2Space | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"T2Space",
"TopologicalSpace"
] | If the codomain of a topological embedding is a Hausdorff space, then so is its domain.
See also `T2Space.of_continuous_injective`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Homeomorph.t2Space [TopologicalSpace Y] [T2Space X] (h : X ≃ₜ Y) : T2Space Y | h.symm.isEmbedding.t2Space | theorem | Homeomorph.t2Space | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"T2Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ULift.instT2Space [T2Space X] : T2Space (ULift X) | IsEmbedding.uliftDown.t2Space | instance | ULift.instT2Space | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.t2Space {Y : X → Type v} [∀ a, TopologicalSpace (Y a)]
[∀ a, T2Space (Y a)] : T2Space (∀ a, Y a) | inferInstance | instance | Pi.t2Space | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"T2Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Sigma.t2Space {ι} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ a, T2Space (X a)] :
T2Space (Σ i, X i) | by
constructor
rintro ⟨i, x⟩ ⟨j, y⟩ ne
rcases eq_or_ne i j with (rfl | h)
· replace ne : x ≠ y := ne_of_apply_ne _ ne
exact separated_by_isOpenEmbedding .sigmaMk ne
· let _ := (⊥ : TopologicalSpace ι); have : DiscreteTopology ι := ⟨rfl⟩
exact separated_by_continuous (continuous_def.2 fun u _ => isOpen... | instance | Sigma.t2Space | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"DiscreteTopology",
"T2Space",
"TopologicalSpace",
"eq_or_ne",
"isOpen_sigma_fst_preimage",
"separated_by_continuous",
"separated_by_isOpenEmbedding"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t2Setoid : Setoid X | sInf {s | T2Space (Quotient s)} | def | t2Setoid | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"T2Space"
] | The smallest equivalence relation on a topological space giving a T2 quotient. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
T2Quotient | Quotient (t2Setoid X) | def | T2Quotient | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"t2Setoid"
] | The largest T2 quotient of a topological space. This construction is left-adjoint to the
inclusion of T2 spaces into all topological spaces. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mk : X → T2Quotient X | Quotient.mk (t2Setoid X) | def | T2Quotient.mk | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"T2Quotient",
"t2Setoid"
] | The map from a topological space to its largest T2 quotient. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mk_eq {x y : X} : mk x = mk y ↔ ∀ s : Setoid X, T2Space (Quotient s) → s x y | Setoid.quotient_mk_sInf_eq | lemma | T2Quotient.mk_eq | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Setoid.quotient_mk_sInf_eq",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
surjective_mk : Surjective (mk : X → T2Quotient X) | Quotient.mk_surjective | lemma | T2Quotient.surjective_mk | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Quotient.mk_surjective",
"T2Quotient"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_mk : Continuous (mk : X → T2Quotient X) | continuous_quotient_mk' | lemma | T2Quotient.continuous_mk | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Continuous",
"T2Quotient",
"continuous_quotient_mk'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inductionOn {motive : T2Quotient X → Prop} (q : T2Quotient X)
(h : ∀ x, motive (T2Quotient.mk x)) : motive q | Quotient.inductionOn q h | lemma | T2Quotient.inductionOn | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"T2Quotient",
"T2Quotient.mk"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inductionOn₂ [TopologicalSpace Y] {motive : T2Quotient X → T2Quotient Y → Prop}
(q : T2Quotient X) (q' : T2Quotient Y) (h : ∀ x y, motive (mk x) (mk y)) : motive q q' | Quotient.inductionOn₂ q q' h | lemma | T2Quotient.inductionOn₂ | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"T2Quotient",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compatible {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y]
{f : X → Y} (hf : Continuous f) : letI _ | t2Setoid X
∀ (a b : X), a ≈ b → f a = f b := by
change t2Setoid X ≤ Setoid.ker f
exact sInf_le <| .of_injective_continuous
(Setoid.kerLift_injective _) (hf.quotient_lift fun _ _ ↦ id) | lemma | T2Quotient.compatible | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Continuous",
"Setoid.ker",
"Setoid.kerLift_injective",
"T2Space",
"TopologicalSpace",
"t2Setoid"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lift {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y]
{f : X → Y} (hf : Continuous f) : T2Quotient X → Y | Quotient.lift f (T2Quotient.compatible hf) | def | T2Quotient.lift | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Continuous",
"T2Quotient",
"T2Quotient.compatible",
"T2Space",
"TopologicalSpace"
] | The universal property of the largest T2 quotient of a topological space `X`: any continuous
map from `X` to a T2 space `Y` uniquely factors through `T2Quotient X`. This declaration builds the
factored map. Its continuity is `T2Quotient.continuous_lift`, the fact that it indeed factors the
original map is `T2Quotient.l... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuous_lift {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y]
{f : X → Y} (hf : Continuous f) : Continuous (T2Quotient.lift hf) | continuous_coinduced_dom.mpr hf | lemma | T2Quotient.continuous_lift | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Continuous",
"T2Quotient.lift",
"T2Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lift_mk {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y]
{f : X → Y} (hf : Continuous f) (x : X) : lift hf (mk x) = f x | Quotient.lift_mk (s := t2Setoid X) f (T2Quotient.compatible hf) x | lemma | T2Quotient.lift_mk | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Continuous",
"Quotient.lift_mk",
"T2Quotient.compatible",
"T2Space",
"TopologicalSpace",
"t2Setoid"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
unique_lift {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y]
{f : X → Y} (hf : Continuous f) {g : T2Quotient X → Y} (hfg : g ∘ mk = f) :
g = lift hf | by
apply surjective_mk X |>.right_cancellable |>.mp <| funext _
simp [← hfg] | lemma | T2Quotient.unique_lift | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Continuous",
"T2Quotient",
"T2Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_eq [T2Space X] {f g : Y → X} (hf : Continuous f) (hg : Continuous g) :
IsClosed { y : Y | f y = g y } | continuous_iff_isClosed.mp (hf.prodMk hg) _ isClosed_diagonal | theorem | isClosed_eq | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Continuous",
"IsClosed",
"T2Space",
"isClosed_diagonal"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.isClosed_eq [T2Space Y] {f g : X → Y} {s : Set X} (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 isClosed_diagonal | theorem | IsClosed.isClosed_eq | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"ContinuousOn",
"IsClosed",
"Set",
"T2Space",
"isClosed_diagonal"
] | If functions `f` and `g` are continuous on a closed set `s`,
then the set of points `x ∈ s` such that `f x = g x` is a closed set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpen_ne_fun [T2Space X] {f g : Y → X} (hf : Continuous f) (hg : Continuous g) :
IsOpen { y : Y | f y ≠ g y } | isOpen_compl_iff.mpr <| isClosed_eq hf hg | theorem | isOpen_ne_fun | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Continuous",
"IsOpen",
"T2Space",
"isClosed_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.EqOn.closure [T2Space X] {s : Set Y} {f g : Y → X} (h : EqOn f g s)
(hf : Continuous f) (hg : Continuous g) : EqOn f g (closure s) | closure_minimal h (isClosed_eq hf hg) | theorem | Set.EqOn.closure | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Continuous",
"Set",
"T2Space",
"closure",
"closure_minimal",
"isClosed_eq"
] | If two continuous maps are equal on `s`, then they are equal on the closure of `s`. See also
`Set.EqOn.of_subset_closure` for a more general version. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Continuous.ext_on [T2Space X] {s : Set Y} (hs : Dense s) {f g : Y → X} (hf : Continuous f)
(hg : Continuous g) (h : EqOn f g s) : f = g | funext fun x => h.closure hf hg (hs x) | theorem | Continuous.ext_on | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Continuous",
"Dense",
"Set",
"T2Space"
] | If two continuous functions are equal on a dense set, then they are equal. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
eqOn_closure₂' [T2Space Z] {s : Set X} {t : Set Y} {f g : X → Y → Z}
(h : ∀ x ∈ s, ∀ y ∈ t, f x y = g x y) (hf₁ : ∀ x, Continuous (f x))
(hf₂ : ∀ y, Continuous fun x => f x y) (hg₁ : ∀ x, Continuous (g x))
(hg₂ : ∀ y, Continuous fun x => g x y) : ∀ x ∈ closure s, ∀ y ∈ closure t, f x y = g x y | suffices closure s ⊆ ⋂ y ∈ closure t, { x | f x y = g x y } by simpa only [subset_def, mem_iInter]
(closure_minimal fun x hx => mem_iInter₂.2 <| Set.EqOn.closure (h x hx) (hf₁ _) (hg₁ _)) <|
isClosed_biInter fun _ _ => isClosed_eq (hf₂ _) (hg₂ _) | theorem | eqOn_closure₂' | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Continuous",
"Set",
"Set.EqOn.closure",
"T2Space",
"closure",
"closure_minimal",
"isClosed_biInter",
"isClosed_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eqOn_closure₂ [T2Space Z] {s : Set X} {t : Set Y} {f g : X → Y → Z}
(h : ∀ x ∈ s, ∀ y ∈ t, f x y = g x y) (hf : Continuous (uncurry f))
(hg : Continuous (uncurry g)) : ∀ x ∈ closure s, ∀ y ∈ closure t, f x y = g x y | eqOn_closure₂' h hf.uncurry_left hf.uncurry_right hg.uncurry_left hg.uncurry_right | theorem | eqOn_closure₂ | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Continuous",
"Set",
"T2Space",
"closure",
"eqOn_closure₂'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.EqOn.of_subset_closure [T2Space Y] {s t : Set X} {f g : X → Y} (h : EqOn f g s)
(hf : ContinuousOn f t) (hg : ContinuousOn g t) (hst : s ⊆ t) (hts : t ⊆ closure s) :
EqOn f g t | by
intro x hx
have : (𝓝[s] x).NeBot := mem_closure_iff_clusterPt.mp (hts hx)
exact
tendsto_nhds_unique_of_eventuallyEq ((hf x hx).mono_left <| nhdsWithin_mono _ hst)
((hg x hx).mono_left <| nhdsWithin_mono _ hst) (h.eventuallyEq_of_mem self_mem_nhdsWithin) | theorem | Set.EqOn.of_subset_closure | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"ContinuousOn",
"Set",
"T2Space",
"closure",
"nhdsWithin_mono",
"self_mem_nhdsWithin",
"tendsto_nhds_unique_of_eventuallyEq"
] | If `f x = g x` for all `x ∈ s` and `f`, `g` are continuous on `t`, `s ⊆ t ⊆ closure s`, then
`f x = g x` for all `x ∈ t`. See also `Set.EqOn.closure`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Function.LeftInverse.isClosed_range [T2Space X] {f : X → Y} {g : Y → X}
(h : Function.LeftInverse f g) (hf : Continuous f) (hg : Continuous g) : IsClosed (range g) | have : EqOn (g ∘ f) id (closure <| range g) :=
h.rightInvOn_range.eqOn.closure (hg.comp hf) continuous_id
isClosed_of_closure_subset fun x hx => ⟨f x, this hx⟩ | theorem | Function.LeftInverse.isClosed_range | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Continuous",
"IsClosed",
"T2Space",
"closure",
"continuous_id",
"isClosed_of_closure_subset"
] | Retract subspaces of Hausdorff spaces are closed. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Function.LeftInverse.isClosedEmbedding [T2Space X] {f : X → Y} {g : Y → X}
(h : Function.LeftInverse f g) (hf : Continuous f) (hg : Continuous g) : IsClosedEmbedding g | ⟨.of_leftInverse h hf hg, h.isClosed_range hf hg⟩ | theorem | Function.LeftInverse.isClosedEmbedding | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Continuous",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SeparatedNhds.of_isCompact_isCompact [T2Space X] {s t : Set X} (hs : IsCompact s)
(ht : IsCompact t) (hst : Disjoint s t) : SeparatedNhds s t | by
simp only [SeparatedNhds, prod_subset_compl_diagonal_iff_disjoint.symm] at hst ⊢
exact generalized_tube_lemma hs ht isClosed_diagonal.isOpen_compl hst | theorem | SeparatedNhds.of_isCompact_isCompact | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Disjoint",
"IsCompact",
"SeparatedNhds",
"Set",
"T2Space",
"generalized_tube_lemma"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SeparatedNhds.of_isClosed_isCompact_closure_compl_isClosed [R1Space X] {s : Set X}
{t : Set X} (H1 : IsClosed s) (H2 : IsCompact (closure sᶜ)) (H3 : IsClosed t)
(H4 : Disjoint s t) : SeparatedNhds s t | by
-- Since `t` is a closed subset of the compact set `closure sᶜ`, it is compact.
have ht : IsCompact t := .of_isClosed_subset H2 H3 <| H4.subset_compl_left.trans subset_closure
-- we split `s` into its frontier and its interior.
rw [← diff_union_of_subset (interior_subset (s := s))]
-- since `t ⊆ sᶜ`, which... | lemma | SeparatedNhds.of_isClosed_isCompact_closure_compl_isClosed | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Disjoint",
"IsClosed",
"IsCompact",
"R1Space",
"SeparatedNhds",
"Set",
"closure",
"frontier_eq_closure_inter_closure",
"interior_subset",
"isClosed_closure",
"isOpen_interior",
"le_rfl",
"subset_closure"
] | In a `R1Space X`, for disjoint closed sets `s t` such that `closure sᶜ` is compact,
there are neighbourhoods that separate `s` and `t`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
SeparatedNhds.of_finset_finset [T2Space X] (s t : Finset X) (h : Disjoint s t) :
SeparatedNhds (s : Set X) t | .of_isCompact_isCompact s.finite_toSet.isCompact t.finite_toSet.isCompact <| mod_cast h | theorem | SeparatedNhds.of_finset_finset | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Disjoint",
"Finset",
"SeparatedNhds",
"Set",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SeparatedNhds.of_finite [T2Space X] {s t : Set X} (hs : s.Finite) (ht : t.Finite)
(h : Disjoint s t) : SeparatedNhds s t | by
rw [← hs.coe_toFinset, ← ht.coe_toFinset]
exact SeparatedNhds.of_finset_finset _ _ (Finite.disjoint_toFinset.2 h) | theorem | SeparatedNhds.of_finite | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Disjoint",
"SeparatedNhds",
"SeparatedNhds.of_finset_finset",
"Set",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SeparatedNhds.of_singleton_finset [T2Space X] {x : X} {s : Finset X} (h : x ∉ s) :
SeparatedNhds ({x} : Set X) s | mod_cast .of_finset_finset {x} s (Finset.disjoint_singleton_left.mpr h) | theorem | SeparatedNhds.of_singleton_finset | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Finset",
"SeparatedNhds",
"Set",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.isClosed [T2Space X] {s : Set X} (hs : IsCompact s) : IsClosed s | isClosed_iff_forall_filter.2 fun _x _f _ hfs hfx =>
let ⟨_y, hy, hfy⟩ := hs.exists_clusterPt hfs
mem_of_eq_of_mem (eq_of_nhds_neBot (hfy.mono hfx).neBot).symm hy | theorem | IsCompact.isClosed | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"IsClosed",
"IsCompact",
"Set",
"T2Space",
"eq_of_nhds_neBot",
"symm"
] | In a `T2Space`, every compact set is closed. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.preimage_continuous [CompactSpace X] [T2Space Y] {f : X → Y} {s : Set Y}
(hs : IsCompact s) (hf : Continuous f) : IsCompact (f ⁻¹' s) | (hs.isClosed.preimage hf).isCompact | theorem | IsCompact.preimage_continuous | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"CompactSpace",
"Continuous",
"IsCompact",
"Set",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.isCompact_iff {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)]
[∀ i, T2Space (X i)] {s : Set (Π i, X i)} :
IsCompact s ↔ IsClosed s ∧ ∀ i, IsCompact (eval i '' s) | by
constructor <;> intro H
· exact ⟨H.isClosed, fun i ↦ H.image <| continuous_apply i⟩
· exact IsCompact.of_isClosed_subset (isCompact_univ_pi H.2) H.1 (subset_pi_eval_image univ s) | lemma | Pi.isCompact_iff | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"IsClosed",
"IsCompact",
"IsCompact.of_isClosed_subset",
"Set",
"T2Space",
"TopologicalSpace",
"continuous_apply",
"isCompact_univ_pi"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.isCompact_closure_iff {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)]
[∀ i, R1Space (X i)] {s : Set (Π i, X i)} :
IsCompact (closure s) ↔ ∀ i, IsCompact (closure <| eval i '' s) | by
simp_rw [← exists_isCompact_superset_iff, Pi.exists_compact_superset_iff, image_subset_iff] | lemma | Pi.isCompact_closure_iff | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"IsCompact",
"Pi.exists_compact_superset_iff",
"R1Space",
"Set",
"TopologicalSpace",
"closure",
"exists_isCompact_superset_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_subset_nhds_of_isCompact [T2Space X] {ι : Type*} [Nonempty ι] {V : ι → Set X}
(hV : Directed (· ⊇ ·) V) (hV_cpct : ∀ i, IsCompact (V i)) {U : Set X}
(hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x) : ∃ i, V i ⊆ U | exists_subset_nhds_of_isCompact' hV hV_cpct (fun i => (hV_cpct i).isClosed) hU | theorem | exists_subset_nhds_of_isCompact | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Directed",
"IsCompact",
"Set",
"T2Space",
"exists_subset_nhds_of_isCompact'"
] | If `V : ι → Set X` is a decreasing family of compact sets then any neighborhood of
`⋂ i, V i` contains some `V i`. This is a version of `exists_subset_nhds_of_isCompact'` where we
don't need to assume each `V i` closed because it follows from compactness since `X` is
assumed to be Hausdorff. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
CompactExhaustion.isClosed [T2Space X] (K : CompactExhaustion X) (n : ℕ) : IsClosed (K n) | (K.isCompact n).isClosed | theorem | CompactExhaustion.isClosed | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"CompactExhaustion",
"IsClosed",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.inter [T2Space X] {s t : Set X} (hs : IsCompact s) (ht : IsCompact t) :
IsCompact (s ∩ t) | hs.inter_right <| ht.isClosed | theorem | IsCompact.inter | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"IsCompact",
"Set",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_closure_of_isCompact [T2Space Y] {s : Set X} (hs : IsCompact (closure s)) {f : X → Y}
(hf : ContinuousOn f (closure s)) : f '' closure s = closure (f '' s) | Subset.antisymm hf.image_closure <|
closure_minimal (image_mono subset_closure) (hs.image_of_continuousOn hf).isClosed | theorem | image_closure_of_isCompact | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"ContinuousOn",
"IsCompact",
"Set",
"T2Space",
"closure",
"closure_minimal",
"subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.ne_iff_eventually_ne [T2Space Y] {x : X} {f g : X → Y}
(hf : ContinuousAt f x) (hg : ContinuousAt g x) :
f x ≠ g x ↔ ∀ᶠ x in 𝓝 x, f x ≠ g x | by
constructor <;> intro hfg
· obtain ⟨Uf, Ug, h₁U, h₂U, h₃U, h₄U, h₅U⟩ := t2_separation hfg
rw [Set.disjoint_iff_inter_eq_empty] at h₅U
filter_upwards [inter_mem
(hf.preimage_mem_nhds (IsOpen.mem_nhds h₁U h₃U))
(hg.preimage_mem_nhds (IsOpen.mem_nhds h₂U h₄U))]
intro x hx
simp only [Set.... | theorem | ContinuousAt.ne_iff_eventually_ne | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"ContinuousAt",
"IsOpen.mem_nhds",
"Set.disjoint_iff_inter_eq_empty",
"Set.mem_inter_iff",
"Set.mem_preimage",
"T2Space",
"t2_separation"
] | Two continuous maps into a Hausdorff space disagree at a point iff they disagree in a
neighborhood. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ContinuousAt.eventuallyEq_nhds_iff_eventuallyEq_nhdsNE [T2Space Y] {x : X} {f g : X → Y}
(hf : ContinuousAt f x) (hg : ContinuousAt g x) [(𝓝[≠] x).NeBot] :
f =ᶠ[𝓝[≠] x] g ↔ f =ᶠ[𝓝 x] g | by
constructor <;> intro hfg
· apply eventuallyEq_nhds_of_eventuallyEq_nhdsNE hfg
by_contra hCon
obtain ⟨a, ha⟩ : {x | f x ≠ g x ∧ f x = g x}.Nonempty := by
have h₁ := (eventually_nhdsWithin_of_eventually_nhds
((hf.ne_iff_eventually_ne hg).1 hCon)).and hfg
have h₂ : ∅ ∉ 𝓝[≠] x := by exa... | theorem | ContinuousAt.eventuallyEq_nhds_iff_eventuallyEq_nhdsNE | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"ContinuousAt",
"T2Space",
"eventuallyEq_nhds_of_eventuallyEq_nhdsNE",
"eventually_nhdsWithin_of_eventually_nhds",
"nhdsWithin_le_nhds"
] | **Local identity principle** for continuous maps: Two continuous maps into a Hausdorff space
agree in a punctured neighborhood of a non-isolated point iff they agree in a neighborhood. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Continuous.isClosedMap [CompactSpace X] [T2Space Y] {f : X → Y}
(h : Continuous f) : IsClosedMap f | fun _s hs => (hs.isCompact.image h).isClosed | theorem | Continuous.isClosedMap | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"CompactSpace",
"Continuous",
"IsClosedMap",
"T2Space"
] | A continuous map from a compact space to a Hausdorff space is a closed map. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Continuous.isClosedEmbedding [CompactSpace X] [T2Space Y] {f : X → Y} (h : Continuous f)
(hf : Function.Injective f) : IsClosedEmbedding f | .of_continuous_injective_isClosedMap h hf h.isClosedMap | theorem | Continuous.isClosedEmbedding | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"CompactSpace",
"Continuous",
"T2Space"
] | A continuous injective map from a compact space to a Hausdorff space is a closed embedding. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsQuotientMap.of_surjective_continuous [CompactSpace X] [T2Space Y] {f : X → Y}
(hsurj : Surjective f) (hcont : Continuous f) : IsQuotientMap f | hcont.isClosedMap.isQuotientMap hcont hsurj | theorem | IsQuotientMap.of_surjective_continuous | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"CompactSpace",
"Continuous",
"T2Space"
] | A continuous surjective map from a compact space to a Hausdorff space is a quotient map. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isPreirreducible_iff_forall_mem_subset_closure_singleton [R1Space X] {S : Set X} :
IsPreirreducible S ↔ ∀ x ∈ S, S ⊆ closure {x} | by
constructor
· intro h x hx y hy
by_contra e
obtain ⟨U, V, hU, hV, hxU, hyV, h'⟩ := r1_separation fun h => e h.specializes.mem_closure
exact ((h U V hU hV ⟨x, hx, hxU⟩ ⟨y, hy, hyV⟩).mono inter_subset_right).not_disjoint h'
· intro h u v hu hv ⟨x, hxs, hxu⟩ ⟨y, hys, hyv⟩
exact ⟨x, hxs, hxu, (spec... | theorem | isPreirreducible_iff_forall_mem_subset_closure_singleton | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"IsPreirreducible",
"R1Space",
"Set",
"closure",
"r1_separation"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isPreirreducible_iff_subsingleton [T2Space X] {S : Set X} :
IsPreirreducible S ↔ S.Subsingleton | by
simp [isPreirreducible_iff_forall_mem_subset_closure_singleton, Set.Subsingleton, eq_comm] | theorem | isPreirreducible_iff_subsingleton | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"IsPreirreducible",
"Set",
"Set.Subsingleton",
"T2Space",
"isPreirreducible_iff_forall_mem_subset_closure_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsPreirreducible.subsingleton [T2Space X] {S : Set X} (h : IsPreirreducible S) :
S.Subsingleton | isPreirreducible_iff_subsingleton.1 h | lemma | IsPreirreducible.subsingleton | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"IsPreirreducible",
"Set",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isIrreducible_iff_singleton [T2Space X] {S : Set X} : IsIrreducible S ↔ ∃ x, S = {x} | by
rw [IsIrreducible, isPreirreducible_iff_subsingleton,
exists_eq_singleton_iff_nonempty_subsingleton] | theorem | isIrreducible_iff_singleton | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"IsIrreducible",
"Set",
"T2Space",
"isPreirreducible_iff_subsingleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
not_preirreducible_nontrivial_t2 (X) [TopologicalSpace X] [PreirreducibleSpace X]
[Nontrivial X] [T2Space X] : False | (PreirreducibleSpace.isPreirreducible_univ (X := X)).subsingleton.not_nontrivial nontrivial_univ | theorem | not_preirreducible_nontrivial_t2 | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Nontrivial",
"PreirreducibleSpace",
"T2Space",
"TopologicalSpace"
] | There does not exist a nontrivial preirreducible T₂ space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
t2Space_antitone {X : Type*} : Antitone (@T2Space X) | fun inst₁ inst₂ h_top h_t2 ↦ @T2Space.of_injective_continuous _ _ inst₁ inst₂
h_t2 _ Function.injective_id <| continuous_id_of_le h_top | theorem | t2Space_antitone | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Antitone",
"T2Space",
"T2Space.of_injective_continuous",
"continuous_id_of_le"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
totallySeparatedSpace_of_cardinalMk_lt_continuum (h : Cardinal.mk X < Cardinal.continuum) :
TotallySeparatedSpace X | totallySeparatedSpace_of_t0_of_basis_clopen <|
CompletelyRegularSpace.isTopologicalBasis_clopens_of_cardinalMk_lt_continuum h | theorem | CompletelyRegularSpace.totallySeparatedSpace_of_cardinalMk_lt_continuum | Topology.Separation | Mathlib/Topology/Separation/Lemmas.lean | [] | [
"Cardinal.continuum",
"Cardinal.mk",
"CompletelyRegularSpace.isTopologicalBasis_clopens_of_cardinalMk_lt_continuum",
"TotallySeparatedSpace",
"totallySeparatedSpace_of_t0_of_basis_clopen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Set.Countable.totallySeparatedSpace {s : Set X} (h : s.Countable) :
TotallySeparatedSpace s | have : _root_.Countable s := h
inferInstanceAs (TotallySeparatedSpace s) | lemma | Set.Countable.totallySeparatedSpace | Topology.Separation | Mathlib/Topology/Separation/Lemmas.lean | [] | [
"Set",
"TotallySeparatedSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.Countable.isTotallyDisconnected [MetricSpace X] {s : Set X} (hs : s.Countable) :
IsTotallyDisconnected s | by
rw [← totallyDisconnectedSpace_subtype_iff]
have : Countable s := hs
infer_instance | theorem | Set.Countable.isTotallyDisconnected | Topology.Separation | Mathlib/Topology/Separation/Lemmas.lean | [] | [
"Countable",
"IsTotallyDisconnected",
"MetricSpace",
"Set",
"totallyDisconnectedSpace_subtype_iff"
] | Countable subsets of metric spaces are totally disconnected. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
two_pow_mk_le_two_pow_mk_dense [NormalSpace X] {s d : Set X} (hs : IsClosed s)
[DiscreteTopology s] (hd : Dense d) : (2 : Cardinal) ^ #s ≤ 2 ^ #d | by
have h_closed (t) (ht : t ∈ 𝒫 s) : IsClosed t := by
rw [← inter_eq_self_of_subset_right ht, ← Subtype.image_preimage_val]
exact hs.isClosedMap_subtype_val _ (isClosed_discrete _)
choose U V hU hV hUs hVs hUV using fun t : 𝒫 s ↦
normal_separation (h_closed t t.2) (h_closed _ diff_subset) disjoint_sd... | theorem | IsClosed.two_pow_mk_le_two_pow_mk_dense | Topology.Separation | Mathlib/Topology/Separation/NotNormal.lean | [] | [
"Cardinal",
"Dense",
"DiscreteTopology",
"IsClosed",
"NormalSpace",
"Set",
"SetCoe.ext",
"Subtype.image_preimage_val",
"antisymm",
"isClosed_discrete",
"normal_separation",
"trans"
] | Let `s` be a closed set in a normal space and `d` be a dense set. If the induced topology on `s`
is discrete, then `𝒫 s` has cardinality less than or equal to `𝒫 d`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mk_lt_two_pow_mk_dense [NormalSpace X] {s d : Set X} (hs : IsClosed s)
[DiscreteTopology s] (hd : Dense d) : #s < 2 ^ #d | (#s).cantor.trans_le <| hs.two_pow_mk_le_two_pow_mk_dense hd | theorem | IsClosed.mk_lt_two_pow_mk_dense | Topology.Separation | Mathlib/Topology/Separation/NotNormal.lean | [] | [
"Dense",
"DiscreteTopology",
"IsClosed",
"NormalSpace",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
two_pow_mk_lt_continuum [NormalSpace X] {s : Set X} (hs : IsClosed s)
[DiscreteTopology s] : 2 ^ #s ≤ 𝔠 | have ⟨d, hd_countable, hd_dense⟩ := exists_countable_dense X
calc
2 ^ #s ≤ 2 ^ #d := hs.two_pow_mk_le_two_pow_mk_dense hd_dense
_ ≤ 2 ^ ℵ₀ := power_le_power_left two_ne_zero hd_countable.le_aleph0
_ = 𝔠 := two_power_aleph0 | theorem | IsClosed.two_pow_mk_lt_continuum | Topology.Separation | Mathlib/Topology/Separation/NotNormal.lean | [] | [
"DiscreteTopology",
"IsClosed",
"NormalSpace",
"Set",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mk_lt_continuum [NormalSpace X] {s : Set X} (hs : IsClosed s)
[DiscreteTopology s] : #s < 𝔠 | (#s).cantor.trans_le hs.two_pow_mk_lt_continuum | theorem | IsClosed.mk_lt_continuum | Topology.Separation | Mathlib/Topology/Separation/NotNormal.lean | [] | [
"DiscreteTopology",
"IsClosed",
"NormalSpace",
"Set"
] | Let `s` be a closed set in a separable normal space. If the induced topology on `s` is discrete,
then `s` has cardinality less than continuum. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
not_normal_of_continuum_le_mk {s : Set X} (hs : IsClosed s) [DiscreteTopology s]
(hmk : 𝔠 ≤ #s) : ¬NormalSpace X | fun _ ↦ hs.mk_lt_continuum.not_ge hmk | theorem | IsClosed.not_normal_of_continuum_le_mk | Topology.Separation | Mathlib/Topology/Separation/NotNormal.lean | [] | [
"DiscreteTopology",
"IsClosed",
"NormalSpace",
"Set"
] | Let `s` be a closed set in a separable space. If the induced topology on `s` is discrete and `s`
has cardinality at least continuum, then the ambient space is not a normal space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
perfectlyNormalSpace_iff_forall_isClosed_preimage_zero :
PerfectlyNormalSpace X ↔ ∀ s, IsClosed s → ∃ f : C(X, ℝ), s = f ⁻¹' {0} ∧
∀ x, f x ∈ Icc (0 : ℝ) 1 | where
mp h s hs := by
-- write `s` as the intersection of a sequence of open sets `U n`
obtain ⟨U, ho, hu⟩ := isGδ_iff_eq_iInter_nat.1 hs.isGδ
have (n : ℕ) : Disjoint s (U n)ᶜ := by
apply HasSubset.Subset.disjoint_compl_right
grw [hu, iInter_subset]
-- for each `n`, construct a continuous ... | theorem | perfectlyNormalSpace_iff_forall_isClosed_preimage_zero | Topology.Separation | Mathlib/Topology/Separation/PerfectlyNormal.lean | [] | [
"ContinuousMap.coe_mk",
"Disjoint",
"IsClosed",
"PerfectlyNormalSpace",
"SeparatedNhds",
"SeparatedNhds.of_finite",
"Topology.IsInducing.perfectlyNormalSpace",
"continuous_tsum",
"div_eq_one_iff_eq",
"exists_continuous_zero_one_of_isClosed",
"le_antisymm",
"summable_geometric_two'",
"tendsto... | A topological space is perfectly normal iff every closed set is the zero set of a continuous
function taking values in the unit interval. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
totallySeparatedSpace_of_t0_of_basis_clopen [T0Space X]
(h : IsTopologicalBasis { s : Set X | IsClopen s }) : TotallySeparatedSpace X | by
constructor
rintro x - y - hxy
choose U hU using exists_isOpen_xor_mem hxy
obtain ⟨hU₀, hU₁⟩ := hU
rcases hU₁ with hx | hy
· choose V hV using h.isOpen_iff.mp hU₀ x hx.1
exact ⟨V, Vᶜ, hV.1.isOpen, hV.1.compl.isOpen, hV.2.1, notMem_subset hV.2.2 hx.2,
(union_compl_self V).superset, disjoint_comp... | theorem | totallySeparatedSpace_of_t0_of_basis_clopen | Topology.Separation | Mathlib/Topology/Separation/Profinite.lean | [] | [
"IsClopen",
"Set",
"T0Space",
"TotallySeparatedSpace",
"exists_isOpen_xor_mem"
] | A T0 space with a clopen basis is totally separated. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nhds_basis_clopen (x : X) : (𝓝 x).HasBasis (fun s : Set X => x ∈ s ∧ IsClopen s) id | ⟨fun U => by
constructor
· have hx : connectedComponent x = {x} :=
totallyDisconnectedSpace_iff_connectedComponent_singleton.mp ‹_› x
rw [connectedComponent_eq_iInter_isClopen] at hx
intro hU
let N := { s // IsClopen s ∧ x ∈ s }
rsuffices ⟨⟨s, hs, hs'⟩, hs''⟩ : ∃ s : N, s.val ⊆ U... | theorem | nhds_basis_clopen | Topology.Separation | Mathlib/Topology/Separation/Profinite.lean | [] | [
"Directed",
"IsClopen",
"IsClosed",
"Set",
"connectedComponent",
"connectedComponent_eq_iInter_isClopen",
"exists_subset_nhds_of_compactSpace",
"isClopen_univ",
"mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isTopologicalBasis_isClopen : IsTopologicalBasis { s : Set X | IsClopen s } | by
apply isTopologicalBasis_of_isOpen_of_nhds fun U (hU : IsClopen U) => hU.2
intro x U hxU U_op
have : U ∈ 𝓝 x := IsOpen.mem_nhds U_op hxU
rcases (nhds_basis_clopen x).mem_iff.mp this with ⟨V, ⟨hxV, hV⟩, hVU : V ⊆ U⟩
use V
tauto | theorem | isTopologicalBasis_isClopen | Topology.Separation | Mathlib/Topology/Separation/Profinite.lean | [] | [
"IsClopen",
"IsOpen.mem_nhds",
"Set",
"nhds_basis_clopen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compact_exists_isClopen_in_isOpen {x : X} {U : Set X} (is_open : IsOpen U) (memU : x ∈ U) :
∃ V : Set X, IsClopen V ∧ x ∈ V ∧ V ⊆ U | isTopologicalBasis_isClopen.mem_nhds_iff.1 (is_open.mem_nhds memU) | theorem | compact_exists_isClopen_in_isOpen | Topology.Separation | Mathlib/Topology/Separation/Profinite.lean | [] | [
"IsClopen",
"IsOpen",
"Set"
] | Every member of an open set in a compact Hausdorff totally disconnected space
is contained in a clopen set contained in the open set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
loc_compact_Haus_tot_disc_of_zero_dim [TotallyDisconnectedSpace H] :
IsTopologicalBasis { s : Set H | IsClopen s } | by
refine isTopologicalBasis_of_isOpen_of_nhds (fun u hu => hu.2) fun x U memU hU => ?_
obtain ⟨s, comp, xs, sU⟩ := exists_compact_subset hU memU
let u : Set s := ((↑) : s → H) ⁻¹' interior s
have u_open_in_s : IsOpen u := isOpen_interior.preimage continuous_subtype_val
lift x to s using interior_subset xs
... | theorem | loc_compact_Haus_tot_disc_of_zero_dim | Topology.Separation | Mathlib/Topology/Separation/Profinite.lean | [] | [
"CompactSpace",
"IsClopen",
"IsOpen",
"Set",
"Set.inter_eq_self_of_subset_right",
"Set.range",
"Set.range_comp",
"Subtype.image_preimage_coe",
"Subtype.range_coe",
"TotallyDisconnectedSpace",
"compact_exists_isClopen_in_isOpen",
"continuous_subtype_val",
"exists_compact_subset",
"interior"... | A locally compact Hausdorff totally disconnected space has a basis with clopen elements. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
loc_compact_t2_tot_disc_iff_tot_sep :
TotallyDisconnectedSpace H ↔ TotallySeparatedSpace H | by
constructor
· intro h
exact totallySeparatedSpace_of_t0_of_basis_clopen loc_compact_Haus_tot_disc_of_zero_dim
apply TotallySeparatedSpace.totallyDisconnectedSpace | theorem | loc_compact_t2_tot_disc_iff_tot_sep | Topology.Separation | Mathlib/Topology/Separation/Profinite.lean | [] | [
"TotallyDisconnectedSpace",
"TotallySeparatedSpace",
"loc_compact_Haus_tot_disc_of_zero_dim",
"totallySeparatedSpace_of_t0_of_basis_clopen"
] | A locally compact Hausdorff space is totally disconnected
if and only if it is totally separated. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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