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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Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete {ι : Type*} {p : ι → Prop}
{t : ι → Set X} {s : Set X} (hs : IsDiscrete s) {x : X} (hb : (𝓝 x).HasBasis p t)
(hx : x ∈ s) : ∃ i, p i ∧ t i ∩ s = {x} | by
rcases (nhdsWithin_hasBasis hb s).mem_iff.1 (singleton_mem_nhdsWithin_of_mem_discrete hs hx) with
⟨i, hi, hix⟩
exact ⟨i, hi, hix.antisymm <| singleton_subset_iff.2 ⟨mem_of_mem_nhds <| hb.mem_of_mem hi, hx⟩⟩ | theorem | Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsDiscrete",
"Set",
"nhdsWithin_hasBasis",
"singleton_mem_nhdsWithin_of_mem_discrete"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_inter_eq_singleton_of_mem_discrete {s : Set X} (hs : IsDiscrete s) {x : X}
(hx : x ∈ s) : ∃ U ∈ 𝓝 x, U ∩ s = {x} | by
simpa using (𝓝 x).basis_sets.exists_inter_eq_singleton_of_mem_discrete hs hx | theorem | nhds_inter_eq_singleton_of_mem_discrete | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsDiscrete",
"Set"
] | A point `x` in a discrete subset `s` of a topological space admits a neighbourhood
that only meets `s` at `x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpen_inter_eq_singleton_of_mem_discrete {s : Set X} (hs : IsDiscrete s) {x : X}
(hx : x ∈ s) : ∃ U : Set X, IsOpen U ∧ U ∩ s = {x} | by
obtain ⟨U, hU_nhds, hU_inter⟩ := nhds_inter_eq_singleton_of_mem_discrete hs hx
obtain ⟨t, ht_sub, ht_open, ht_x⟩ := mem_nhds_iff.mp hU_nhds
grind | theorem | isOpen_inter_eq_singleton_of_mem_discrete | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsDiscrete",
"IsOpen",
"Set",
"nhds_inter_eq_singleton_of_mem_discrete"
] | Let `x` be a point in a discrete subset `s` of a topological space, then there exists an open
set that only meets `s` at `x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
disjoint_nhdsWithin_of_mem_discrete {s : Set X} (hs : IsDiscrete s) {x : X} (hx : x ∈ s) :
∃ U ∈ 𝓝[≠] x, Disjoint U s | let ⟨V, h, h'⟩ := nhds_inter_eq_singleton_of_mem_discrete hs hx
⟨{x}ᶜ ∩ V, inter_mem_nhdsWithin _ h,
disjoint_iff_inter_eq_empty.mpr (by rw [inter_assoc, h', compl_inter_self])⟩ | theorem | disjoint_nhdsWithin_of_mem_discrete | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Disjoint",
"IsDiscrete",
"Set",
"inter_mem_nhdsWithin",
"nhds_inter_eq_singleton_of_mem_discrete"
] | For point `x` in a discrete subset `s` of a topological space, there is a set `U`
such that
1. `U` is a punctured neighborhood of `x` (i.e. `U ∪ {x}` is a neighbourhood of `x`),
2. `U` is disjoint from `s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isClosedEmbedding_update {ι : Type*} {β : ι → Type*}
[DecidableEq ι] [(i : ι) → TopologicalSpace (β i)]
(x : (i : ι) → β i) (i : ι) [(i : ι) → T1Space (β i)] :
IsClosedEmbedding (update x i) | by
refine .of_continuous_injective_isClosedMap (continuous_const.update i continuous_id)
(update_injective x i) fun s hs ↦ ?_
rw [update_image]
apply isClosed_set_pi
simp [forall_update_iff, hs] | theorem | isClosedEmbedding_update | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T1Space",
"TopologicalSpace",
"continuous_id",
"isClosed_set_pi"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsNE_le_cofinite {α : Type*} [TopologicalSpace α] [T1Space α] (a : α) :
𝓝[≠] a ≤ cofinite | by
refine le_cofinite_iff_compl_singleton_mem.mpr fun x ↦ ?_
rcases eq_or_ne a x with rfl | hx
exacts [self_mem_nhdsWithin, eventually_ne_nhdsWithin hx] | lemma | nhdsNE_le_cofinite | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T1Space",
"TopologicalSpace",
"eq_or_ne",
"eventually_ne_nhdsWithin",
"self_mem_nhdsWithin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Function.update_eventuallyEq_nhdsNE
{α β : Type*} [TopologicalSpace α] [T1Space α] [DecidableEq α] (f : α → β) (a a' : α) (b : β) :
Function.update f a b =ᶠ[𝓝[≠] a'] f | (Function.update_eventuallyEq_cofinite f a b).filter_mono (nhdsNE_le_cofinite a') | lemma | Function.update_eventuallyEq_nhdsNE | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Function.update",
"Function.update_eventuallyEq_cofinite",
"T1Space",
"TopologicalSpace",
"nhdsNE_le_cofinite"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
R1Space (X : Type*) [TopologicalSpace X] : Prop where
specializes_or_disjoint_nhds (x y : X) : Specializes x y ∨ Disjoint (𝓝 x) (𝓝 y) | class | R1Space | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Disjoint",
"Specializes",
"TopologicalSpace"
] | A topological space is called a *preregular* (a.k.a. R₁) space,
if any two topologically distinguishable points have disjoint neighbourhoods. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
disjoint_nhds_nhds_iff_not_specializes : Disjoint (𝓝 x) (𝓝 y) ↔ ¬x ⤳ y | ⟨fun hd hspec ↦ hspec.not_disjoint hd, (specializes_or_disjoint_nhds _ _).resolve_left⟩ | theorem | disjoint_nhds_nhds_iff_not_specializes | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Disjoint"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
specializes_iff_not_disjoint : x ⤳ y ↔ ¬Disjoint (𝓝 x) (𝓝 y) | disjoint_nhds_nhds_iff_not_specializes.not_left.symm | theorem | specializes_iff_not_disjoint | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Disjoint"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
disjoint_nhds_nhds_iff_not_inseparable : Disjoint (𝓝 x) (𝓝 y) ↔ ¬Inseparable x y | by
rw [disjoint_nhds_nhds_iff_not_specializes, specializes_iff_inseparable] | theorem | disjoint_nhds_nhds_iff_not_inseparable | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Disjoint",
"Inseparable",
"disjoint_nhds_nhds_iff_not_specializes",
"specializes_iff_inseparable"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
r1Space_iff_inseparable_or_disjoint_nhds {X : Type*} [TopologicalSpace X] :
R1Space X ↔ ∀ x y : X, Inseparable x y ∨ Disjoint (𝓝 x) (𝓝 y) | ⟨fun _h x y ↦ (specializes_or_disjoint_nhds x y).imp_left Specializes.inseparable, fun h ↦
⟨fun x y ↦ (h x y).imp_left Inseparable.specializes⟩⟩ | theorem | r1Space_iff_inseparable_or_disjoint_nhds | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Disjoint",
"Inseparable",
"R1Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Inseparable.of_nhds_neBot {x y : X} (h : NeBot (𝓝 x ⊓ 𝓝 y)) :
Inseparable x y | (r1Space_iff_inseparable_or_disjoint_nhds.mp ‹_› _ _).resolve_right fun h' => h.ne h'.eq_bot | theorem | Inseparable.of_nhds_neBot | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Inseparable"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
r1_separation {x y : X} (h : ¬Inseparable x y) :
∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v | by
rw [← disjoint_nhds_nhds_iff_not_inseparable,
(nhds_basis_opens x).disjoint_iff (nhds_basis_opens y)] at h
obtain ⟨u, ⟨hxu, hu⟩, v, ⟨hyv, hv⟩, huv⟩ := h
exact ⟨u, v, hu, hv, hxu, hyv, huv⟩ | theorem | r1_separation | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Disjoint",
"Inseparable",
"IsOpen",
"Set",
"disjoint_iff",
"disjoint_nhds_nhds_iff_not_inseparable",
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_nhds_unique_inseparable {f : Y → X} {l : Filter Y} {a b : X} [NeBot l]
(ha : Tendsto f l (𝓝 a)) (hb : Tendsto f l (𝓝 b)) : Inseparable a b | .of_nhds_neBot <| neBot_of_le <| le_inf ha hb | theorem | tendsto_nhds_unique_inseparable | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Filter",
"Inseparable"
] | Limits are unique up to separability.
A weaker version of `tendsto_nhds_unique` for `R1Space`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isClosed_setOf_specializes : IsClosed { p : X × X | p.1 ⤳ p.2 } | by
simp only [← isOpen_compl_iff, compl_setOf, ← disjoint_nhds_nhds_iff_not_specializes,
isOpen_setOf_disjoint_nhds_nhds] | theorem | isClosed_setOf_specializes | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsClosed",
"disjoint_nhds_nhds_iff_not_specializes",
"isOpen_compl_iff",
"isOpen_setOf_disjoint_nhds_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_setOf_inseparable : IsClosed { p : X × X | Inseparable p.1 p.2 } | by
simp only [← specializes_iff_inseparable, isClosed_setOf_specializes] | theorem | isClosed_setOf_inseparable | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Inseparable",
"IsClosed",
"isClosed_setOf_specializes",
"specializes_iff_inseparable"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.mem_closure_iff_exists_inseparable {K : Set X} (hK : IsCompact K) :
y ∈ closure K ↔ ∃ x ∈ K, Inseparable x y | by
refine ⟨fun hy ↦ ?_, fun ⟨x, hxK, hxy⟩ ↦
(hxy.mem_closed_iff isClosed_closure).1 <| subset_closure hxK⟩
contrapose! hy
have : Disjoint (𝓝 y) (𝓝ˢ K) := hK.disjoint_nhdsSet_right.2 fun x hx ↦
(disjoint_nhds_nhds_iff_not_inseparable.2 (hy x hx)).symm
simpa only [disjoint_iff, notMem_closure_iff_nhdsWi... | theorem | IsCompact.mem_closure_iff_exists_inseparable | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Disjoint",
"Inseparable",
"IsCompact",
"Set",
"closure",
"disjoint_iff",
"isClosed_closure",
"notMem_closure_iff_nhdsWithin_eq_bot",
"principal_le_nhdsSet",
"subset_closure",
"symm"
] | In an R₁ space, a point belongs to the closure of a compact set `K`
if and only if it is topologically inseparable from some point of `K`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.closure_eq_biUnion_inseparable {K : Set X} (hK : IsCompact K) :
closure K = ⋃ x ∈ K, {y | Inseparable x y} | by
ext; simp [hK.mem_closure_iff_exists_inseparable] | theorem | IsCompact.closure_eq_biUnion_inseparable | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Inseparable",
"IsCompact",
"Set",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.closure_eq_biUnion_closure_singleton {K : Set X} (hK : IsCompact K) :
closure K = ⋃ x ∈ K, closure {x} | by
simp only [hK.closure_eq_biUnion_inseparable, ← specializes_iff_inseparable,
specializes_iff_mem_closure, setOf_mem_eq] | theorem | IsCompact.closure_eq_biUnion_closure_singleton | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsCompact",
"Set",
"closure",
"specializes_iff_inseparable",
"specializes_iff_mem_closure"
] | In an R₁ space, the closure of a compact set is the union of the closures of its points. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.closure_subset_of_isOpen {K : Set X} (hK : IsCompact K)
{U : Set X} (hU : IsOpen U) (hKU : K ⊆ U) : closure K ⊆ U | by
rw [hK.closure_eq_biUnion_inseparable, iUnion₂_subset_iff]
exact fun x hx y hxy ↦ (hxy.mem_open_iff hU).1 (hKU hx) | theorem | IsCompact.closure_subset_of_isOpen | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsCompact",
"IsOpen",
"Set",
"closure"
] | In an R₁ space, if a compact set `K` is contained in an open set `U`,
then its closure is also contained in `U`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.closure {K : Set X} (hK : IsCompact K) : IsCompact (closure K) | by
refine isCompact_of_finite_subcover fun U hUo hKU ↦ ?_
rcases hK.elim_finite_subcover U hUo (subset_closure.trans hKU) with ⟨t, ht⟩
exact ⟨t, hK.closure_subset_of_isOpen (isOpen_biUnion fun _ _ ↦ hUo _) ht⟩ | theorem | IsCompact.closure | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsCompact",
"Set",
"closure",
"isCompact_of_finite_subcover",
"isOpen_biUnion"
] | The closure of a compact set in an R₁ space is a compact set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.closure_of_subset {s K : Set X} (hK : IsCompact K) (h : s ⊆ K) :
IsCompact (closure s) | hK.closure.of_isClosed_subset isClosed_closure (closure_mono h) | theorem | IsCompact.closure_of_subset | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsCompact",
"Set",
"closure",
"closure_mono",
"isClosed_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_isCompact_superset_iff {s : Set X} :
(∃ K, IsCompact K ∧ s ⊆ K) ↔ IsCompact (closure s) | ⟨fun ⟨_K, hK, hsK⟩ => hK.closure_of_subset hsK, fun h => ⟨closure s, h, subset_closure⟩⟩ | theorem | exists_isCompact_superset_iff | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsCompact",
"Set",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SeparatedNhds.of_isCompact_isCompact_isClosed {K L : Set X} (hK : IsCompact K)
(hL : IsCompact L) (h'L : IsClosed L) (hd : Disjoint K L) : SeparatedNhds K L | by
simp_rw [separatedNhds_iff_disjoint, hK.disjoint_nhdsSet_left, hL.disjoint_nhdsSet_right,
disjoint_nhds_nhds_iff_not_inseparable]
intro x hx y hy h
exact absurd ((h.mem_closed_iff h'L).2 hy) <| disjoint_left.1 hd hx | theorem | SeparatedNhds.of_isCompact_isCompact_isClosed | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Disjoint",
"IsClosed",
"IsCompact",
"SeparatedNhds",
"Set",
"disjoint_nhds_nhds_iff_not_inseparable",
"separatedNhds_iff_disjoint"
] | If `K` and `L` are disjoint compact sets in an R₁ topological space
and `L` is also closed, then `K` and `L` have disjoint neighborhoods. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.binary_compact_cover {K U V : Set X}
(hK : IsCompact K) (hU : IsOpen U) (hV : IsOpen V) (h2K : K ⊆ U ∪ V) :
∃ K₁ K₂ : Set X, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂ | by
have hK' : IsCompact (closure K) := hK.closure
have : SeparatedNhds (closure K \ U) (closure K \ V) := by
apply SeparatedNhds.of_isCompact_isCompact_isClosed (hK'.diff hU) (hK'.diff hV)
(isClosed_closure.sdiff hV)
rw [disjoint_iff_inter_eq_empty, diff_inter_diff, diff_eq_empty]
exact hK.closure... | theorem | IsCompact.binary_compact_cover | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsCompact",
"IsOpen",
"SeparatedNhds",
"SeparatedNhds.of_isCompact_isCompact_isClosed",
"Set",
"closure",
"subset_closure"
] | If a compact set is covered by two open sets, then we can cover it by two compact subsets. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.finite_compact_cover {s : Set X} (hs : IsCompact s) {ι : Type*}
(t : Finset ι) (U : ι → Set X) (hU : ∀ i ∈ t, IsOpen (U i)) (hsC : s ⊆ ⋃ i ∈ t, U i) :
∃ K : ι → Set X, (∀ i, IsCompact (K i)) ∧ (∀ i, K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i | by
classical
induction t using Finset.induction generalizing U s with
| empty =>
refine ⟨fun _ => ∅, fun _ => isCompact_empty, fun i => empty_subset _, ?_⟩
simpa only [subset_empty_iff, Finset.notMem_empty, iUnion_false, iUnion_empty] using hsC
| insert x t hx ih =>
simp only [Finset.set_biUnion_ins... | theorem | IsCompact.finite_compact_cover | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Finset",
"Finset.forall_mem_insert",
"Finset.induction",
"Finset.notMem_empty",
"Finset.set_biUnion_insert",
"Finset.set_biUnion_insert_update",
"IsCompact",
"IsOpen",
"Set",
"eq_or_ne",
"isCompact_empty",
"isOpen_biUnion"
] | For every finite open cover `Uᵢ` of a compact set, there exists a compact cover `Kᵢ ⊆ Uᵢ`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
R1Space.of_continuous_specializes_imp [TopologicalSpace Y] {f : Y → X} (hc : Continuous f)
(hspec : ∀ x y, f x ⤳ f y → x ⤳ y) : R1Space Y | where
specializes_or_disjoint_nhds x y := (specializes_or_disjoint_nhds (f x) (f y)).imp (hspec x y) <|
((hc.tendsto _).disjoint · (hc.tendsto _)) | theorem | R1Space.of_continuous_specializes_imp | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Continuous",
"R1Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Topology.IsInducing.r1Space [TopologicalSpace Y] {f : Y → X} (hf : IsInducing f) :
R1Space Y | .of_continuous_specializes_imp hf.continuous fun _ _ ↦ hf.specializes_iff.1 | theorem | Topology.IsInducing.r1Space | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"R1Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
R1Space.induced (f : Y → X) : @R1Space Y (.induced f ‹_›) | @IsInducing.r1Space _ _ _ _ (.induced f _) f (.induced f) | theorem | R1Space.induced | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"R1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
R1Space.sInf {X : Type*} {T : Set (TopologicalSpace X)}
(hT : ∀ t ∈ T, @R1Space X t) : @R1Space X (sInf T) | by
let _ := sInf T
refine ⟨fun x y ↦ ?_⟩
simp only [Specializes, nhds_sInf]
by_cases! hTd : ∃ t ∈ T, Disjoint (@nhds X t x) (@nhds X t y)
· rcases hTd with ⟨t, htT, htd⟩
exact .inr <| htd.mono (iInf₂_le t htT) (iInf₂_le t htT)
· exact .inl <| iInf₂_mono fun t ht ↦ ((hT t ht).1 x y).resolve_right (hTd t ... | theorem | R1Space.sInf | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Disjoint",
"R1Space",
"Set",
"Specializes",
"TopologicalSpace",
"nhds_sInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
R1Space.iInf {ι X : Type*} {t : ι → TopologicalSpace X}
(ht : ∀ i, @R1Space X (t i)) : @R1Space X (iInf t) | .sInf <| forall_mem_range.2 ht | theorem | R1Space.iInf | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"R1Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
R1Space.inf {X : Type*} {t₁ t₂ : TopologicalSpace X}
(h₁ : @R1Space X t₁) (h₂ : @R1Space X t₂) : @R1Space X (t₁ ⊓ t₂) | by
rw [inf_eq_iInf]
apply R1Space.iInf
simp [*] | theorem | R1Space.inf | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"R1Space",
"R1Space.iInf",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_mem_nhds_isCompact_mapsTo_of_isCompact_mem_nhds
{X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [R1Space Y] {f : X → Y} {x : X}
{K : Set X} {s : Set Y} (hf : Continuous f) (hs : s ∈ 𝓝 (f x)) (hKc : IsCompact K)
(hKx : K ∈ 𝓝 x) : ∃ L ∈ 𝓝 x, IsCompact L ∧ MapsTo f L s | by
have hc : IsCompact (f '' K \ interior s) := (hKc.image hf).diff isOpen_interior
obtain ⟨U, V, Uo, Vo, hxU, hV, hd⟩ : SeparatedNhds {f x} (f '' K \ interior s) := by
simp_rw [separatedNhds_iff_disjoint, nhdsSet_singleton, hc.disjoint_nhdsSet_right,
disjoint_nhds_nhds_iff_not_inseparable]
rintro y ⟨... | theorem | exists_mem_nhds_isCompact_mapsTo_of_isCompact_mem_nhds | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Continuous",
"IsCompact",
"R1Space",
"SeparatedNhds",
"Set",
"TopologicalSpace",
"disjoint_nhds_nhds_iff_not_inseparable",
"interior",
"interior_subset",
"isOpen_interior",
"mem_interior_iff_mem_nhds",
"nhdsSet_singleton",
"separatedNhds_iff_disjoint"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.isCompact_isClosed_basis_nhds {x : X} {L : Set X} (hLc : IsCompact L)
(hxL : L ∈ 𝓝 x) : (𝓝 x).HasBasis (fun K ↦ K ∈ 𝓝 x ∧ IsCompact K ∧ IsClosed K) (·) | hasBasis_self.2 fun _U hU ↦
let ⟨K, hKx, hKc, hKU⟩ := exists_mem_nhds_isCompact_mapsTo_of_isCompact_mem_nhds
continuous_id (interior_mem_nhds.2 hU) hLc hxL
⟨closure K, mem_of_superset hKx subset_closure, ⟨hKc.closure, isClosed_closure⟩,
(hKc.closure_subset_of_isOpen isOpen_interior hKU).trans interi... | theorem | IsCompact.isCompact_isClosed_basis_nhds | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsClosed",
"IsCompact",
"Set",
"continuous_id",
"exists_mem_nhds_isCompact_mapsTo_of_isCompact_mem_nhds",
"isOpen_interior",
"subset_closure",
"trans"
] | If a point in an R₁ space has a compact neighborhood,
then it has a basis of compact closed neighborhoods. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.coclosedCompact_eq_cocompact : coclosedCompact X = cocompact X | by
refine le_antisymm ?_ cocompact_le_coclosedCompact
rw [hasBasis_coclosedCompact.le_basis_iff hasBasis_cocompact]
exact fun K hK ↦ ⟨closure K, ⟨isClosed_closure, hK.closure⟩, compl_subset_compl.2 subset_closure⟩ | theorem | Filter.coclosedCompact_eq_cocompact | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"le_antisymm"
] | In an R₁ space, the filters `coclosedCompact` and `cocompact` are equal. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Bornology.relativelyCompact_eq_inCompact :
Bornology.relativelyCompact X = Bornology.inCompact X | Bornology.ext _ _ Filter.coclosedCompact_eq_cocompact | theorem | Bornology.relativelyCompact_eq_inCompact | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Bornology.ext",
"Bornology.inCompact",
"Bornology.relativelyCompact",
"Filter.coclosedCompact_eq_cocompact"
] | In an R₁ space, the bornologies `relativelyCompact` and `inCompact` are equal. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isCompact_isClosed_basis_nhds (x : X) :
(𝓝 x).HasBasis (fun K => K ∈ 𝓝 x ∧ IsCompact K ∧ IsClosed K) (·) | let ⟨_L, hLc, hLx⟩ := exists_compact_mem_nhds x
hLc.isCompact_isClosed_basis_nhds hLx | theorem | isCompact_isClosed_basis_nhds | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsClosed",
"IsCompact"
] | In a (weakly) locally compact R₁ space, compact closed neighborhoods of a point `x`
form a basis of neighborhoods of `x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_mem_nhds_isCompact_isClosed (x : X) : ∃ K ∈ 𝓝 x, IsCompact K ∧ IsClosed K | (isCompact_isClosed_basis_nhds x).ex_mem | theorem | exists_mem_nhds_isCompact_isClosed | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsClosed",
"IsCompact",
"isCompact_isClosed_basis_nhds"
] | In a (weakly) locally compact R₁ space, each point admits a compact closed neighborhood. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_isOpen_superset_and_isCompact_closure {K : Set X} (hK : IsCompact K) :
∃ V, IsOpen V ∧ K ⊆ V ∧ IsCompact (closure V) | by
rcases exists_compact_superset hK with ⟨K', hK', hKK'⟩
exact ⟨interior K', isOpen_interior, hKK', hK'.closure_of_subset interior_subset⟩ | theorem | exists_isOpen_superset_and_isCompact_closure | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsCompact",
"IsOpen",
"Set",
"closure",
"exists_compact_superset",
"isOpen_interior"
] | In a weakly locally compact R₁ space,
every compact set has an open neighborhood with compact closure. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_isOpen_mem_isCompact_closure (x : X) :
∃ U : Set X, IsOpen U ∧ x ∈ U ∧ IsCompact (closure U) | by
simpa only [singleton_subset_iff]
using exists_isOpen_superset_and_isCompact_closure isCompact_singleton | theorem | exists_isOpen_mem_isCompact_closure | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsCompact",
"IsOpen",
"Set",
"closure",
"exists_isOpen_superset_and_isCompact_closure",
"isCompact_singleton"
] | In a weakly locally compact R₁ space,
every point has an open neighborhood with compact closure. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
CompletelyRegularSpace (X : Type u) [TopologicalSpace X] : Prop where
completely_regular : ∀ (x : X), ∀ K : Set X, IsClosed K → x ∉ K →
∃ f : X → I, Continuous f ∧ f x = 0 ∧ EqOn f 1 K | class | CompletelyRegularSpace | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"Continuous",
"IsClosed",
"Set",
"TopologicalSpace"
] | A space is completely regular if points can be separated from closed sets via
continuous functions to the unit interval. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
completelyRegularSpace_iff_isOpen : CompletelyRegularSpace X ↔
∀ (x : X), ∀ K : Set X, IsOpen K → x ∈ K →
∃ f : X → I, Continuous f ∧ f x = 0 ∧ EqOn f 1 Kᶜ | by
conv_lhs => tactic =>
simp_rw +singlePass [completelyRegularSpace_iff, compl_surjective.forall, isClosed_compl_iff,
mem_compl_iff, not_not] | lemma | completelyRegularSpace_iff_isOpen | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"CompletelyRegularSpace",
"Continuous",
"IsOpen",
"Set",
"isClosed_compl_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CompletelyRegularSpace.completely_regular_isOpen [CompletelyRegularSpace X] :
∀ (x : X), ∀ K : Set X, IsOpen K → x ∈ K →
∃ f : X → I, Continuous f ∧ f x = 0 ∧ EqOn f 1 Kᶜ | completelyRegularSpace_iff_isOpen.mp inferInstance | lemma | CompletelyRegularSpace.completely_regular_isOpen | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"CompletelyRegularSpace",
"Continuous",
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CompletelyRegularSpace.instRegularSpace [CompletelyRegularSpace X] :
RegularSpace X | by
rw [regularSpace_iff]
intro s a hs ha
obtain ⟨f, cf, hf, hhf⟩ := CompletelyRegularSpace.completely_regular a s hs ha
apply disjoint_of_map (f := f)
apply Disjoint.mono (cf.tendsto_nhdsSet_nhds hhf) cf.continuousAt
exact disjoint_nhds_nhds.mpr (hf.symm ▸ zero_ne_one).symm | instance | CompletelyRegularSpace.instRegularSpace | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"CompletelyRegularSpace",
"Disjoint.mono",
"RegularSpace",
"symm",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
NormalSpace.instCompletelyRegularSpace [NormalSpace X] [R0Space X] :
CompletelyRegularSpace X | by
rw [completelyRegularSpace_iff]
intro x K hK hx
have cx : IsClosed (closure {x}) := isClosed_closure
have d : Disjoint (closure {x}) K := by
rw [Set.disjoint_iff]
intro a ⟨hax, haK⟩
exact hx ((specializes_iff_mem_closure.mpr hax).symm.mem_closed hK haK)
let ⟨⟨f, cf⟩, hfx, hfK, hficc⟩ := exists_... | instance | NormalSpace.instCompletelyRegularSpace | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"CompletelyRegularSpace",
"Continuous",
"Disjoint",
"IsClosed",
"NormalSpace",
"R0Space",
"Set.disjoint_iff",
"closure",
"exists_continuous_zero_one_of_isClosed",
"isClosed_closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Topology.IsInducing.completelyRegularSpace
{Y : Type v} [TopologicalSpace Y] [CompletelyRegularSpace Y]
{f : X → Y} (hf : IsInducing f) : CompletelyRegularSpace X | where
completely_regular x K hK hxK := by
rw [hf.isClosed_iff] at hK
obtain ⟨K, hK, rfl⟩ := hK
rw [mem_preimage] at hxK
obtain ⟨g, hcf, egfx, hgK⟩ := CompletelyRegularSpace.completely_regular _ _ hK hxK
refine ⟨g ∘ f, hcf.comp hf.continuous, egfx, ?_⟩
conv => arg 2; equals (1 : Y → ↥I) ∘ f => ... | lemma | Topology.IsInducing.completelyRegularSpace | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"CompletelyRegularSpace",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
completelyRegularSpace_induced
{X Y : Type*} {t : TopologicalSpace Y} (ht : @CompletelyRegularSpace Y t)
(f : X → Y) : @CompletelyRegularSpace X (t.induced f) | @IsInducing.completelyRegularSpace _ (t.induced f) _ t _ _ (IsInducing.induced f) | lemma | completelyRegularSpace_induced | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"CompletelyRegularSpace",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
completelyRegularSpace_iInf {ι X : Type*} {t : ι → TopologicalSpace X}
(ht : ∀ i, @CompletelyRegularSpace X (t i)) : @CompletelyRegularSpace X (⨅ i, t i) | by
letI := (⨅ i, t i) -- register this as default topological space to reduce `@`s
rw [completelyRegularSpace_iff_isOpen]
intro x K hK hxK
simp_rw [← hK.mem_nhds_iff, nhds_iInf, mem_iInf, exists_finite_iff_finset,
Finset.coe_sort_coe] at hxK; clear hK
obtain ⟨I', V, hV, rfl⟩ := hxK
simp only [mem_nhds_i... | lemma | completelyRegularSpace_iInf | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"CompletelyRegularSpace",
"CompletelyRegularSpace.completely_regular_isOpen",
"Continuous.finset_sup",
"Finset.coe_sort_coe",
"Finset.sup_eq_top_iff",
"Pi.one_apply",
"TopologicalSpace",
"completelyRegularSpace_iff_isOpen",
"continuous_iInf_dom",
"mem_nhds_iff",
"nhds_iInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
completelyRegularSpace_inf {X : Type*} {t₁ t₂ : TopologicalSpace X}
(ht₁ : @CompletelyRegularSpace X t₁) (ht₂ : @CompletelyRegularSpace X t₂) :
@CompletelyRegularSpace X (t₁ ⊓ t₂) | by
rw [inf_eq_iInf]; apply completelyRegularSpace_iInf; simp [*] | lemma | completelyRegularSpace_inf | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"CompletelyRegularSpace",
"TopologicalSpace",
"completelyRegularSpace_iInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isInducing_stoneCechUnit [CompletelyRegularSpace X] :
IsInducing (stoneCechUnit : X → StoneCech X) | by
rw [isInducing_iff_nhds]
intro x
apply le_antisymm
· rw [← map_le_iff_le_comap]; exact continuous_stoneCechUnit.continuousAt
· simp_rw [le_nhds_iff, ((nhds_basis_opens _).comap _).mem_iff, and_assoc]
intro U hxU hU
obtain ⟨f, hf, efx, hfU⟩ :=
CompletelyRegularSpace.completely_regular_isOpen x... | lemma | isInducing_stoneCechUnit | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"CompletelyRegularSpace",
"CompletelyRegularSpace.completely_regular_isOpen",
"StoneCech",
"comap",
"continuous_stoneCechExtend",
"le_antisymm",
"le_nhds_iff",
"nhds_basis_opens",
"stoneCechExtend_extends",
"stoneCechExtend_stoneCechUnit",
"stoneCechUnit"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isDenseInducing_stoneCechUnit [CompletelyRegularSpace X] :
IsDenseInducing (stoneCechUnit : X → StoneCech X) | where
toIsInducing := isInducing_stoneCechUnit
dense := denseRange_stoneCechUnit | lemma | isDenseInducing_stoneCechUnit | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"CompletelyRegularSpace",
"IsDenseInducing",
"StoneCech",
"denseRange_stoneCechUnit",
"isInducing_stoneCechUnit",
"stoneCechUnit"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
completelyRegularSpace_iff_isInducing_stoneCechUnit :
CompletelyRegularSpace X ↔ IsInducing (stoneCechUnit : X → StoneCech X) | where
mp _ := isInducing_stoneCechUnit
mpr hs := hs.completelyRegularSpace | lemma | completelyRegularSpace_iff_isInducing_stoneCechUnit | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"CompletelyRegularSpace",
"StoneCech",
"isInducing_stoneCechUnit",
"stoneCechUnit"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CompletelyRegularSpace.of_isTopologicalBasis_clopens
(h : TopologicalSpace.IsTopologicalBasis {s : Set X | IsClopen s}) :
CompletelyRegularSpace X | where
completely_regular x K hK hx := by
obtain ⟨s, hs, hx, hsK⟩ := h.exists_subset_of_mem_open hx hK.isOpen_compl
refine ⟨sᶜ.indicator 1, ?_, by simpa, fun x hx ↦ indicator_of_mem ?_ _⟩
· exact hs.compl.continuous_indicator continuous_const
· exact (mem_compl_iff s x).mpr fun hs ↦ hsK hs hx | theorem | CompletelyRegularSpace.of_isTopologicalBasis_clopens | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"CompletelyRegularSpace",
"IsClopen",
"Set",
"TopologicalSpace.IsTopologicalBasis",
"continuous_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CompletelyRegularSpace.isTopologicalBasis_clopens_of_cardinalMk_lt_continuum
[CompletelyRegularSpace X] (hX : Cardinal.mk X < continuum) :
IsTopologicalBasis {s : Set X | IsClopen s} | by
refine isTopologicalBasis_of_isOpen_of_nhds (fun x s ↦ IsClopen.isOpen s) (fun x s hxs hs ↦ ?_)
choose f hf using completely_regular_isOpen x s hs hxs
obtain ⟨hfc, hf₀, hf₁⟩ := hf
let R := Set.range f
have hR : lift.{u, 0} (Cardinal.mk R) < lift.{0, u} continuum := by
simpa [R] using mk_range_le_lift.t... | theorem | CompletelyRegularSpace.isTopologicalBasis_clopens_of_cardinalMk_lt_continuum | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"Cardinal.mk",
"CompletelyRegularSpace",
"IsClopen",
"IsClopen.isOpen",
"Set",
"Set.range",
"le_iff_lt_or_eq",
"unitInterval",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
T35Space (X : Type u) [TopologicalSpace X] : Prop extends T0Space X, CompletelyRegularSpace X | class | T35Space | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"CompletelyRegularSpace",
"T0Space",
"TopologicalSpace"
] | A T₃.₅ space is a completely regular space that is also T₀. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
T35Space.instT3space [T35Space X] : T3Space X | where | instance | T35Space.instT3space | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"T35Space",
"T3Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
T4Space.instT35Space [T4Space X] : T35Space X | where | instance | T4Space.instT35Space | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"T35Space",
"T4Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Topology.IsEmbedding.t35Space
{Y : Type v} [TopologicalSpace Y] [T35Space Y]
{f : X → Y} (hf : IsEmbedding f) : T35Space X | @T35Space.mk _ _ hf.t0Space hf.isInducing.completelyRegularSpace | lemma | Topology.IsEmbedding.t35Space | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"T35Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
separatesPoints_continuous_of_t35Space [T35Space X] :
SeparatesPoints {f : X → ℝ | Continuous f} | by
intro x y x_ne_y
obtain ⟨f, f_cont, f_zero, f_one⟩ :=
CompletelyRegularSpace.completely_regular x {y} isClosed_singleton x_ne_y
exact ⟨fun x ↦ f x, continuous_subtype_val.comp f_cont, by simp_all⟩ | lemma | separatesPoints_continuous_of_t35Space | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"Continuous",
"T35Space",
"isClosed_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
separatesPoints_continuous_of_t35Space_Icc [T35Space X] :
SeparatesPoints {f : X → I | Continuous f} | by
intro x y x_ne_y
obtain ⟨f, f_cont, f_zero, f_one⟩ :=
CompletelyRegularSpace.completely_regular x {y} isClosed_singleton x_ne_y
exact ⟨f, f_cont, by simp_all⟩ | lemma | separatesPoints_continuous_of_t35Space_Icc | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"Continuous",
"T35Space",
"isClosed_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
injective_stoneCechUnit_of_t35Space [T35Space X] :
Function.Injective (stoneCechUnit : X → StoneCech X) | by
intro a b hab
contrapose hab
obtain ⟨f, fc, fab⟩ := separatesPoints_continuous_of_t35Space_Icc hab
exact fun q ↦ fab (eq_if_stoneCechUnit_eq fc q) | lemma | injective_stoneCechUnit_of_t35Space | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"StoneCech",
"T35Space",
"eq_if_stoneCechUnit_eq",
"separatesPoints_continuous_of_t35Space_Icc",
"stoneCechUnit"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isEmbedding_stoneCechUnit [T35Space X] :
IsEmbedding (stoneCechUnit : X → StoneCech X) | where
toIsInducing := isInducing_stoneCechUnit
injective := injective_stoneCechUnit_of_t35Space | lemma | isEmbedding_stoneCechUnit | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"StoneCech",
"T35Space",
"injective_stoneCechUnit_of_t35Space",
"isInducing_stoneCechUnit",
"stoneCechUnit"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isDenseEmbedding_stoneCechUnit [T35Space X] :
IsDenseEmbedding (stoneCechUnit : X → StoneCech X) | where
toIsDenseInducing := isDenseInducing_stoneCechUnit
injective := injective_stoneCechUnit_of_t35Space | lemma | isDenseEmbedding_stoneCechUnit | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"IsDenseEmbedding",
"StoneCech",
"T35Space",
"injective_stoneCechUnit_of_t35Space",
"isDenseInducing_stoneCechUnit",
"stoneCechUnit"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t35Space_iff_isEmbedding_stoneCechUnit :
T35Space X ↔ IsEmbedding (stoneCechUnit : X → StoneCech X) | where
mp _ := isEmbedding_stoneCechUnit
mpr hs := hs.t35Space | lemma | t35Space_iff_isEmbedding_stoneCechUnit | Topology.Separation | Mathlib/Topology/Separation/CompletelyRegular.lean | [
"Mathlib.Topology.Algebra.Indicator"
] | [
"StoneCech",
"T35Space",
"isEmbedding_stoneCechUnit",
"stoneCechUnit"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PreconnectedSpace.trivial_of_discrete [PreconnectedSpace X] [DiscreteTopology X] :
Subsingleton X | by
by_contra! ⟨x, y, hxy⟩
rw [Ne, ← mem_singleton_iff, (isClopen_discrete _).eq_univ <| singleton_nonempty y] at hxy
exact hxy (mem_univ x) | theorem | PreconnectedSpace.trivial_of_discrete | Topology.Separation | Mathlib/Topology/Separation/Connected.lean | [] | [
"DiscreteTopology",
"PreconnectedSpace",
"isClopen_discrete"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsPreconnected.infinite_of_nontrivial [T1Space X] {s : Set X} (h : IsPreconnected s)
(hs : s.Nontrivial) : s.Infinite | by
refine mt (fun hf => (subsingleton_coe s).mp ?_) (not_subsingleton_iff.mpr hs)
haveI := @Finite.instDiscreteTopology s _ _ hf.to_subtype
exact @PreconnectedSpace.trivial_of_discrete _ _ (Subtype.preconnectedSpace h) _ | theorem | IsPreconnected.infinite_of_nontrivial | Topology.Separation | Mathlib/Topology/Separation/Connected.lean | [] | [
"Finite.instDiscreteTopology",
"IsPreconnected",
"PreconnectedSpace.trivial_of_discrete",
"Set",
"Subtype.preconnectedSpace",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PreconnectedSpace.infinite [PreconnectedSpace X] [Nontrivial X] [T1Space X] : Infinite X | infinite_univ_iff.mp <| isPreconnected_univ.infinite_of_nontrivial nontrivial_univ | theorem | PreconnectedSpace.infinite | Topology.Separation | Mathlib/Topology/Separation/Connected.lean | [] | [
"Infinite",
"Nontrivial",
"PreconnectedSpace",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
subsingleton_iff_discrete_and_indiscrete :
Subsingleton X ↔ DiscreteTopology X ∧ IndiscreteTopology X | ⟨fun _ ↦ ⟨inferInstance, inferInstance⟩, fun ⟨_, _⟩ ↦ PreconnectedSpace.trivial_of_discrete⟩ | theorem | subsingleton_iff_discrete_and_indiscrete | Topology.Separation | Mathlib/Topology/Separation/Connected.lean | [] | [
"DiscreteTopology",
"IndiscreteTopology"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_finite_clopen_cover (hU : IsOpenCover U) : ∃ (n : ℕ) (V : Fin n → Clopens X),
(∀ j, ∃ i, (V j : Set X) ⊆ U i) ∧ univ ⊆ ⋃ j, (V j : Set X) | by
-- Choose an index `r x` for each point in `X` such that `∀ x, x ∈ U (r x)`.
choose r hr using hU.exists_mem
-- Choose a clopen neighbourhood `V x` of each `x` contained in `U (r x)`.
choose V hV hVx hVU using fun x ↦ compact_exists_isClopen_in_isOpen (U _).isOpen (hr x)
-- Apply compactness to extract a f... | lemma | TopologicalSpace.IsOpenCover.exists_finite_clopen_cover | Topology.Separation | Mathlib/Topology/Separation/DisjointCover.lean | [] | [
"Set",
"Set.mem_iUnion_of_mem",
"compact_exists_isClopen_in_isOpen"
] | Any open cover of a profinite space can be refined to a finite cover by clopens. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_finite_nonempty_disjoint_clopen_cover (hU : IsOpenCover U) :
∃ (n : ℕ) (W : Fin n → Clopens X), (∀ j, W j ≠ ⊥ ∧ ∃ i, (W j : Set X) ⊆ U i)
∧ (univ : Set X) ⊆ ⋃ j, ↑(W j) ∧ Pairwise (Disjoint on W) | by
classical
obtain ⟨n, V, hVle, hVun⟩ := hU.exists_finite_clopen_cover
obtain ⟨W, hWle, hWun, hWd⟩ := Fintype.exists_disjointed_le V
simp only [← SetLike.coe_set_eq, Clopens.coe_finset_sup, Finset.mem_univ, iUnion_true] at hWun
let t : Finset (Fin n) := {j | W j ≠ ⊥}
refine ⟨#t, fun k ↦ W (t.equivFin.symm ... | lemma | TopologicalSpace.IsOpenCover.exists_finite_nonempty_disjoint_clopen_cover | Topology.Separation | Mathlib/Topology/Separation/DisjointCover.lean | [] | [
"Disjoint",
"Finset",
"Finset.mem_univ",
"Fintype.exists_disjointed_le",
"Pairwise",
"Set",
"Set.nonempty_iff_ne_empty",
"SetLike.coe_ne_coe",
"SetLike.coe_set_eq",
"subset_trans"
] | Any open cover of a profinite space can be refined to a finite cover by pairwise disjoint
nonempty clopens. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_open_prod_subset_of_mem_nhds_diagonal (hS : S ∈ nhdsSet (diagonal X)) (x : X) :
∃ U : Set X, IsOpen U ∧ x ∈ U ∧ U ×ˢ U ⊆ S | by
have : S ∈ 𝓝 (x, x) := mem_nhdsSet_iff_forall.mp hS _ rfl
obtain ⟨u, v, huo, hux, hvo, hvx, H⟩ := by rwa [mem_nhds_prod_iff'] at this
exact ⟨_, huo.inter hvo, ⟨hux, hvx⟩, fun p hp ↦ H ⟨hp.1.1, hp.2.2⟩⟩ | lemma | TopologicalSpace.exists_open_prod_subset_of_mem_nhds_diagonal | Topology.Separation | Mathlib/Topology/Separation/DisjointCover.lean | [] | [
"IsOpen",
"Set",
"mem_nhds_prod_iff'",
"nhdsSet"
] | If `S` is any neighbourhood of the diagonal in a topological space `X`, any point of `X` has an
open neighbourhood `U` such that `U ×ˢ U ⊆ S`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_finite_open_cover_prod_subset_of_mem_nhds_diagonal_of_compact
(hS : S ∈ nhdsSet (diagonal X)) :
∃ (t : Finset X) (U : t → Opens X), IsOpenCover U ∧ ∀ i, (U i : Set X) ×ˢ U i ⊆ S | by
choose U hUo hUx hUp using exists_open_prod_subset_of_mem_nhds_diagonal hS
obtain ⟨t, ht⟩ := isCompact_univ.elim_finite_subcover _ hUo (fun x _ ↦ mem_iUnion.mpr ⟨_, hUx x⟩)
refine ⟨t, fun i ↦ ⟨_, hUo i⟩, .of_sets _ ?_, (hUp ·)⟩
simpa [iUnion_subtype, ← univ_subset_iff] using ht | lemma | TopologicalSpace.exists_finite_open_cover_prod_subset_of_mem_nhds_diagonal_of_compact | Topology.Separation | Mathlib/Topology/Separation/DisjointCover.lean | [] | [
"Finset",
"Set",
"nhdsSet"
] | If `S` is any neighbourhood of the diagonal in a compact topological space `X`, then there
exists a finite cover of `X` by opens `U i` such that `U i ×ˢ U i ⊆ S` for all `i`.
That the indexing set is a finset of `X` is an artifact of the proof; it could be any finite type. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_finite_disjoint_nonempty_clopen_cover_of_mem_nhds_diagonal_of_profinite
(hS : S ∈ nhdsSet (diagonal X)) :
∃ (n : ℕ) (D : Fin n → Clopens X), (∀ i, D i ≠ ⊥) ∧ (∀ i, ∀ y ∈ D i, ∀ z ∈ D i, (y, z) ∈ S)
∧ (univ : Set X) ⊆ ⋃ i, D i ∧ Pairwise (Disjoint on D) | by
obtain ⟨t, U, hUc, hUS⟩ := exists_finite_open_cover_prod_subset_of_mem_nhds_diagonal_of_compact hS
-- Now refine it to a disjoint covering.
obtain ⟨n, W, hW₁, hW₂, hW₃⟩ := hUc.exists_finite_nonempty_disjoint_clopen_cover
refine ⟨n, W, fun j ↦ (hW₁ j).1, fun j y hy z hz ↦ ?_, hW₂, hW₃⟩
exact match (hW₁ j).2... | lemma | TopologicalSpace.exists_finite_disjoint_nonempty_clopen_cover_of_mem_nhds_diagonal_of_profinite | Topology.Separation | Mathlib/Topology/Separation/DisjointCover.lean | [] | [
"Disjoint",
"Pairwise",
"Set",
"nhdsSet"
] | If `S` is any neighbourhood of the diagonal in a profinite topological space `X`, then there
exists a finite cover of `X` by disjoint nonempty clopens `U i` with `U i ×ˢ U i ⊆ S` for all `i`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_disjoint_nonempty_clopen_cover_of_mem_nhds_diagonal (hS : S ∈ nhdsSet (diagonal V)) :
∃ (n : ℕ) (D : Fin n → Clopens X), (∀ i, D i ≠ ⊥) ∧ (∀ i, ∀ y ∈ D i, ∀ z ∈ D i, (f y, f z) ∈ S)
∧ (univ : Set X) ⊆ ⋃ i, D i ∧ Pairwise (Disjoint on D) | by
have : (f.prodMap f) ⁻¹' S ∈ nhdsSet (diagonal X) := by
rw [mem_nhdsSet_iff_forall] at hS ⊢
rintro ⟨x, y⟩ (rfl : x = y)
exact (map_continuous _).continuousAt.preimage_mem_nhds (hS _ rfl)
exact exists_finite_disjoint_nonempty_clopen_cover_of_mem_nhds_diagonal_of_profinite this | lemma | ContinuousMap.exists_disjoint_nonempty_clopen_cover_of_mem_nhds_diagonal | Topology.Separation | Mathlib/Topology/Separation/DisjointCover.lean | [] | [
"Disjoint",
"Pairwise",
"Set",
"mem_nhdsSet_iff_forall",
"nhdsSet"
] | For any continuous function `f : X → V`, with `X` profinite, and `S` a neighbourhood of the
diagonal in `V × V`, there exists a finite cover of `X` by pairwise-disjoint nonempty clopens, on
each of which `f` varies within `S`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_finite_approximation_of_mem_nhds_diagonal (hS : S ∈ nhdsSet (diagonal V)) :
∃ (n : ℕ) (g : X → Fin n) (h : Fin n → V), Continuous g ∧ ∀ x, (f x, h (g x)) ∈ S | by
obtain ⟨n, E, hEne, hES, hEuniv, hEdis⟩ :=
exists_disjoint_nonempty_clopen_cover_of_mem_nhds_diagonal f hS
have h_uniq (x) : ∃! i, x ∈ E i := by
refine match mem_iUnion.mp (hEuniv <| mem_univ x) with
| ⟨i, hi⟩ => ⟨i, hi, fun j hj ↦ hEdis.eq ?_⟩
simpa [← Clopens.coe_disjoint, not_disjoint_iff] u... | lemma | ContinuousMap.exists_finite_approximation_of_mem_nhds_diagonal | Topology.Separation | Mathlib/Topology/Separation/DisjointCover.lean | [] | [
"Continuous",
"Set.ext",
"SetLike.coe_set_eq",
"nhdsSet"
] | For any continuous function `f : X → V`, with `X` profinite, and `S` a neighbourhood of the
diagonal in `V × V`, the function `f` can be `S`-approximated by a function factoring through
`Fin n`, for some `n`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_finite_sum_const_mulIndicator_approximation_of_mem_nhds_diagonal [CommMonoid V]
(hS : S ∈ nhdsSet (diagonal V)) :
∃ (n : ℕ) (U : Fin n → Clopens X) (v : Fin n → V),
∀ x, (f x, ∏ n, mulIndicator (U n) (fun _ ↦ v n) x) ∈ S | by
obtain ⟨n, g, h, hg, hgh⟩ := exists_finite_approximation_of_mem_nhds_diagonal f hS
refine ⟨n, fun i ↦ ⟨_, (isClopen_discrete {i}).preimage hg⟩, h, fun x ↦ ?_⟩
convert! hgh x
exact (Fintype.prod_eq_single _ fun i hi ↦ mulIndicator_of_notMem hi.symm _).trans
(mulIndicator_of_mem rfl _) | lemma | ContinuousMap.exists_finite_sum_const_mulIndicator_approximation_of_mem_nhds_diagonal | Topology.Separation | Mathlib/Topology/Separation/DisjointCover.lean | [] | [
"CommMonoid",
"Fintype.prod_eq_single",
"isClopen_discrete",
"nhdsSet",
"trans"
] | If `f` is a continuous map from a profinite space to a topological space with a commutative monoid
structure, then we can approximate `f` by finite products of indicator functions of clopen sets.
(Note no compatibility is assumed between the monoid structure on `V` and the topology.) | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsGδ.compl_singleton (x : X) [T1Space X] : IsGδ ({x}ᶜ : Set X) | isOpen_compl_singleton.isGδ | theorem | IsGδ.compl_singleton | Topology.Separation | Mathlib/Topology/Separation/GDelta.lean | [] | [
"IsGδ",
"Set",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.Countable.isGδ_compl {s : Set X} [T1Space X] (hs : s.Countable) : IsGδ sᶜ | by
rw [← biUnion_of_singleton s, compl_iUnion₂]
exact .biInter hs fun x _ => .compl_singleton x | theorem | Set.Countable.isGδ_compl | Topology.Separation | Mathlib/Topology/Separation/GDelta.lean | [] | [
"IsGδ",
"Set",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.Finite.isGδ_compl {s : Set X} [T1Space X] (hs : s.Finite) : IsGδ sᶜ | hs.countable.isGδ_compl | theorem | Set.Finite.isGδ_compl | Topology.Separation | Mathlib/Topology/Separation/GDelta.lean | [] | [
"IsGδ",
"Set",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.Subsingleton.isGδ_compl {s : Set X} [T1Space X] (hs : s.Subsingleton) : IsGδ sᶜ | hs.finite.isGδ_compl | theorem | Set.Subsingleton.isGδ_compl | Topology.Separation | Mathlib/Topology/Separation/GDelta.lean | [] | [
"IsGδ",
"Set",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Finset.isGδ_compl [T1Space X] (s : Finset X) : IsGδ (sᶜ : Set X) | s.finite_toSet.isGδ_compl | theorem | Finset.isGδ_compl | Topology.Separation | Mathlib/Topology/Separation/GDelta.lean | [] | [
"Finset",
"IsGδ",
"Set",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsGδ.singleton [FirstCountableTopology X] [T1Space X] (x : X) :
IsGδ ({x} : Set X) | by
rcases (nhds_basis_opens x).exists_antitone_subbasis with ⟨U, hU, h_basis⟩
rw [← biInter_basis_nhds h_basis.toHasBasis]
exact .biInter (to_countable _) fun n _ => (hU n).2.isGδ | theorem | IsGδ.singleton | Topology.Separation | Mathlib/Topology/Separation/GDelta.lean | [] | [
"FirstCountableTopology",
"IsGδ",
"Set",
"T1Space",
"biInter_basis_nhds",
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.Finite.isGδ [FirstCountableTopology X] {s : Set X} [T1Space X] (hs : s.Finite) :
IsGδ s | Finite.induction_on _ hs .empty fun _ _ ↦ .union (.singleton _) | theorem | Set.Finite.isGδ | Topology.Separation | Mathlib/Topology/Separation/GDelta.lean | [] | [
"FirstCountableTopology",
"IsGδ",
"Set",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PerfectlyNormalSpace (X : Type u) [TopologicalSpace X] : Prop extends NormalSpace X where
closed_gdelta : ∀ ⦃h : Set X⦄, IsClosed h → IsGδ h | class | PerfectlyNormalSpace | Topology.Separation | Mathlib/Topology/Separation/GDelta.lean | [] | [
"IsClosed",
"IsGδ",
"NormalSpace",
"Set",
"TopologicalSpace"
] | A topological space `X` is a *perfectly normal space* provided it is normal and
closed sets are Gδ. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Disjoint.hasSeparatingCover_closed_gdelta_right {s t : Set X} [NormalSpace X]
(st_dis : Disjoint s t) (t_cl : IsClosed t) (t_gd : IsGδ t) : HasSeparatingCover s t | by
obtain ⟨T, T_open, T_count, T_int⟩ := t_gd
rcases T.eq_empty_or_nonempty with rfl | T_nonempty
· rw [T_int, sInter_empty] at st_dis
rw [(s.disjoint_univ).mp st_dis]
exact t.hasSeparatingCover_empty_left
obtain ⟨g, g_surj⟩ := T_count.exists_surjective T_nonempty
choose g' g'_open clt_sub_g' clg'_sub... | theorem | Disjoint.hasSeparatingCover_closed_gdelta_right | Topology.Separation | Mathlib/Topology/Separation/GDelta.lean | [] | [
"Disjoint",
"HasSeparatingCover",
"IsClosed",
"IsGδ",
"NormalSpace",
"Set",
"antisymm",
"closure",
"closure_compl",
"isClosed_closure",
"normal_exists_closure_subset",
"subset_closure",
"trans"
] | Lemma that allows the easy conclusion that perfectly normal spaces are completely normal. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsClosed.isGδ [PerfectlyNormalSpace X] {s : Set X} (hs : IsClosed s) : IsGδ s | PerfectlyNormalSpace.closed_gdelta hs | theorem | IsClosed.isGδ | Topology.Separation | Mathlib/Topology/Separation/GDelta.lean | [] | [
"IsClosed",
"IsGδ",
"PerfectlyNormalSpace",
"Set"
] | In a perfectly normal space, all closed sets are Gδ. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Topology.IsInducing.perfectlyNormalSpace [PerfectlyNormalSpace Y] {e : X → Y}
(he : IsInducing e) : PerfectlyNormalSpace X | where
toNormalSpace := he.completelyNormalSpace.toNormalSpace
closed_gdelta _ hs := (he.isClosed_iff.1 hs).elim fun _ ht =>
ht.2 ▸ ht.1.isGδ.preimage he.continuous | theorem | Topology.IsInducing.perfectlyNormalSpace | Topology.Separation | Mathlib/Topology/Separation/GDelta.lean | [] | [
"PerfectlyNormalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
T6Space (X : Type u) [TopologicalSpace X] : Prop extends T0Space X, PerfectlyNormalSpace X | class | T6Space | Topology.Separation | Mathlib/Topology/Separation/GDelta.lean | [] | [
"PerfectlyNormalSpace",
"T0Space",
"TopologicalSpace"
] | A T₆ space is a perfectly normal T₀ space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Topology.IsEmbedding.t6Space [T6Space Y] {e : X → Y}
(he : IsEmbedding e) : T6Space X | where
toPerfectlyNormalSpace := he.perfectlyNormalSpace
toT0Space := he.t0Space | theorem | Topology.IsEmbedding.t6Space | Topology.Separation | Mathlib/Topology/Separation/GDelta.lean | [] | [
"T6Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
T2Space (X : Type u) [TopologicalSpace X] : Prop where
/-- Every two points in a Hausdorff space admit disjoint open neighbourhoods. -/
t2 : Pairwise fun x y => ∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v | class | T2Space | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Disjoint",
"IsOpen",
"Pairwise",
"Set",
"TopologicalSpace"
] | A T₂ space, also known as a Hausdorff space, is one in which for every
`x ≠ y` there exists disjoint open sets around `x` and `y`. This is
the most widely used of the separation axioms. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t2_separation [T2Space X] {x y : X} (h : x ≠ y) :
∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v | T2Space.t2 h | theorem | t2_separation | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Disjoint",
"IsOpen",
"Set",
"T2Space"
] | Two different points can be separated by open sets. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
t2Space_iff_disjoint_nhds : T2Space X ↔ Pairwise fun x y : X => Disjoint (𝓝 x) (𝓝 y) | by
refine (t2Space_iff X).trans (forall₃_congr fun x y _ => ?_)
simp only [(nhds_basis_opens x).disjoint_iff (nhds_basis_opens y), ← exists_and_left,
and_assoc, and_comm, and_left_comm] | theorem | t2Space_iff_disjoint_nhds | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Disjoint",
"Pairwise",
"T2Space",
"disjoint_iff",
"nhds_basis_opens",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
disjoint_nhds_nhds [T2Space X] {x y : X} : Disjoint (𝓝 x) (𝓝 y) ↔ x ≠ y | ⟨fun hd he => by simp [he, nhds_neBot.ne] at hd, (t2Space_iff_disjoint_nhds.mp ‹_› ·)⟩ | theorem | disjoint_nhds_nhds | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Disjoint",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pairwise_disjoint_nhds [T2Space X] : Pairwise (Disjoint on (𝓝 : X → Filter X)) | fun _ _ =>
disjoint_nhds_nhds.2 | theorem | pairwise_disjoint_nhds | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Disjoint",
"Filter",
"Pairwise",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.pairwiseDisjoint_nhds [T2Space X] (s : Set X) : s.PairwiseDisjoint 𝓝 | pairwise_disjoint_nhds.set_pairwise s | theorem | Set.pairwiseDisjoint_nhds | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Set",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.Finite.t2_separation [T2Space X] {s : Set X} (hs : s.Finite) :
∃ U : X → Set X, (∀ x, x ∈ U x ∧ IsOpen (U x)) ∧ s.PairwiseDisjoint U | s.pairwiseDisjoint_nhds.exists_mem_filter_basis hs nhds_basis_opens | theorem | Set.Finite.t2_separation | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"IsOpen",
"Set",
"T2Space",
"nhds_basis_opens"
] | Points of a finite set can be separated by open sets from each other. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
SeparationQuotient.t2Space_iff : T2Space (SeparationQuotient X) ↔ R1Space X | by
simp only [t2Space_iff_disjoint_nhds, Pairwise, surjective_mk.forall₂, ne_eq, mk_eq_mk,
r1Space_iff_inseparable_or_disjoint_nhds, ← disjoint_comap_iff surjective_mk, comap_mk_nhds_mk,
← or_iff_not_imp_left] | theorem | SeparationQuotient.t2Space_iff | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"Pairwise",
"R1Space",
"SeparationQuotient",
"T2Space",
"r1Space_iff_inseparable_or_disjoint_nhds",
"t2Space_iff_disjoint_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SeparationQuotient.t2Space [R1Space X] : T2Space (SeparationQuotient X) | t2Space_iff.2 ‹_› | instance | SeparationQuotient.t2Space | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"R1Space",
"SeparationQuotient",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
R1Space.t2Space_iff_t0Space [R1Space X] : T2Space X ↔ T0Space X | by
constructor <;> intro <;> infer_instance | theorem | R1Space.t2Space_iff_t0Space | Topology.Separation | Mathlib/Topology/Separation/Hausdorff.lean | [] | [
"R1Space",
"T0Space",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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