fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
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 | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | eqOn_closure₂ | null |
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_eventuallyE... | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Set.EqOn.of_subset_closure | 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`. |
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... | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Function.LeftInverse.isClosed_range | null |
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 | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Function.LeftInverse.isClosedEmbedding | null |
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 | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | SeparatedNhds.of_isCompact_isCompact | null |
SeparatedNhds.of_isClosed_isCompact_closure_compl_isClosed [T2Space 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
have ht : IsCompact t := .of_isClosed_subset H2 H3 <| H4.subset_compl_left.trans subset_closure
rw [←... | lemma | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | SeparatedNhds.of_isClosed_isCompact_closure_compl_isClosed | In a `T2Space X`, for disjoint closed sets `s t` such that `closure sᶜ` is compact,
there are neighbourhoods that separate `s` and `t`. |
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 | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | SeparatedNhds.of_finset_finset | null |
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 | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | SeparatedNhds.of_singleton_finset | null |
@[aesop 50% apply, grind ←]
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 | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | IsCompact.isClosed | In a `T2Space`, every compact set is closed. |
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 | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | IsCompact.preimage_continuous | null |
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 (isC... | lemma | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Pi.isCompact_iff | null |
Pi.isCompact_closure_iff {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)]
[∀ i, T2Space (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 | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Pi.isCompact_closure_iff | null |
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 | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | 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. |
CompactExhaustion.isClosed [T2Space X] (K : CompactExhaustion X) (n : ℕ) : IsClosed (K n) :=
(K.isCompact n).isClosed | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | CompactExhaustion.isClosed | null |
IsCompact.inter [T2Space X] {s t : Set X} (hs : IsCompact s) (ht : IsCompact t) :
IsCompact (s ∩ t) :=
hs.inter_right <| ht.isClosed | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | IsCompact.inter | null |
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 | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | image_closure_of_isCompact | null |
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... | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | ContinuousAt.ne_iff_eventually_ne | Two continuous maps into a Hausdorff space agree at a point iff they agree in a
neighborhood. |
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, h... | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | ContinuousAt.eventuallyEq_nhds_iff_eventuallyEq_nhdsNE | **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. |
protected Continuous.isClosedMap [CompactSpace X] [T2Space Y] {f : X → Y}
(h : Continuous f) : IsClosedMap f := fun _s hs => (hs.isCompact.image h).isClosed | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Continuous.isClosedMap | A continuous map from a compact space to a Hausdorff space is a closed map. |
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 | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Continuous.isClosedEmbedding | A continuous injective map from a compact space to a Hausdorff space is a closed embedding. |
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 | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | IsQuotientMap.of_surjective_continuous | A continuous surjective map from a compact space to a Hausdorff space is a quotient map. |
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... | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | isPreirreducible_iff_forall_mem_subset_closure_singleton | null |
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 | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | isPreirreducible_iff_subsingleton | null |
protected IsPreirreducible.subsingleton [T2Space X] {S : Set X} (h : IsPreirreducible S) :
S.Subsingleton :=
isPreirreducible_iff_subsingleton.1 h | lemma | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | IsPreirreducible.subsingleton | null |
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 | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | isIrreducible_iff_singleton | null |
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 | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | not_preirreducible_nontrivial_t2 | There does not exist a nontrivial preirreducible T₂ space. |
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 | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | t2Space_antitone | null |
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 | Topology | [
"Mathlib.Topology.GDelta.MetrizableSpace",
"Mathlib.Topology.Separation.CompletelyRegular",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Separation/Lemmas.lean | totallySeparatedSpace_of_cardinalMk_lt_continuum | null |
protected _root_.Set.Countable.totallySeparatedSpace {s : Set X} (h : s.Countable) :
TotallySeparatedSpace s :=
have : _root_.Countable s := h
inferInstanceAs (TotallySeparatedSpace s) | lemma | Topology | [
"Mathlib.Topology.GDelta.MetrizableSpace",
"Mathlib.Topology.Separation.CompletelyRegular",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Separation/Lemmas.lean | _root_.Set.Countable.totallySeparatedSpace | null |
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 | Topology | [
"Mathlib.Topology.GDelta.MetrizableSpace",
"Mathlib.Topology.Separation.CompletelyRegular",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Separation/Lemmas.lean | Set.Countable.isTotallyDisconnected | Countable subsets of metric spaces are totally disconnected. |
IsClosed.mk_lt_continuum [NormalSpace X] {s : Set X} (hs : IsClosed s)
[DiscreteTopology s] : #s < 𝔠 := by
by_contra! h
rcases exists_countable_dense X with ⟨t, htc, htd⟩
haveI := htc.to_subtype
refine (Cardinal.cantor 𝔠).not_ge ?_
calc
2 ^ 𝔠 ≤ #C(s, ℝ) := by
rw [ContinuousMap.equivFnOfDiscre... | theorem | Topology | [
"Mathlib.Analysis.Real.Cardinality",
"Mathlib.Topology.TietzeExtension"
] | Mathlib/Topology/Separation/NotNormal.lean | IsClosed.mk_lt_continuum | 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.
The proof follows
https://en.wikipedia.org/wiki/Moore_plane#Proof_that_the_Moore_plane_is_not_normal |
IsClosed.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 | Topology | [
"Mathlib.Analysis.Real.Cardinality",
"Mathlib.Topology.TietzeExtension"
] | Mathlib/Topology/Separation/NotNormal.lean | IsClosed.not_normal_of_continuum_le_mk | 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. |
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 ... | theorem | Topology | [
"Mathlib.Topology.Separation.Regular",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Separation/Profinite.lean | totallySeparatedSpace_of_t0_of_basis_clopen | A T0 space with a clopen basis is totally separated. |
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 ... | theorem | Topology | [
"Mathlib.Topology.Separation.Regular",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Separation/Profinite.lean | nhds_basis_clopen | null |
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
taut... | theorem | Topology | [
"Mathlib.Topology.Separation.Regular",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Separation/Profinite.lean | isTopologicalBasis_isClopen | null |
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 | Topology | [
"Mathlib.Topology.Separation.Regular",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Separation/Profinite.lean | compact_exists_isClopen_in_isOpen | Every member of an open set in a compact Hausdorff totally disconnected space
is contained in a clopen set contained in the open set. |
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
... | theorem | Topology | [
"Mathlib.Topology.Separation.Regular",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Separation/Profinite.lean | loc_compact_Haus_tot_disc_of_zero_dim | A locally compact Hausdorff totally disconnected space has a basis with clopen elements. |
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 | Topology | [
"Mathlib.Topology.Separation.Regular",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Separation/Profinite.lean | loc_compact_t2_tot_disc_iff_tot_sep | A locally compact Hausdorff space is totally disconnected
if and only if it is totally separated. |
@[mk_iff]
RegularSpace (X : Type u) [TopologicalSpace X] : Prop where
/-- If `a` is a point that does not belong to a closed set `s`, then `a` and `s` admit disjoint
neighborhoods. -/
regular : ∀ {s : Set X} {a}, IsClosed s → a ∉ s → Disjoint (𝓝ˢ s) (𝓝 a) | class | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | RegularSpace | A topological space is called a *regular space* if for any closed set `s` and `a ∉ s`, there
exist disjoint open sets `U ⊇ s` and `V ∋ a`. We formulate this condition in terms of `Disjoint`ness
of filters `𝓝ˢ s` and `𝓝 a`. |
regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
List.TFAE [RegularSpace X,
∀ (s : Set X) x, x ∉ closure s → Disjoint (𝓝ˢ s) (𝓝 x),
∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s,
∀ (x : X) (s : Set X), s ∈ 𝓝 x → ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s,
∀ x : X, (𝓝 x).lift' c... | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | regularSpace_TFAE | null |
RegularSpace.of_lift'_closure_le (h : ∀ x : X, (𝓝 x).lift' closure ≤ 𝓝 x) :
RegularSpace X :=
Iff.mpr ((regularSpace_TFAE X).out 0 4) h | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | RegularSpace.of_lift'_closure_le | null |
RegularSpace.of_lift'_closure (h : ∀ x : X, (𝓝 x).lift' closure = 𝓝 x) : RegularSpace X :=
Iff.mpr ((regularSpace_TFAE X).out 0 5) h | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | RegularSpace.of_lift'_closure | null |
RegularSpace.of_hasBasis {ι : X → Sort*} {p : ∀ a, ι a → Prop} {s : ∀ a, ι a → Set X}
(h₁ : ∀ a, (𝓝 a).HasBasis (p a) (s a)) (h₂ : ∀ a i, p a i → IsClosed (s a i)) :
RegularSpace X :=
.of_lift'_closure fun a => (h₁ a).lift'_closure_eq_self (h₂ a) | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | RegularSpace.of_hasBasis | null |
RegularSpace.of_exists_mem_nhds_isClosed_subset
(h : ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s) : RegularSpace X :=
Iff.mpr ((regularSpace_TFAE X).out 0 3) h | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | RegularSpace.of_exists_mem_nhds_isClosed_subset | null |
SeparatedNhds.of_isCompact_isClosed {s t : Set X}
(hs : IsCompact s) (ht : IsClosed t) (hst : Disjoint s t) : SeparatedNhds s t := by
simpa only [separatedNhds_iff_disjoint, hs.disjoint_nhdsSet_left, disjoint_nhds_nhdsSet,
ht.closure_eq, disjoint_left] using hst | lemma | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | SeparatedNhds.of_isCompact_isClosed | A weakly locally compact R₁ space is regular. -/
instance (priority := 100) [WeaklyLocallyCompactSpace X] [R1Space X] : RegularSpace X :=
.of_hasBasis isCompact_isClosed_basis_nhds fun _ _ ⟨_, _, h⟩ ↦ h
section
variable [RegularSpace X] {x : X} {s : Set X}
theorem disjoint_nhdsSet_nhds : Disjoint (𝓝ˢ s) (𝓝 x) ↔ x... |
IsClosed.HasSeparatingCover {s t : Set X} [LindelofSpace X] [RegularSpace X]
(s_cl : IsClosed s) (t_cl : IsClosed t) (st_dis : Disjoint s t) : HasSeparatingCover s t := by
rcases isEmpty_or_nonempty X with empty_X | nonempty_X
· rw [subset_eq_empty (t := s) (fun ⦃_⦄ _ ↦ trivial) (univ_eq_empty_iff.mpr empty_X)]... | lemma | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | IsClosed.HasSeparatingCover | This technique to witness `HasSeparatingCover` in regular Lindelöf topological spaces
will be used to prove regular Lindelöf spaces are normal. |
exists_compact_closed_between [LocallyCompactSpace X] [RegularSpace X]
{K U : Set X} (hK : IsCompact K) (hU : IsOpen U) (h_KU : K ⊆ U) :
∃ L, IsCompact L ∧ IsClosed L ∧ K ⊆ interior L ∧ L ⊆ U :=
let ⟨L, L_comp, KL, LU⟩ := exists_compact_between hK hU h_KU
⟨closure L, L_comp.closure, isClosed_closure, KL.tra... | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | exists_compact_closed_between | In a (possibly non-Hausdorff) locally compact regular space, for every containment `K ⊆ U` of
a compact set `K` in an open set `U`, there is a compact closed neighborhood `L`
such that `K ⊆ L ⊆ U`: equivalently, there is a compact closed set `L` such
that `K ⊆ interior L` and `L ⊆ U`. |
exists_open_between_and_isCompact_closure [LocallyCompactSpace X] [RegularSpace X]
{K U : Set X} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) :
∃ V, IsOpen V ∧ K ⊆ V ∧ closure V ⊆ U ∧ IsCompact (closure V) := by
rcases exists_compact_closed_between hK hU hKU with ⟨L, L_compact, L_closed, KL, LU⟩
have A ... | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | exists_open_between_and_isCompact_closure | In a locally compact regular space, given a compact set `K` inside an open set `U`, we can find
an open set `V` between these sets with compact closure: `K ⊆ V` and the closure of `V` is
inside `U`. |
T25Space (X : Type u) [TopologicalSpace X] : Prop where
/-- Given two distinct points in a T₂.₅ space, their filters of closed neighborhoods are
disjoint. -/
t2_5 : ∀ ⦃x y : X⦄, x ≠ y → Disjoint ((𝓝 x).lift' closure) ((𝓝 y).lift' closure)
@[simp] | class | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | T25Space | A T₂.₅ space, also known as a Urysohn space, is a topological space
where for every pair `x ≠ y`, there are two open sets, with the intersection of closures
empty, one containing `x` and the other `y` . |
disjoint_lift'_closure_nhds [T25Space X] {x y : X} :
Disjoint ((𝓝 x).lift' closure) ((𝓝 y).lift' closure) ↔ x ≠ y :=
⟨fun h hxy => by simp [hxy, nhds_neBot.ne] at h, fun h => T25Space.t2_5 h⟩ | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | disjoint_lift'_closure_nhds | null |
exists_nhds_disjoint_closure [T25Space X] {x y : X} (h : x ≠ y) :
∃ s ∈ 𝓝 x, ∃ t ∈ 𝓝 y, Disjoint (closure s) (closure t) :=
((𝓝 x).basis_sets.lift'_closure.disjoint_iff (𝓝 y).basis_sets.lift'_closure).1 <|
disjoint_lift'_closure_nhds.2 h | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | exists_nhds_disjoint_closure | null |
exists_open_nhds_disjoint_closure [T25Space X] {x y : X} (h : x ≠ y) :
∃ u : Set X,
x ∈ u ∧ IsOpen u ∧ ∃ v : Set X, y ∈ v ∧ IsOpen v ∧ Disjoint (closure u) (closure v) := by
simpa only [exists_prop, and_assoc] using
((nhds_basis_opens x).lift'_closure.disjoint_iff (nhds_basis_opens y).lift'_closure).1
... | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | exists_open_nhds_disjoint_closure | null |
T25Space.of_injective_continuous [TopologicalSpace Y] [T25Space Y] {f : X → Y}
(hinj : Injective f) (hcont : Continuous f) : T25Space X where
t2_5 x y hne := (tendsto_lift'_closure_nhds hcont x).disjoint (t2_5 <| hinj.ne hne)
(tendsto_lift'_closure_nhds hcont y) | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | T25Space.of_injective_continuous | null |
Topology.IsEmbedding.t25Space [TopologicalSpace Y] [T25Space Y] {f : X → Y}
(hf : IsEmbedding f) : T25Space X :=
.of_injective_continuous hf.injective hf.continuous | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | Topology.IsEmbedding.t25Space | null |
protected Homeomorph.t25Space [TopologicalSpace Y] [T25Space X] (h : X ≃ₜ Y) : T25Space Y :=
h.symm.isEmbedding.t25Space | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | Homeomorph.t25Space | null |
Subtype.instT25Space [T25Space X] {p : X → Prop} : T25Space {x // p x} :=
IsEmbedding.subtypeVal.t25Space | instance | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | Subtype.instT25Space | null |
T3Space (X : Type u) [TopologicalSpace X] : Prop extends T0Space X, RegularSpace X | class | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | T3Space | A T₃ space is a T₀ space which is a regular space. Any T₃ space is a T₁ space, a T₂ space, and
a T₂.₅ space. |
RegularSpace.t3Space_iff_t0Space [RegularSpace X] : T3Space X ↔ T0Space X := by
constructor <;> intro <;> infer_instance | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | RegularSpace.t3Space_iff_t0Space | null |
protected Topology.IsEmbedding.t3Space [TopologicalSpace Y] [T3Space Y] {f : X → Y}
(hf : IsEmbedding f) : T3Space X :=
{ toT0Space := hf.t0Space
toRegularSpace := hf.isInducing.regularSpace } | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | Topology.IsEmbedding.t3Space | null |
protected Homeomorph.t3Space [TopologicalSpace Y] [T3Space X] (h : X ≃ₜ Y) : T3Space Y :=
h.symm.isEmbedding.t3Space | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | Homeomorph.t3Space | null |
Subtype.t3Space [T3Space X] {p : X → Prop} : T3Space (Subtype p) :=
IsEmbedding.subtypeVal.t3Space | instance | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | Subtype.t3Space | null |
ULift.instT3Space [T3Space X] : T3Space (ULift X) :=
IsEmbedding.uliftDown.t3Space | instance | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | ULift.instT3Space | null |
disjoint_nested_nhds [T3Space X] {x y : X} (h : x ≠ y) :
∃ U₁ ∈ 𝓝 x, ∃ V₁ ∈ 𝓝 x, ∃ U₂ ∈ 𝓝 y, ∃ V₂ ∈ 𝓝 y,
IsClosed V₁ ∧ IsClosed V₂ ∧ IsOpen U₁ ∧ IsOpen U₂ ∧ V₁ ⊆ U₁ ∧ V₂ ⊆ U₂ ∧ Disjoint U₁ U₂ := by
rcases t2_separation h with ⟨U₁, U₂, U₁_op, U₂_op, x_in, y_in, H⟩
rcases exists_mem_nhds_isClosed_subset... | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | disjoint_nested_nhds | Given two points `x ≠ y`, we can find neighbourhoods `x ∈ V₁ ⊆ U₁` and `y ∈ V₂ ⊆ U₂`,
with the `Vₖ` closed and the `Uₖ` open, such that the `Uₖ` are disjoint. |
NormalSpace (X : Type u) [TopologicalSpace X] : Prop where
/-- Two disjoint sets in a normal space admit disjoint neighbourhoods. -/
normal : ∀ s t : Set X, IsClosed s → IsClosed t → Disjoint s t → SeparatedNhds s t | class | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | NormalSpace | The `SeparationQuotient` of a regular space is a T₃ space. -/
instance [RegularSpace X] : T3Space (SeparationQuotient X) where
regular {s a} hs ha := by
rcases surjective_mk a with ⟨a, rfl⟩
rw [← disjoint_comap_iff surjective_mk, comap_mk_nhds_mk, comap_mk_nhdsSet]
exact RegularSpace.regular (hs.preimage ... |
normal_separation [NormalSpace X] {s t : Set X} (H1 : IsClosed s) (H2 : IsClosed t)
(H3 : Disjoint s t) : SeparatedNhds s t :=
NormalSpace.normal s t H1 H2 H3 | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | normal_separation | null |
disjoint_nhdsSet_nhdsSet [NormalSpace X] {s t : Set X} (hs : IsClosed s) (ht : IsClosed t)
(hd : Disjoint s t) : Disjoint (𝓝ˢ s) (𝓝ˢ t) :=
(normal_separation hs ht hd).disjoint_nhdsSet | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | disjoint_nhdsSet_nhdsSet | null |
normal_exists_closure_subset [NormalSpace X] {s t : Set X} (hs : IsClosed s) (ht : IsOpen t)
(hst : s ⊆ t) : ∃ u, IsOpen u ∧ s ⊆ u ∧ closure u ⊆ t := by
have : Disjoint s tᶜ := Set.disjoint_left.mpr fun x hxs hxt => hxt (hst hxs)
rcases normal_separation hs (isClosed_compl_iff.2 ht) this with
⟨s', t', hs', ... | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | normal_exists_closure_subset | null |
protected Topology.IsClosedEmbedding.normalSpace [TopologicalSpace Y] [NormalSpace Y]
{f : X → Y} (hf : IsClosedEmbedding f) : NormalSpace X where
normal s t hs ht hst := by
have H : SeparatedNhds (f '' s) (f '' t) :=
NormalSpace.normal (f '' s) (f '' t) (hf.isClosedMap s hs) (hf.isClosedMap t ht)
... | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | Topology.IsClosedEmbedding.normalSpace | If the codomain of a closed embedding is a normal space, then so is the domain. |
protected Homeomorph.normalSpace [TopologicalSpace Y] [NormalSpace X] (h : X ≃ₜ Y) :
NormalSpace Y :=
h.symm.isClosedEmbedding.normalSpace | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | Homeomorph.normalSpace | null |
T4Space (X : Type u) [TopologicalSpace X] : Prop extends T1Space X, NormalSpace X | class | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | T4Space | A regular topological space with a Lindelöf topology is a normal space. A consequence of e.g.
Corollaries 20.8 and 20.10 of [Willard's *General Topology*][zbMATH02107988] (without the
assumption of Hausdorff). -/
instance (priority := 100) NormalSpace.of_regularSpace_lindelofSpace
[RegularSpace X] [LindelofSpace X]... |
protected Topology.IsClosedEmbedding.t4Space [TopologicalSpace Y] [T4Space Y] {f : X → Y}
(hf : IsClosedEmbedding f) : T4Space X where
toT1Space := hf.isEmbedding.t1Space
toNormalSpace := hf.normalSpace | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | Topology.IsClosedEmbedding.t4Space | If the codomain of a closed embedding is a T₄ space, then so is the domain. |
protected Homeomorph.t4Space [TopologicalSpace Y] [T4Space X] (h : X ≃ₜ Y) : T4Space Y :=
h.symm.isClosedEmbedding.t4Space | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | Homeomorph.t4Space | null |
ULift.instT4Space [T4Space X] : T4Space (ULift X) := IsClosedEmbedding.uliftDown.t4Space | instance | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | ULift.instT4Space | null |
CompletelyNormalSpace (X : Type u) [TopologicalSpace X] : Prop where
/-- If `closure s` is disjoint with `t`, and `s` is disjoint with `closure t`, then `s` and `t`
admit disjoint neighbourhoods. -/
completely_normal :
∀ ⦃s t : Set X⦄, Disjoint (closure s) t → Disjoint s (closure t) → Disjoint (𝓝ˢ s) (𝓝ˢ t)... | class | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | CompletelyNormalSpace | The `SeparationQuotient` of a normal space is a normal space. -/
instance [NormalSpace X] : NormalSpace (SeparationQuotient X) where
normal s t hs ht hd := separatedNhds_iff_disjoint.2 <| by
rw [← disjoint_comap_iff surjective_mk, comap_mk_nhdsSet, comap_mk_nhdsSet]
exact disjoint_nhdsSet_nhdsSet (hs.preimage... |
T5Space (X : Type u) [TopologicalSpace X] : Prop extends T1Space X, CompletelyNormalSpace X | class | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | T5Space | A completely normal space is a normal space. -/
instance (priority := 100) CompletelyNormalSpace.toNormalSpace
[CompletelyNormalSpace X] : NormalSpace X where
normal s t hs ht hd := separatedNhds_iff_disjoint.2 <|
completely_normal (by rwa [hs.closure_eq]) (by rwa [ht.closure_eq])
theorem Topology.IsEmbeddin... |
Topology.IsEmbedding.t5Space [TopologicalSpace Y] [T5Space Y] {e : X → Y}
(he : IsEmbedding e) : T5Space X where
__ := he.t1Space
completely_normal := by
have := he.completelyNormalSpace
exact completely_normal | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | Topology.IsEmbedding.t5Space | null |
protected Homeomorph.t5Space [TopologicalSpace Y] [T5Space X] (h : X ≃ₜ Y) : T5Space Y :=
h.symm.isClosedEmbedding.t5Space | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | Homeomorph.t5Space | null |
connectedComponent_eq_iInter_isClopen [T2Space X] [CompactSpace X] (x : X) :
connectedComponent x = ⋂ s : { s : Set X // IsClopen s ∧ x ∈ s }, s := by
apply Subset.antisymm connectedComponent_subset_iInter_isClopen
refine IsPreconnected.subset_connectedComponent ?_ (mem_iInter.2 fun s => s.2.2)
have hs : @IsC... | theorem | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | connectedComponent_eq_iInter_isClopen | A `T₅` space is a `T₄` space. -/
instance (priority := 100) T5Space.toT4Space [T5Space X] : T4Space X where
-- follows from type-class inference
/-- A subspace of a T₅ space is a T₅ space. -/
instance [T5Space X] {p : X → Prop} : T5Space { x // p x } :=
IsEmbedding.subtypeVal.t5Space
instance ULift.instT5Space [T... |
@[stacks 0900 "The Stacks entry proves profiniteness."]
ConnectedComponents.t2 [T2Space X] [CompactSpace X] : T2Space (ConnectedComponents X) := by
refine ⟨ConnectedComponents.surjective_coe.forall₂.2 fun a b ne => ?_⟩
rw [ConnectedComponents.coe_ne_coe] at ne
have h := connectedComponent_disjoint ne
rw [connec... | instance | Topology | [
"Mathlib.Tactic.StacksAttribute",
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Separation.Hausdorff",
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Separation/Regular.lean | ConnectedComponents.t2 | `ConnectedComponents X` is Hausdorff when `X` is Hausdorff and compact |
SeparatedNhds : Set X → Set X → Prop := fun s t : Set X =>
∃ U V : Set X, IsOpen U ∧ IsOpen V ∧ s ⊆ U ∧ t ⊆ V ∧ Disjoint U V | def | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | SeparatedNhds | `SeparatedNhds` is a predicate on pairs of sub`Set`s of a topological space. It holds if the two
sub`Set`s are contained in disjoint open sets. |
separatedNhds_iff_disjoint {s t : Set X} : SeparatedNhds s t ↔ Disjoint (𝓝ˢ s) (𝓝ˢ t) := by
simp only [(hasBasis_nhdsSet s).disjoint_iff (hasBasis_nhdsSet t), SeparatedNhds, ←
exists_and_left, and_assoc, and_comm, and_left_comm]
alias ⟨SeparatedNhds.disjoint_nhdsSet, _⟩ := separatedNhds_iff_disjoint | theorem | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | separatedNhds_iff_disjoint | null |
HasSeparatingCover : Set X → Set X → Prop := fun s t ↦
∃ u : ℕ → Set X, s ⊆ ⋃ n, u n ∧ ∀ n, IsOpen (u n) ∧ Disjoint (closure (u n)) t | def | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | HasSeparatingCover | `HasSeparatingCover`s can be useful witnesses for `SeparatedNhds`. |
hasSeparatingCovers_iff_separatedNhds {s t : Set X} :
HasSeparatingCover s t ∧ HasSeparatingCover t s ↔ SeparatedNhds s t := by
constructor
· rintro ⟨⟨u, u_cov, u_props⟩, ⟨v, v_cov, v_props⟩⟩
have open_lemma : ∀ (u₀ a : ℕ → Set X), (∀ n, IsOpen (u₀ n)) →
IsOpen (⋃ n, u₀ n \ closure (a n)) := fun _ _ u... | theorem | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | hasSeparatingCovers_iff_separatedNhds | Used to prove that a regular topological space with Lindelöf topology is a normal space,
and a perfectly normal space is a completely normal space. |
Set.hasSeparatingCover_empty_left (s : Set X) : HasSeparatingCover ∅ s :=
⟨fun _ ↦ ∅, empty_subset (⋃ _, ∅),
fun _ ↦ ⟨isOpen_empty, by simp only [closure_empty, empty_disjoint]⟩⟩ | theorem | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | Set.hasSeparatingCover_empty_left | null |
Set.hasSeparatingCover_empty_right (s : Set X) : HasSeparatingCover s ∅ :=
⟨fun _ ↦ univ, (subset_univ s).trans univ.iUnion_const.symm.subset,
fun _ ↦ ⟨isOpen_univ, by apply disjoint_empty⟩⟩ | theorem | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | Set.hasSeparatingCover_empty_right | null |
HasSeparatingCover.mono {s₁ s₂ t₁ t₂ : Set X} (sc_st : HasSeparatingCover s₂ t₂)
(s_sub : s₁ ⊆ s₂) (t_sub : t₁ ⊆ t₂) : HasSeparatingCover s₁ t₁ := by
obtain ⟨u, u_cov, u_props⟩ := sc_st
exact
⟨u,
s_sub.trans u_cov,
fun n ↦
⟨(u_props n).1,
disjoint_of_subset (fun ⦃_⦄ a ↦ a) t_sub (u_... | theorem | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | HasSeparatingCover.mono | null |
@[symm]
symm : SeparatedNhds s t → SeparatedNhds t s := fun ⟨U, V, oU, oV, aU, bV, UV⟩ =>
⟨V, U, oV, oU, bV, aU, Disjoint.symm UV⟩ | theorem | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | symm | null |
comm (s t : Set X) : SeparatedNhds s t ↔ SeparatedNhds t s :=
⟨symm, symm⟩ | theorem | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | comm | null |
preimage [TopologicalSpace Y] {f : X → Y} {s t : Set Y} (h : SeparatedNhds s t)
(hf : Continuous f) : SeparatedNhds (f ⁻¹' s) (f ⁻¹' t) :=
let ⟨U, V, oU, oV, sU, tV, UV⟩ := h
⟨f ⁻¹' U, f ⁻¹' V, oU.preimage hf, oV.preimage hf, preimage_mono sU, preimage_mono tV,
UV.preimage f⟩ | theorem | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | preimage | null |
protected disjoint (h : SeparatedNhds s t) : Disjoint s t :=
let ⟨_, _, _, _, hsU, htV, hd⟩ := h; hd.mono hsU htV | theorem | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | disjoint | null |
disjoint_closure_left (h : SeparatedNhds s t) : Disjoint (closure s) t :=
let ⟨_U, _V, _, hV, hsU, htV, hd⟩ := h
(hd.closure_left hV).mono (closure_mono hsU) htV | theorem | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | disjoint_closure_left | null |
disjoint_closure_right (h : SeparatedNhds s t) : Disjoint s (closure t) :=
h.symm.disjoint_closure_left.symm
@[simp] theorem empty_right (s : Set X) : SeparatedNhds s ∅ :=
⟨_, _, isOpen_univ, isOpen_empty, fun a _ => mem_univ a, Subset.rfl, disjoint_empty _⟩
@[simp] theorem empty_left (s : Set X) : SeparatedNhds ∅ ... | theorem | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | disjoint_closure_right | null |
mono (h : SeparatedNhds s₂ t₂) (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) : SeparatedNhds s₁ t₁ :=
let ⟨U, V, hU, hV, hsU, htV, hd⟩ := h
⟨U, V, hU, hV, hs.trans hsU, ht.trans htV, hd⟩ | theorem | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | mono | null |
union_left : SeparatedNhds s u → SeparatedNhds t u → SeparatedNhds (s ∪ t) u := by
simpa only [separatedNhds_iff_disjoint, nhdsSet_union, disjoint_sup_left] using And.intro | theorem | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | union_left | null |
union_right (ht : SeparatedNhds s t) (hu : SeparatedNhds s u) : SeparatedNhds s (t ∪ u) :=
(ht.symm.union_left hu.symm).symm | theorem | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | union_right | null |
isOpen_left_of_isOpen_union (hst : SeparatedNhds s t) (hst' : IsOpen (s ∪ t)) : IsOpen s := by
obtain ⟨u, v, hu, hv, hsu, htv, huv⟩ := hst
suffices s = (s ∪ t) ∩ u from this ▸ hst'.inter hu
rw [union_inter_distrib_right, (huv.symm.mono_left htv).inter_eq, union_empty,
inter_eq_left.2 hsu] | lemma | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | isOpen_left_of_isOpen_union | null |
isOpen_right_of_isOpen_union (hst : SeparatedNhds s t) (hst' : IsOpen (s ∪ t)) : IsOpen t :=
hst.symm.isOpen_left_of_isOpen_union (union_comm _ _ ▸ hst') | lemma | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | isOpen_right_of_isOpen_union | null |
isOpen_union_iff (hst : SeparatedNhds s t) : IsOpen (s ∪ t) ↔ IsOpen s ∧ IsOpen t :=
⟨fun h ↦ ⟨hst.isOpen_left_of_isOpen_union h, hst.isOpen_right_of_isOpen_union h⟩,
fun ⟨h1, h2⟩ ↦ h1.union h2⟩ | lemma | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | isOpen_union_iff | null |
isClosed_left_of_isClosed_union (hst : SeparatedNhds s t) (hst' : IsClosed (s ∪ t)) :
IsClosed s := by
obtain ⟨u, v, hu, hv, hsu, htv, huv⟩ := hst
rw [← isOpen_compl_iff] at hst' ⊢
suffices sᶜ = (s ∪ t)ᶜ ∪ v from this ▸ hst'.union hv
rw [← compl_inj_iff, Set.compl_union, compl_compl, compl_compl, union_inte... | lemma | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | isClosed_left_of_isClosed_union | null |
isClosed_right_of_isClosed_union (hst : SeparatedNhds s t) (hst' : IsClosed (s ∪ t)) :
IsClosed t :=
hst.symm.isClosed_left_of_isClosed_union (union_comm _ _ ▸ hst') | lemma | Topology | [
"Mathlib.Topology.Continuous",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Separation/SeparatedNhds.lean | isClosed_right_of_isClosed_union | null |
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