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_eventuallyEq ((hf x hx).mono_left <| nhdsWithin_mono _ hst)
((hg x hx).mono_left <| nhdsWithin_mono _ hst) (h.eventuallyEq_of_mem self_mem_nhdsWithin) | theorem | 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 => ⟨f x, this 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 [← diff_union_of_subset (interior_subset (s := s))]
refine .union_left ?_ ⟨interior s, sᶜ, isOpen_interior, H1.isOpen_compl, le_rfl,
H4.subset_compl_left, disjoint_compl_right.mono_left interior_subset⟩
rw [← H1.frontier_eq, frontier_eq_closure_inter_closure, H1.closure_eq]
refine .of_isCompact_isCompact ?_ ht (disjoint_of_subset_left inter_subset_left H4)
exact H2.of_isClosed_subset (H1.inter isClosed_closure) inter_subset_right | 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 (isCompact_univ_pi H.2) H.1 (subset_pi_eval_image univ s) | 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_upwards [inter_mem
(hf.preimage_mem_nhds (IsOpen.mem_nhds h₁U h₃U))
(hg.preimage_mem_nhds (IsOpen.mem_nhds h₂U h₄U))]
intro x hx
simp only [Set.mem_inter_iff, Set.mem_preimage] at hx
by_contra H
rw [H] at hx
have : g x ∈ Uf ∩ Ug := hx
simp [h₅U] at this
· obtain ⟨t, h₁t, h₂t, h₃t⟩ := eventually_nhds_iff.1 hfg
exact h₁t x h₃t | 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, ha⟩ : {x | f x ≠ g x ∧ f x = g x}.Nonempty := by
have h₁ := (eventually_nhdsWithin_of_eventually_nhds
((hf.ne_iff_eventually_ne hg).1 hCon)).and hfg
have h₂ : ∅ ∉ 𝓝[≠] x := by exact empty_notMem (𝓝[≠] x)
simp_all
simp at ha
· exact hfg.filter_mono nhdsWithin_le_nhds
@[deprecated (since := "2025-05-22")]
alias ContinuousAt.eventuallyEq_nhd_iff_eventuallyEq_nhdNE :=
ContinuousAt.eventuallyEq_nhds_iff_eventuallyEq_nhdsNE | 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⟩ ⟨y, hy, hyV⟩).mono inter_subset_right).not_disjoint h'
· intro h u v hu hv ⟨x, hxs, hxu⟩ ⟨y, hys, hyv⟩
exact ⟨x, hxs, hxu, (specializes_iff_mem_closure.mpr (h x hxs hys)).mem_open hv hyv⟩ | 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.equivFnOfDiscrete.cardinal_eq, mk_arrow, mk_real, lift_continuum,
lift_uzero]
exact (power_le_power_left two_ne_zero h).trans (power_le_power_right (nat_lt_continuum 2).le)
_ ≤ #C(X, ℝ) := by
choose f hf using ContinuousMap.exists_restrict_eq (Y := ℝ) hs
have hfi : Injective f := LeftInverse.injective hf
exact mk_le_of_injective hfi
_ ≤ #C(t, ℝ) := mk_le_of_injective <| ContinuousMap.injective_restrict htd
_ ≤ #(t → ℝ) := mk_le_of_injective DFunLike.coe_injective
_ ≤ 𝔠 := by
rw [mk_arrow, mk_real, lift_uzero, lift_continuum, continuum, ← power_mul]
exact power_le_power_left two_ne_zero mk_le_aleph0 | 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 x hx.1
exact ⟨V, Vᶜ, hV.1.isOpen, hV.1.compl.isOpen, hV.2.1, notMem_subset hV.2.2 hx.2,
(union_compl_self V).superset, disjoint_compl_right⟩
· choose V hV using h.isOpen_iff.mp hU₀ y hy.1
exact ⟨Vᶜ, V, hV.1.compl.isOpen, hV.1.isOpen, notMem_subset hV.2.2 hy.2, hV.2.1,
(union_comm _ _ ▸ union_compl_self V).superset, disjoint_compl_left⟩
@[deprecated (since := "2025-09-11")]
alias totallySeparatedSpace_of_t1_of_basis_clopen := totallySeparatedSpace_of_t0_of_basis_clopen
variable [T2Space X] [CompactSpace X] [TotallyDisconnectedSpace X] | 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 N := { s // IsClopen s ∧ x ∈ s }
rsuffices ⟨⟨s, hs, hs'⟩, hs''⟩ : ∃ s : N, s.val ⊆ U
· exact ⟨s, ⟨hs', hs⟩, hs''⟩
haveI : Nonempty N := ⟨⟨univ, isClopen_univ, mem_univ x⟩⟩
have hNcl : ∀ s : N, IsClosed s.val := fun s => s.property.1.1
have hdir : Directed Superset fun s : N => s.val := by
rintro ⟨s, hs, hxs⟩ ⟨t, ht, hxt⟩
exact ⟨⟨s ∩ t, hs.inter ht, ⟨hxs, hxt⟩⟩, inter_subset_left, inter_subset_right⟩
have h_nhds : ∀ y ∈ ⋂ s : N, s.val, U ∈ 𝓝 y := fun y y_in => by
rw [hx, mem_singleton_iff] at y_in
rwa [y_in]
exact exists_subset_nhds_of_compactSpace hdir hNcl h_nhds
· rintro ⟨V, ⟨hxV, -, V_op⟩, hUV : V ⊆ U⟩
rw [mem_nhds_iff]
exact ⟨V, hUV, V_op, hxV⟩⟩ | 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
tauto | 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
have u_open_in_s : IsOpen u := isOpen_interior.preimage continuous_subtype_val
lift x to s using interior_subset xs
haveI : CompactSpace s := isCompact_iff_compactSpace.1 comp
obtain ⟨V : Set s, VisClopen, Vx, V_sub⟩ := compact_exists_isClopen_in_isOpen u_open_in_s xs
have VisClopen' : IsClopen (((↑) : s → H) '' V) := by
refine ⟨comp.isClosed.isClosedEmbedding_subtypeVal.isClosed_iff_image_isClosed.1 VisClopen.1,
?_⟩
let v : Set u := ((↑) : u → s) ⁻¹' V
have : ((↑) : u → H) = ((↑) : s → H) ∘ ((↑) : u → s) := rfl
have f0 : IsEmbedding ((↑) : u → H) := IsEmbedding.subtypeVal.comp IsEmbedding.subtypeVal
have f1 : IsOpenEmbedding ((↑) : u → H) := by
refine ⟨f0, ?_⟩
· have : Set.range ((↑) : u → H) = interior s := by
rw [this, Set.range_comp, Subtype.range_coe, Subtype.image_preimage_coe]
apply Set.inter_eq_self_of_subset_right interior_subset
rw [this]
apply isOpen_interior
have f2 : IsOpen v := VisClopen.2.preimage continuous_subtype_val
have f3 : ((↑) : s → H) '' V = ((↑) : u → H) '' v := by
rw [this, image_comp, Subtype.image_preimage_coe, inter_eq_self_of_subset_right V_sub]
rw [f3]
apply f1.isOpenMap v f2
use (↑) '' V, VisClopen', by simp [Vx], Subset.trans (by simp) sU | 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' closure ≤ 𝓝 x,
∀ x : X, (𝓝 x).lift' closure = 𝓝 x] := by
tfae_have 1 ↔ 5 := by
rw [regularSpace_iff, (@compl_surjective (Set X) _).forall, forall_swap]
simp only [isClosed_compl_iff, mem_compl_iff, Classical.not_not, @and_comm (_ ∈ _),
(nhds_basis_opens _).lift'_closure.le_basis_iff (nhds_basis_opens _), and_imp,
(nhds_basis_opens _).disjoint_iff_right, ← subset_interior_iff_mem_nhdsSet,
interior_compl, compl_subset_compl]
tfae_have 5 → 6 := fun h a => (h a).antisymm (𝓝 _).le_lift'_closure
tfae_have 6 → 4
| H, a, s, hs => by
rw [← H] at hs
rcases (𝓝 a).basis_sets.lift'_closure.mem_iff.mp hs with ⟨U, hU, hUs⟩
exact ⟨closure U, mem_of_superset hU subset_closure, isClosed_closure, hUs⟩
tfae_have 4 → 2
| H, s, a, ha => by
have ha' : sᶜ ∈ 𝓝 a := by rwa [← mem_interior_iff_mem_nhds, interior_compl]
rcases H _ _ ha' with ⟨U, hU, hUc, hUs⟩
refine disjoint_of_disjoint_of_mem disjoint_compl_left ?_ hU
rwa [← subset_interior_iff_mem_nhdsSet, hUc.isOpen_compl.interior_eq, subset_compl_comm]
tfae_have 2 → 3 := by
refine fun H a s => ⟨fun hd has => mem_closure_iff_nhds_ne_bot.mp has ?_, H s a⟩
exact (hd.symm.mono_right <| @principal_le_nhdsSet _ _ s).eq_bot
tfae_have 3 → 1 := fun H => ⟨fun hs ha => (H _ _).mpr <| hs.closure_eq.symm ▸ ha⟩
tfae_finish | 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 ∉ closure s := by
have h := (regularSpace_TFAE X).out 0 2
exact h.mp ‹_› _ _
theorem disjoint_nhds_nhdsSet : Disjoint (𝓝 x) (𝓝ˢ s) ↔ x ∉ closure s :=
disjoint_comm.trans disjoint_nhdsSet_nhds
/-- A regular space is R₁. -/
instance (priority := 100) : R1Space X where
specializes_or_disjoint_nhds _ _ := or_iff_not_imp_left.2 fun h ↦ by
rwa [← nhdsSet_singleton, disjoint_nhdsSet_nhds, ← specializes_iff_mem_closure]
theorem exists_mem_nhds_isClosed_subset {x : X} {s : Set X} (h : s ∈ 𝓝 x) :
∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s := by
have h' := (regularSpace_TFAE X).out 0 3
exact h'.mp ‹_› _ _ h
theorem closed_nhds_basis (x : X) : (𝓝 x).HasBasis (fun s : Set X => s ∈ 𝓝 x ∧ IsClosed s) id :=
hasBasis_self.2 fun _ => exists_mem_nhds_isClosed_subset
theorem lift'_nhds_closure (x : X) : (𝓝 x).lift' closure = 𝓝 x :=
(closed_nhds_basis x).lift'_closure_eq_self fun _ => And.right
theorem Filter.HasBasis.nhds_closure {ι : Sort*} {x : X} {p : ι → Prop} {s : ι → Set X}
(h : (𝓝 x).HasBasis p s) : (𝓝 x).HasBasis p fun i => closure (s i) :=
lift'_nhds_closure x ▸ h.lift'_closure
theorem hasBasis_nhds_closure (x : X) : (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x) closure :=
(𝓝 x).basis_sets.nhds_closure
theorem hasBasis_opens_closure (x : X) : (𝓝 x).HasBasis (fun s => x ∈ s ∧ IsOpen s) closure :=
(nhds_basis_opens x).nhds_closure
theorem IsCompact.exists_isOpen_closure_subset {K U : Set X} (hK : IsCompact K) (hU : U ∈ 𝓝ˢ K) :
∃ V, IsOpen V ∧ K ⊆ V ∧ closure V ⊆ U := by
have hd : Disjoint (𝓝ˢ K) (𝓝ˢ Uᶜ) := by
simpa [hK.disjoint_nhdsSet_left, disjoint_nhds_nhdsSet,
← subset_interior_iff_mem_nhdsSet] using hU
rcases ((hasBasis_nhdsSet _).disjoint_iff (hasBasis_nhdsSet _)).1 hd
with ⟨V, ⟨hVo, hKV⟩, W, ⟨hW, hUW⟩, hVW⟩
refine ⟨V, hVo, hKV, Subset.trans ?_ (compl_subset_comm.1 hUW)⟩
exact closure_minimal hVW.subset_compl_right hW.isClosed_compl
theorem IsCompact.lift'_closure_nhdsSet {K : Set X} (hK : IsCompact K) :
(𝓝ˢ K).lift' closure = 𝓝ˢ K := by
refine le_antisymm (fun U hU ↦ ?_) (le_lift'_closure _)
rcases hK.exists_isOpen_closure_subset hU with ⟨V, hVo, hKV, hVU⟩
exact mem_of_superset (mem_lift' <| hVo.mem_nhdsSet.2 hKV) hVU
theorem TopologicalSpace.IsTopologicalBasis.nhds_basis_closure {B : Set (Set X)}
(hB : IsTopologicalBasis B) (x : X) :
(𝓝 x).HasBasis (fun s : Set X => x ∈ s ∧ s ∈ B) closure := by
simpa only [and_comm] using hB.nhds_hasBasis.nhds_closure
theorem TopologicalSpace.IsTopologicalBasis.exists_closure_subset {B : Set (Set X)}
(hB : IsTopologicalBasis B) {x : X} {s : Set X} (h : s ∈ 𝓝 x) :
∃ t ∈ B, x ∈ t ∧ closure t ⊆ s := by
simpa only [exists_prop, and_assoc] using hB.nhds_hasBasis.nhds_closure.mem_iff.mp h
protected theorem Topology.IsInducing.regularSpace [TopologicalSpace Y] {f : Y → X}
(hf : IsInducing f) : RegularSpace Y :=
.of_hasBasis
(fun b => by rw [hf.nhds_eq_comap b]; exact (closed_nhds_basis _).comap _)
fun b s hs => by exact hs.2.preimage hf.continuous
theorem regularSpace_induced (f : Y → X) : @RegularSpace Y (induced f ‹_›) :=
letI := induced f ‹_›
(IsInducing.induced f).regularSpace
theorem regularSpace_sInf {X} {T : Set (TopologicalSpace X)} (h : ∀ t ∈ T, @RegularSpace X t) :
@RegularSpace X (sInf T) := by
let _ := sInf T
have : ∀ a, (𝓝 a).HasBasis
(fun If : Σ I : Set T, I → Set X =>
If.1.Finite ∧ ∀ i : If.1, If.2 i ∈ @nhds X i a ∧ @IsClosed X i (If.2 i))
fun If => ⋂ i : If.1, If.snd i := fun a ↦ by
rw [nhds_sInf, ← iInf_subtype'']
exact .iInf fun t : T => @closed_nhds_basis X t (h t t.2) a
refine .of_hasBasis this fun a If hIf => isClosed_iInter fun i => ?_
exact (hIf.2 i).2.mono (sInf_le (i : T).2)
theorem regularSpace_iInf {ι X} {t : ι → TopologicalSpace X} (h : ∀ i, @RegularSpace X (t i)) :
@RegularSpace X (iInf t) :=
regularSpace_sInf <| forall_mem_range.mpr h
theorem RegularSpace.inf {X} {t₁ t₂ : TopologicalSpace X} (h₁ : @RegularSpace X t₁)
(h₂ : @RegularSpace X t₂) : @RegularSpace X (t₁ ⊓ t₂) := by
rw [inf_eq_iInf]
exact regularSpace_iInf (Bool.forall_bool.2 ⟨h₂, h₁⟩)
instance {p : X → Prop} : RegularSpace (Subtype p) :=
IsEmbedding.subtypeVal.isInducing.regularSpace
instance [TopologicalSpace Y] [RegularSpace Y] : RegularSpace (X × Y) :=
(regularSpace_induced (@Prod.fst X Y)).inf (regularSpace_induced (@Prod.snd X Y))
instance {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i, RegularSpace (X i)] :
RegularSpace (∀ i, X i) :=
regularSpace_iInf fun _ => regularSpace_induced _
/-- In a regular space, if a compact set and a closed set are disjoint, then they have disjoint
neighborhoods. |
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)]
exact hasSeparatingCovers_iff_separatedNhds.mpr (SeparatedNhds.empty_left t) |>.1
have (a : X) : ∃ n : Set X, IsOpen n ∧ Disjoint (closure n) t ∧ (a ∈ s → a ∈ n) := by
wlog ains : a ∈ s
· exact ⟨∅, isOpen_empty, SeparatedNhds.empty_left t |>.disjoint_closure_left, fun a ↦ ains a⟩
obtain ⟨n, nna, ncl, nsubkc⟩ := ((regularSpace_TFAE X).out 0 3 :).mp ‹RegularSpace X› a tᶜ <|
t_cl.compl_mem_nhds (disjoint_left.mp st_dis ains)
exact
⟨interior n,
isOpen_interior,
disjoint_left.mpr fun ⦃_⦄ ain ↦
nsubkc <| (IsClosed.closure_subset_iff ncl).mpr interior_subset ain,
fun _ ↦ mem_interior_iff_mem_nhds.mpr nna⟩
choose u u_open u_dis u_nhds using this
obtain ⟨f, f_cov⟩ := s_cl.isLindelof.indexed_countable_subcover
u u_open (fun a ainh ↦ mem_iUnion.mpr ⟨a, u_nhds a ainh⟩)
exact ⟨u ∘ f, f_cov, fun n ↦ ⟨u_open (f n), u_dis (f n)⟩⟩ | 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.trans <| interior_mono subset_closure,
L_comp.closure_subset_of_isOpen hU LU⟩ | 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 : closure (interior L) ⊆ L := by
apply (closure_mono interior_subset).trans (le_of_eq L_closed.closure_eq)
refine ⟨interior L, isOpen_interior, KL, A.trans LU, ?_⟩
exact L_compact.closure_of_subset interior_subset | 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
(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_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 (U₁_op.mem_nhds x_in) with ⟨V₁, V₁_in, V₁_closed, h₁⟩
rcases exists_mem_nhds_isClosed_subset (U₂_op.mem_nhds y_in) with ⟨V₂, V₂_in, V₂_closed, h₂⟩
exact ⟨U₁, mem_of_superset V₁_in h₁, V₁, V₁_in, U₂, mem_of_superset V₂_in h₂, V₂, V₂_in,
V₁_closed, V₂_closed, U₁_op, U₂_op, h₁, h₂, H⟩
open SeparationQuotient | 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 continuous_mk) ha
end T3
section NormalSpace
/-- A topological space is said to be a *normal space* if any two disjoint closed sets
have disjoint open neighborhoods. |
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', ht', hss', htt', hs't'⟩
refine ⟨s', hs', hss', Subset.trans (closure_minimal ?_ (isClosed_compl_iff.2 ht'))
(compl_subset_comm.1 htt')⟩
exact fun x hxs hxt => hs't'.le_bot ⟨hxs, hxt⟩ | 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)
(disjoint_image_of_injective hf.injective hst)
exact (H.preimage hf.continuous).mono (subset_preimage_image _ _) (subset_preimage_image _ _) | 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] : NormalSpace X where
normal _ _ hcl kcl hkdis :=
hasSeparatingCovers_iff_separatedNhds.mp
⟨hcl.HasSeparatingCover kcl hkdis, kcl.HasSeparatingCover hcl (Disjoint.symm hkdis)⟩
instance (priority := 100) NormalSpace.of_regularSpace_secondCountableTopology
[RegularSpace X] [SecondCountableTopology X] : NormalSpace X :=
of_regularSpace_lindelofSpace
end NormalSpace
section Normality
/-- A T₄ space is a normal T₁ space. |
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)
export CompletelyNormalSpace (completely_normal) | 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 continuous_mk) (ht.preimage continuous_mk)
(hd.preimage mk)
end SeparationQuotient
variable (X)
end Normality
section CompletelyNormal
/-- A topological space `X` is a *completely normal space* provided that for any two sets `s`, `t`
such that if both `closure s` is disjoint with `t`, and `s` is disjoint with `closure t`,
then there exist disjoint neighbourhoods of `s` and `t`. |
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.IsEmbedding.completelyNormalSpace [TopologicalSpace Y] [CompletelyNormalSpace Y]
{e : X → Y} (he : IsEmbedding e) : CompletelyNormalSpace X := by
refine ⟨fun s t hd₁ hd₂ => ?_⟩
simp only [he.isInducing.nhdsSet_eq_comap]
refine disjoint_comap (completely_normal ?_ ?_)
· rwa [← subset_compl_iff_disjoint_left, image_subset_iff, preimage_compl,
← he.closure_eq_preimage_closure_image, subset_compl_iff_disjoint_left]
· rwa [← subset_compl_iff_disjoint_right, image_subset_iff, preimage_compl,
← he.closure_eq_preimage_closure_image, subset_compl_iff_disjoint_right]
/-- A subspace of a completely normal space is a completely normal space. -/
instance [CompletelyNormalSpace X] {p : X → Prop} : CompletelyNormalSpace { x // p x } :=
IsEmbedding.subtypeVal.completelyNormalSpace
instance ULift.instCompletelyNormalSpace [CompletelyNormalSpace X] :
CompletelyNormalSpace (ULift X) :=
IsEmbedding.uliftDown.completelyNormalSpace
/-- A T₅ space is a completely normal T₁ space. |
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 : @IsClosed X _ (⋂ s : { s : Set X // IsClopen s ∧ x ∈ s }, s) :=
isClosed_iInter fun s => s.2.1.1
rw [isPreconnected_iff_subset_of_fully_disjoint_closed hs]
intro a b ha hb hab ab_disj
rcases normal_separation ha hb ab_disj with ⟨u, v, hu, hv, hau, hbv, huv⟩
obtain ⟨s, H⟩ : ∃ s : Set X, IsClopen s ∧ x ∈ s ∧ s ⊆ u ∪ v := by
/- Now we find a clopen set `s` around `x`, contained in `u ∪ v`. We utilize the fact that
`X \ u ∪ v` will be compact, so there must be some finite intersection of clopen neighbourhoods
of `X` disjoint to it, but a finite intersection of clopen sets is clopen,
so we let this be our `s`. -/
have H1 := (hu.union hv).isClosed_compl.isCompact.inter_iInter_nonempty
(fun s : { s : Set X // IsClopen s ∧ x ∈ s } => s) fun s => s.2.1.1
rw [← not_disjoint_iff_nonempty_inter, imp_not_comm, not_forall] at H1
obtain ⟨si, H2⟩ :=
H1 (disjoint_compl_left_iff_subset.2 <| hab.trans <| union_subset_union hau hbv)
refine ⟨⋂ U ∈ si, Subtype.val U, ?_, ?_, ?_⟩
· exact isClopen_biInter_finset fun s _ => s.2.1
· exact mem_iInter₂.2 fun s _ => s.2.2
· rwa [← disjoint_compl_left_iff_subset, disjoint_iff_inter_eq_empty,
← not_nonempty_iff_eq_empty]
· have H1 := isClopen_inter_of_disjoint_cover_clopen H.1 H.2.2 hu hv huv
rw [union_comm] at H
have H2 := isClopen_inter_of_disjoint_cover_clopen H.1 H.2.2 hv hu huv.symm
by_cases hxu : x ∈ u <;> [left; right]
· suffices ⋂ s : { s : Set X // IsClopen s ∧ x ∈ s }, ↑s ⊆ u
from Disjoint.left_le_of_le_sup_right hab (huv.mono this hbv)
· apply Subset.trans _ s.inter_subset_right
exact iInter_subset (fun s : { s : Set X // IsClopen s ∧ x ∈ s } => s.1)
⟨s ∩ u, H1, mem_inter H.2.1 hxu⟩
· have h1 : x ∈ v :=
(hab.trans (union_subset_union hau hbv) (mem_iInter.2 fun i => i.2.2)).resolve_left hxu
suffices ⋂ s : { s : Set X // IsClopen s ∧ x ∈ s }, ↑s ⊆ v
from (huv.symm.mono this hau).left_le_of_le_sup_left hab
· refine Subset.trans ?_ s.inter_subset_right
exact iInter_subset (fun s : { s : Set X // IsClopen s ∧ x ∈ s } => s.1)
⟨s ∩ v, H2, mem_inter H.2.1 h1⟩ | 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 [T5Space X] : T5Space (ULift X) :=
IsEmbedding.uliftDown.t5Space
open SeparationQuotient
/-- The `SeparationQuotient` of a completely normal R₀ space is a T₅ space. -/
instance [CompletelyNormalSpace X] [R0Space X] : T5Space (SeparationQuotient X) where
t1 := by
rwa [((t1Space_TFAE (SeparationQuotient X)).out 1 0 :), SeparationQuotient.t1Space_iff]
completely_normal s t hd₁ hd₂ := by
rw [← disjoint_comap_iff surjective_mk, comap_mk_nhdsSet, comap_mk_nhdsSet]
apply completely_normal <;> rw [← preimage_mk_closure]
exacts [hd₁.preimage mk, hd₂.preimage mk]
end CompletelyNormal
/-- In a compact T₂ space, the connected component of a point equals the intersection of all
its clopen neighbourhoods. |
@[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 [connectedComponent_eq_iInter_isClopen b, disjoint_iff_inter_eq_empty] at h
obtain ⟨U, V, hU, ha, hb, rfl⟩ : ∃ (U : Set X) (V : Set (ConnectedComponents X)),
IsClopen U ∧ connectedComponent a ∩ U = ∅ ∧ connectedComponent b ⊆ U ∧ (↑) ⁻¹' V = U := by
have h :=
(isClosed_connectedComponent (α := X)).isCompact.elim_finite_subfamily_closed
_ (fun s : { s : Set X // IsClopen s ∧ b ∈ s } => s.2.1.1) h
obtain ⟨fin_a, ha⟩ := h
set U : Set X := ⋂ (i : { s // IsClopen s ∧ b ∈ s }) (_ : i ∈ fin_a), i
have hU : IsClopen U := isClopen_biInter_finset fun i _ => i.2.1
exact ⟨U, (↑) '' U, hU, ha, subset_iInter₂ fun s _ => s.2.1.connectedComponent_subset s.2.2,
(connectedComponents_preimage_image U).symm ▸ hU.biUnion_connectedComponent_eq⟩
rw [ConnectedComponents.isQuotientMap_coe.isClopen_preimage] at hU
refine ⟨Vᶜ, V, hU.compl.isOpen, hU.isOpen, ?_, hb mem_connectedComponent, disjoint_compl_left⟩
exact fun h => flip Set.Nonempty.ne_empty ha ⟨a, mem_connectedComponent, h⟩ | 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 _ _ u₀i_open ↦
isOpen_iUnion fun i ↦ (u₀i_open i).sdiff isClosed_closure
have cover_lemma : ∀ (h₀ : Set X) (u₀ v₀ : ℕ → Set X),
(h₀ ⊆ ⋃ n, u₀ n) → (∀ n, Disjoint (closure (v₀ n)) h₀) →
(h₀ ⊆ ⋃ n, u₀ n \ closure (⋃ m ≤ n, v₀ m)) :=
fun h₀ u₀ v₀ h₀_cov dis x xinh ↦ by
rcases h₀_cov xinh with ⟨un, ⟨n, rfl⟩, xinun⟩
simp only [mem_iUnion]
refine ⟨n, xinun, ?_⟩
simp_all only [closure_iUnion₂_le_nat, disjoint_right, mem_iUnion,
exists_false, not_false_eq_true]
refine
⟨⋃ n : ℕ, u n \ (closure (⋃ m ≤ n, v m)),
⋃ n : ℕ, v n \ (closure (⋃ m ≤ n, u m)),
open_lemma u (fun n ↦ ⋃ m ≤ n, v m) (fun n ↦ (u_props n).1),
open_lemma v (fun n ↦ ⋃ m ≤ n, u m) (fun n ↦ (v_props n).1),
cover_lemma s u v u_cov (fun n ↦ (v_props n).2),
cover_lemma t v u v_cov (fun n ↦ (u_props n).2),
?_⟩
rw [Set.disjoint_left]
rintro x ⟨un, ⟨n, rfl⟩, xinun⟩
suffices ∀ (m : ℕ), x ∈ v m → x ∈ closure (⋃ m' ∈ {m' | m' ≤ m}, u m') by simpa
intro m xinvm
have n_le_m : n ≤ m := by
by_contra m_gt_n
exact xinun.2 (subset_closure (mem_biUnion (le_of_lt (not_le.mp m_gt_n)) xinvm))
exact subset_closure (mem_biUnion n_le_m xinun.1)
· rintro ⟨U, V, U_open, V_open, h_sub_U, k_sub_V, UV_dis⟩
exact
⟨⟨fun _ ↦ U,
h_sub_U.trans (iUnion_const U).symm.subset,
fun _ ↦
⟨U_open, disjoint_of_subset (fun ⦃a⦄ a ↦ a) k_sub_V (UV_dis.closure_left V_open)⟩⟩,
⟨fun _ ↦ V,
k_sub_V.trans (iUnion_const V).symm.subset,
fun _ ↦
⟨V_open, disjoint_of_subset (fun ⦃a⦄ a ↦ a) h_sub_U (UV_dis.closure_right U_open).symm⟩⟩⟩ | 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_props n).2⟩⟩ | 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 ∅ s :=
(empty_right _).symm | 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_inter_distrib_right,
(disjoint_compl_right.mono_left htv).inter_eq, union_empty, left_eq_inter, subset_compl_comm]
exact (huv.mono_left hsu).subset_compl_left | 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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