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 ⌀ |
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isDenseInducing_stoneCechUnit [CompletelyRegularSpace X] :
IsDenseInducing (stoneCechUnit : X → StoneCech X) where
toIsInducing := isInducing_stoneCechUnit
dense := denseRange_stoneCechUnit | lemma | Topology | [
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.Order.Lattice",
"Mathlib.Analysis.Real.Cardinality"
] | Mathlib/Topology/Separation/CompletelyRegular.lean | isDenseInducing_stoneCechUnit | null |
completelyRegularSpace_iff_isInducing_stoneCechUnit :
CompletelyRegularSpace X ↔ IsInducing (stoneCechUnit : X → StoneCech X) where
mp _ := isInducing_stoneCechUnit
mpr hs := hs.completelyRegularSpace
open TopologicalSpace Cardinal in | lemma | Topology | [
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.Order.Lattice",
"Mathlib.Analysis.Real.Cardinality"
] | Mathlib/Topology/Separation/CompletelyRegular.lean | completelyRegularSpace_iff_isInducing_stoneCechUnit | null |
CompletelyRegularSpace.isTopologicalBasis_clopens_of_cardinalMk_lt_continuum
[CompletelyRegularSpace X] (hX : Cardinal.mk X < continuum) :
IsTopologicalBasis {s : Set X | IsClopen s} := by
refine isTopologicalBasis_of_isOpen_of_nhds (fun x s ↦ IsClopen.isOpen s) (fun x s hxs hs ↦ ?_)
choose f hf using completely_regular_isOpen x s hs hxs
obtain ⟨hfc, hf₀, hf₁⟩ := hf
let R := Set.range f
have hR : lift.{u, 0} (Cardinal.mk R) < lift.{0, u} continuum := by
simpa [R] using mk_range_le_lift.trans_lt (lift_strictMono hX)
rw [lift_continuum, ← lift_continuum.{u, 0}, lift_lt, ← mk_Icc_real zero_lt_one, ← unitInterval]
at hR
obtain ⟨r, hr⟩ : ∃ r : I, r ∈ Rᶜ := compl_nonempty_of_mk_lt_mk hR
have hr' : ∀ (x : X), f x ≠ r := by simpa [R] using hr
have hrclopen : f ⁻¹' Iio r = f ⁻¹' Iic r := by
ext; simp [le_iff_lt_or_eq, hr']
refine ⟨f ⁻¹' Iio r, ⟨hrclopen ▸ isClosed_Iic.preimage hfc, isOpen_Iio.preimage hfc⟩, ?_, ?_⟩
· simp [hf₀, hrclopen]
· refine preimage_subset_iff.mpr (fun x ↦ ?_)
contrapose!; intro hxs
simpa [hf₁ hxs] using le_one' | theorem | Topology | [
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.Order.Lattice",
"Mathlib.Analysis.Real.Cardinality"
] | Mathlib/Topology/Separation/CompletelyRegular.lean | CompletelyRegularSpace.isTopologicalBasis_clopens_of_cardinalMk_lt_continuum | null |
@[mk_iff]
T35Space (X : Type u) [TopologicalSpace X] : Prop extends T0Space X, CompletelyRegularSpace X | class | Topology | [
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.Order.Lattice",
"Mathlib.Analysis.Real.Cardinality"
] | Mathlib/Topology/Separation/CompletelyRegular.lean | T35Space | A T₃.₅ space is a completely regular space that is also T₀. |
T35Space.instT3space [T35Space X] : T3Space X where | instance | Topology | [
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.Order.Lattice",
"Mathlib.Analysis.Real.Cardinality"
] | Mathlib/Topology/Separation/CompletelyRegular.lean | T35Space.instT3space | null |
T4Space.instT35Space [T4Space X] : T35Space X where | instance | Topology | [
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.Order.Lattice",
"Mathlib.Analysis.Real.Cardinality"
] | Mathlib/Topology/Separation/CompletelyRegular.lean | T4Space.instT35Space | null |
Topology.IsEmbedding.t35Space
{Y : Type v} [TopologicalSpace Y] [T35Space Y]
{f : X → Y} (hf : IsEmbedding f) : T35Space X :=
@T35Space.mk _ _ hf.t0Space hf.isInducing.completelyRegularSpace | lemma | Topology | [
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.Order.Lattice",
"Mathlib.Analysis.Real.Cardinality"
] | Mathlib/Topology/Separation/CompletelyRegular.lean | Topology.IsEmbedding.t35Space | null |
separatesPoints_continuous_of_t35Space [T35Space X] :
SeparatesPoints {f : X → ℝ | Continuous f} := by
intro x y x_ne_y
obtain ⟨f, f_cont, f_zero, f_one⟩ :=
CompletelyRegularSpace.completely_regular x {y} isClosed_singleton x_ne_y
exact ⟨fun x ↦ f x, continuous_subtype_val.comp f_cont, by simp_all⟩
@[deprecated (since := "2025-04-13")]
alias separatesPoints_continuous_of_completelyRegularSpace := separatesPoints_continuous_of_t35Space | lemma | Topology | [
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.Order.Lattice",
"Mathlib.Analysis.Real.Cardinality"
] | Mathlib/Topology/Separation/CompletelyRegular.lean | separatesPoints_continuous_of_t35Space | null |
separatesPoints_continuous_of_t35Space_Icc [T35Space X] :
SeparatesPoints {f : X → I | Continuous f} := by
intro x y x_ne_y
obtain ⟨f, f_cont, f_zero, f_one⟩ :=
CompletelyRegularSpace.completely_regular x {y} isClosed_singleton x_ne_y
exact ⟨f, f_cont, by simp_all⟩
@[deprecated (since := "2025-04-13")]
alias separatesPoints_continuous_of_completelyRegularSpace_Icc :=
separatesPoints_continuous_of_t35Space_Icc | lemma | Topology | [
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.Order.Lattice",
"Mathlib.Analysis.Real.Cardinality"
] | Mathlib/Topology/Separation/CompletelyRegular.lean | separatesPoints_continuous_of_t35Space_Icc | null |
injective_stoneCechUnit_of_t35Space [T35Space X] :
Function.Injective (stoneCechUnit : X → StoneCech X) := by
intro a b hab
contrapose hab
obtain ⟨f, fc, fab⟩ := separatesPoints_continuous_of_t35Space_Icc hab
exact fun q ↦ fab (eq_if_stoneCechUnit_eq fc q)
@[deprecated (since := "2025-04-13")]
alias injective_stoneCechUnit_of_completelyRegularSpace := injective_stoneCechUnit_of_t35Space | lemma | Topology | [
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.Order.Lattice",
"Mathlib.Analysis.Real.Cardinality"
] | Mathlib/Topology/Separation/CompletelyRegular.lean | injective_stoneCechUnit_of_t35Space | null |
isEmbedding_stoneCechUnit [T35Space X] :
IsEmbedding (stoneCechUnit : X → StoneCech X) where
toIsInducing := isInducing_stoneCechUnit
injective := injective_stoneCechUnit_of_t35Space | lemma | Topology | [
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.Order.Lattice",
"Mathlib.Analysis.Real.Cardinality"
] | Mathlib/Topology/Separation/CompletelyRegular.lean | isEmbedding_stoneCechUnit | null |
isDenseEmbedding_stoneCechUnit [T35Space X] :
IsDenseEmbedding (stoneCechUnit : X → StoneCech X) where
toIsDenseInducing := isDenseInducing_stoneCechUnit
injective := injective_stoneCechUnit_of_t35Space | lemma | Topology | [
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.Order.Lattice",
"Mathlib.Analysis.Real.Cardinality"
] | Mathlib/Topology/Separation/CompletelyRegular.lean | isDenseEmbedding_stoneCechUnit | null |
t35Space_iff_isEmbedding_stoneCechUnit :
T35Space X ↔ IsEmbedding (stoneCechUnit : X → StoneCech X) where
mp _ := isEmbedding_stoneCechUnit
mpr hs := hs.t35Space | lemma | Topology | [
"Mathlib.Topology.UrysohnsLemma",
"Mathlib.Topology.UnitInterval",
"Mathlib.Topology.Compactification.StoneCech",
"Mathlib.Topology.Order.Lattice",
"Mathlib.Analysis.Real.Cardinality"
] | Mathlib/Topology/Separation/CompletelyRegular.lean | t35Space_iff_isEmbedding_stoneCechUnit | null |
PreconnectedSpace.trivial_of_discrete [PreconnectedSpace X] [DiscreteTopology X] :
Subsingleton X := by
by_contra! h
rcases h with ⟨x, y, hxy⟩
rw [Ne, ← mem_singleton_iff, (isClopen_discrete _).eq_univ <| singleton_nonempty y] at hxy
exact hxy (mem_univ x) | theorem | Topology | [
"Mathlib.Topology.Separation.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Separation/Connected.lean | PreconnectedSpace.trivial_of_discrete | null |
IsPreconnected.infinite_of_nontrivial [T1Space X] {s : Set X} (h : IsPreconnected s)
(hs : s.Nontrivial) : s.Infinite := by
refine mt (fun hf => (subsingleton_coe s).mp ?_) (not_subsingleton_iff.mpr hs)
haveI := @Finite.instDiscreteTopology s _ _ hf.to_subtype
exact @PreconnectedSpace.trivial_of_discrete _ _ (Subtype.preconnectedSpace h) _ | theorem | Topology | [
"Mathlib.Topology.Separation.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Separation/Connected.lean | IsPreconnected.infinite_of_nontrivial | null |
PreconnectedSpace.infinite [PreconnectedSpace X] [Nontrivial X] [T1Space X] : Infinite X :=
infinite_univ_iff.mp <| isPreconnected_univ.infinite_of_nontrivial nontrivial_univ
@[deprecated (since := "2025-03-21")]
alias ConnectedSpace.infinite := PreconnectedSpace.infinite | theorem | Topology | [
"Mathlib.Topology.Separation.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Separation/Connected.lean | PreconnectedSpace.infinite | null |
exists_finite_clopen_cover (hU : IsOpenCover U) : ∃ (n : ℕ) (V : Fin n → Clopens X),
(∀ j, ∃ i, (V j : Set X) ⊆ U i) ∧ univ ⊆ ⋃ j, (V j : Set X) := by
choose r hr using hU.exists_mem
choose V hV hVx hVU using fun x ↦ compact_exists_isClopen_in_isOpen (U _).isOpen (hr x)
obtain ⟨t, ht⟩ : ∃ t, univ ⊆ ⋃ i ∈ t, V i :=
isCompact_univ.elim_finite_subcover V (fun x ↦ (hV x).2) (fun x _ ↦ mem_iUnion.mpr ⟨x, hVx x⟩)
refine ⟨_, fun j ↦ ⟨_, hV (t.equivFin.symm j)⟩, fun j ↦ ⟨_, hVU _⟩, fun x hx ↦ ?_⟩
obtain ⟨m, hm, hm'⟩ := mem_iUnion₂.mp (ht hx)
exact Set.mem_iUnion_of_mem (t.equivFin ⟨m, hm⟩) (by simpa) | lemma | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Order.Disjointed",
"Mathlib.Topology.Separation.Profinite",
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.Sets.OpenCover"
] | Mathlib/Topology/Separation/DisjointCover.lean | exists_finite_clopen_cover | Any open cover of a profinite space can be refined to a finite cover by clopens. |
exists_finite_nonempty_disjoint_clopen_cover (hU : IsOpenCover U) :
∃ (n : ℕ) (W : Fin n → Clopens X), (∀ j, W j ≠ ⊥ ∧ ∃ i, (W j : Set X) ⊆ U i)
∧ (univ : Set X) ⊆ ⋃ j, ↑(W j) ∧ Pairwise (Disjoint on W) := by
classical
obtain ⟨n, V, hVle, hVun⟩ := hU.exists_finite_clopen_cover
obtain ⟨W, hWle, hWun, hWd⟩ := Fintype.exists_disjointed_le V
simp only [← SetLike.coe_set_eq, Clopens.coe_finset_sup, Finset.mem_univ, iUnion_true] at hWun
let t : Finset (Fin n) := {j | W j ≠ ⊥}
refine ⟨#t, fun k ↦ W (t.equivFin.symm k), fun k ↦ ⟨?_, ?_⟩, fun x hx ↦ ?_, ?_⟩
· exact (Finset.mem_filter.mp (t.equivFin.symm k).2).2
· exact match hVle (t.equivFin.symm k) with | ⟨i, hi⟩ => ⟨i, subset_trans (hWle _) hi⟩
· obtain ⟨j, hj⟩ := mem_iUnion.mp <| (hWun ▸ hVun) hx
have : W j ≠ ⊥ := by simpa [← SetLike.coe_ne_coe, ← Set.nonempty_iff_ne_empty] using ⟨x, hj⟩
exact mem_iUnion.mpr ⟨t.equivFin ⟨j, Finset.mem_filter.mpr ⟨Finset.mem_univ _, this⟩⟩, by simpa⟩
· exact hWd.comp_of_injective <| Subtype.val_injective.comp t.equivFin.symm.injective | lemma | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Order.Disjointed",
"Mathlib.Topology.Separation.Profinite",
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.Sets.OpenCover"
] | Mathlib/Topology/Separation/DisjointCover.lean | exists_finite_nonempty_disjoint_clopen_cover | Any open cover of a profinite space can be refined to a finite cover by pairwise disjoint
nonempty clopens. |
exists_open_prod_subset_of_mem_nhds_diagonal (hS : S ∈ nhdsSet (diagonal X)) (x : X) :
∃ U : Set X, IsOpen U ∧ x ∈ U ∧ U ×ˢ U ⊆ S := by
have : S ∈ 𝓝 (x, x) := mem_nhdsSet_iff_forall.mp hS _ rfl
obtain ⟨u, v, huo, hux, hvo, hvx, H⟩ := by rwa [mem_nhds_prod_iff'] at this
exact ⟨_, huo.inter hvo, ⟨hux, hvx⟩, fun p hp ↦ H ⟨hp.1.1, hp.2.2⟩⟩
variable [CompactSpace X] | lemma | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Order.Disjointed",
"Mathlib.Topology.Separation.Profinite",
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.Sets.OpenCover"
] | Mathlib/Topology/Separation/DisjointCover.lean | exists_open_prod_subset_of_mem_nhds_diagonal | If `S` is any neighbourhood of the diagonal in a topological space `X`, any point of `X` has an
open neighbourhood `U` such that `U ×ˢ U ⊆ S`. |
exists_finite_open_cover_prod_subset_of_mem_nhds_diagonal_of_compact
(hS : S ∈ nhdsSet (diagonal X)) :
∃ (t : Finset X) (U : t → Opens X), IsOpenCover U ∧ ∀ i, (U i : Set X) ×ˢ U i ⊆ S := by
choose U hUo hUx hUp using exists_open_prod_subset_of_mem_nhds_diagonal hS
obtain ⟨t, ht⟩ := isCompact_univ.elim_finite_subcover _ hUo (fun x _ ↦ mem_iUnion.mpr ⟨_, hUx x⟩)
refine ⟨t, fun i ↦ ⟨_, hUo i⟩, .of_sets _ ?_, (hUp ·)⟩
simpa [iUnion_subtype, ← univ_subset_iff] using ht
variable [TotallyDisconnectedSpace X] [T2Space X] | lemma | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Order.Disjointed",
"Mathlib.Topology.Separation.Profinite",
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.Sets.OpenCover"
] | Mathlib/Topology/Separation/DisjointCover.lean | exists_finite_open_cover_prod_subset_of_mem_nhds_diagonal_of_compact | If `S` is any neighbourhood of the diagonal in a compact topological space `X`, then there
exists a finite cover of `X` by opens `U i` such that `U i ×ˢ U i ⊆ S` for all `i`.
That the indexing set is a finset of `X` is an artifact of the proof; it could be any finite type. |
private exists_finite_disjoint_nonempty_clopen_cover_of_mem_nhds_diagonal_of_profinite
(hS : S ∈ nhdsSet (diagonal X)) :
∃ (n : ℕ) (D : Fin n → Clopens X), (∀ i, D i ≠ ⊥) ∧ (∀ i, ∀ y ∈ D i, ∀ z ∈ D i, (y, z) ∈ S)
∧ (univ : Set X) ⊆ ⋃ i, D i ∧ Pairwise (Disjoint on D) := by
obtain ⟨t, U, hUc, hUS⟩ := exists_finite_open_cover_prod_subset_of_mem_nhds_diagonal_of_compact hS
obtain ⟨n, W, hW₁, hW₂, hW₃⟩ := hUc.exists_finite_nonempty_disjoint_clopen_cover
refine ⟨n, W, fun j ↦ (hW₁ j).1, fun j y hy z hz ↦ ?_, hW₂, hW₃⟩
exact match (hW₁ j).2 with | ⟨i, hi⟩ => hUS i ⟨hi hy, hi hz⟩ | lemma | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Order.Disjointed",
"Mathlib.Topology.Separation.Profinite",
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.Sets.OpenCover"
] | Mathlib/Topology/Separation/DisjointCover.lean | exists_finite_disjoint_nonempty_clopen_cover_of_mem_nhds_diagonal_of_profinite | If `S` is any neighbourhood of the diagonal in a profinite topological space `X`, then there
exists a finite cover of `X` by disjoint nonempty clopens `U i` with `U i ×ˢ U i ⊆ S` for all `i`. |
exists_disjoint_nonempty_clopen_cover_of_mem_nhds_diagonal (hS : S ∈ nhdsSet (diagonal V)) :
∃ (n : ℕ) (D : Fin n → Clopens X), (∀ i, D i ≠ ⊥) ∧ (∀ i, ∀ y ∈ D i, ∀ z ∈ D i, (f y, f z) ∈ S)
∧ (univ : Set X) ⊆ ⋃ i, D i ∧ Pairwise (Disjoint on D) := by
have : (f.prodMap f) ⁻¹' S ∈ nhdsSet (diagonal X) := by
rw [mem_nhdsSet_iff_forall] at hS ⊢
rintro ⟨x, y⟩ (rfl : x = y)
exact (map_continuous _).continuousAt.preimage_mem_nhds (hS _ rfl)
exact exists_finite_disjoint_nonempty_clopen_cover_of_mem_nhds_diagonal_of_profinite this | lemma | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Order.Disjointed",
"Mathlib.Topology.Separation.Profinite",
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.Sets.OpenCover"
] | Mathlib/Topology/Separation/DisjointCover.lean | exists_disjoint_nonempty_clopen_cover_of_mem_nhds_diagonal | For any continuous function `f : X → V`, with `X` profinite, and `S` a neighbourhood of the
diagonal in `V × V`, there exists a finite cover of `X` by pairwise-disjoint nonempty clopens, on
each of which `f` varies within `S`. |
exists_finite_approximation_of_mem_nhds_diagonal (hS : S ∈ nhdsSet (diagonal V)) :
∃ (n : ℕ) (g : X → Fin n) (h : Fin n → V), Continuous g ∧ ∀ x, (f x, h (g x)) ∈ S := by
obtain ⟨n, E, hEne, hES, hEuniv, hEdis⟩ :=
exists_disjoint_nonempty_clopen_cover_of_mem_nhds_diagonal f hS
have h_uniq (x) : ∃! i, x ∈ E i := by
refine match mem_iUnion.mp (hEuniv <| mem_univ x) with
| ⟨i, hi⟩ => ⟨i, hi, fun j hj ↦ hEdis.eq ?_⟩
simpa [← Clopens.coe_disjoint, not_disjoint_iff] using ⟨x, hj, hi⟩
choose g hg hg' using h_uniq -- for each `x`, `g x` is the unique `i` such that `x ∈ E i`
have h_ex (i) : ∃ x, x ∈ E i := by
simpa [← SetLike.coe_set_eq, ← nonempty_iff_ne_empty] using hEne i
choose r hr using h_ex -- for each `i`, choose an `r i ∈ E i`
refine ⟨n, g, f ∘ r, continuous_discrete_rng.mpr fun j ↦ ?_, fun x ↦ (hES _) _ (hg _) _ (hr _)⟩
convert (E j).isOpen
exact Set.ext fun x ↦ ⟨fun hj ↦ hj ▸ hg x, fun hx ↦ (hg' _ _ hx).symm⟩ | lemma | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Order.Disjointed",
"Mathlib.Topology.Separation.Profinite",
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.Sets.OpenCover"
] | Mathlib/Topology/Separation/DisjointCover.lean | exists_finite_approximation_of_mem_nhds_diagonal | For any continuous function `f : X → V`, with `X` profinite, and `S` a neighbourhood of the
diagonal in `V × V`, the function `f` can be `S`-approximated by a function factoring through
`Fin n`, for some `n`. |
@[to_additive /-- If `f` is a continuous map from a profinite space to a topological space with a
commutative additive monoid structure, then we can approximate `f` by finite sums of indicator
functions of clopen sets.
(Note no compatibility is assumed between the monoid structure on `V` and the topology.) -/]
exists_finite_sum_const_mulIndicator_approximation_of_mem_nhds_diagonal [CommMonoid V]
(hS : S ∈ nhdsSet (diagonal V)) :
∃ (n : ℕ) (U : Fin n → Clopens X) (v : Fin n → V),
∀ x, (f x, ∏ n, mulIndicator (U n) (fun _ ↦ v n) x) ∈ S := by
obtain ⟨n, g, h, hg, hgh⟩ := exists_finite_approximation_of_mem_nhds_diagonal f hS
refine ⟨n, fun i ↦ ⟨_, (isClopen_discrete {i}).preimage hg⟩, h, fun x ↦ ?_⟩
convert hgh x
exact (Fintype.prod_eq_single _ fun i hi ↦ mulIndicator_of_notMem hi.symm _).trans
(mulIndicator_of_mem rfl _) | lemma | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Order.Disjointed",
"Mathlib.Topology.Separation.Profinite",
"Mathlib.Topology.Sets.Closeds",
"Mathlib.Topology.Sets.OpenCover"
] | Mathlib/Topology/Separation/DisjointCover.lean | exists_finite_sum_const_mulIndicator_approximation_of_mem_nhds_diagonal | If `f` is a continuous map from a profinite space to a topological space with a commutative monoid
structure, then we can approximate `f` by finite products of indicator functions of clopen sets.
(Note no compatibility is assumed between the monoid structure on `V` and the topology.) |
IsGδ.compl_singleton (x : X) [T1Space X] : IsGδ ({x}ᶜ : Set X) :=
isOpen_compl_singleton.isGδ | theorem | Topology | [
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation.Regular",
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Separation/GDelta.lean | IsGδ.compl_singleton | null |
Set.Countable.isGδ_compl {s : Set X} [T1Space X] (hs : s.Countable) : IsGδ sᶜ := by
rw [← biUnion_of_singleton s, compl_iUnion₂]
exact .biInter hs fun x _ => .compl_singleton x | theorem | Topology | [
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation.Regular",
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Separation/GDelta.lean | Set.Countable.isGδ_compl | null |
Set.Finite.isGδ_compl {s : Set X} [T1Space X] (hs : s.Finite) : IsGδ sᶜ :=
hs.countable.isGδ_compl | theorem | Topology | [
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation.Regular",
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Separation/GDelta.lean | Set.Finite.isGδ_compl | null |
Set.Subsingleton.isGδ_compl {s : Set X} [T1Space X] (hs : s.Subsingleton) : IsGδ sᶜ :=
hs.finite.isGδ_compl | theorem | Topology | [
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation.Regular",
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Separation/GDelta.lean | Set.Subsingleton.isGδ_compl | null |
Finset.isGδ_compl [T1Space X] (s : Finset X) : IsGδ (sᶜ : Set X) :=
s.finite_toSet.isGδ_compl | theorem | Topology | [
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation.Regular",
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Separation/GDelta.lean | Finset.isGδ_compl | null |
protected IsGδ.singleton [FirstCountableTopology X] [T1Space X] (x : X) :
IsGδ ({x} : Set X) := by
rcases (nhds_basis_opens x).exists_antitone_subbasis with ⟨U, hU, h_basis⟩
rw [← biInter_basis_nhds h_basis.toHasBasis]
exact .biInter (to_countable _) fun n _ => (hU n).2.isGδ | theorem | Topology | [
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation.Regular",
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Separation/GDelta.lean | IsGδ.singleton | null |
Set.Finite.isGδ [FirstCountableTopology X] {s : Set X} [T1Space X] (hs : s.Finite) :
IsGδ s :=
Finite.induction_on _ hs .empty fun _ _ ↦ .union (.singleton _) | theorem | Topology | [
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation.Regular",
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Separation/GDelta.lean | Set.Finite.isGδ | null |
PerfectlyNormalSpace (X : Type u) [TopologicalSpace X] : Prop extends NormalSpace X where
closed_gdelta : ∀ ⦃h : Set X⦄, IsClosed h → IsGδ h | class | Topology | [
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation.Regular",
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Separation/GDelta.lean | PerfectlyNormalSpace | A topological space `X` is a *perfectly normal space* provided it is normal and
closed sets are Gδ. |
Disjoint.hasSeparatingCover_closed_gdelta_right {s t : Set X} [NormalSpace X]
(st_dis : Disjoint s t) (t_cl : IsClosed t) (t_gd : IsGδ t) : HasSeparatingCover s t := by
obtain ⟨T, T_open, T_count, T_int⟩ := t_gd
rcases T.eq_empty_or_nonempty with rfl | T_nonempty
· rw [T_int, sInter_empty] at st_dis
rw [(s.disjoint_univ).mp st_dis]
exact t.hasSeparatingCover_empty_left
obtain ⟨g, g_surj⟩ := T_count.exists_surjective T_nonempty
choose g' g'_open clt_sub_g' clg'_sub_g using fun n ↦ by
apply normal_exists_closure_subset t_cl (T_open (g n).1 (g n).2)
rw [T_int]
exact sInter_subset_of_mem (g n).2
have clg'_int : t = ⋂ i, closure (g' i) := by
apply (subset_iInter fun n ↦ (clt_sub_g' n).trans subset_closure).antisymm
rw [T_int]
refine subset_sInter fun t tinT ↦ ?_
obtain ⟨n, gn⟩ := g_surj ⟨t, tinT⟩
refine iInter_subset_of_subset n <| (clg'_sub_g n).trans ?_
rw [gn]
use fun n ↦ (closure (g' n))ᶜ
constructor
· rw [← compl_iInter, subset_compl_comm, ← clg'_int]
exact st_dis.subset_compl_left
· refine fun n ↦ ⟨isOpen_compl_iff.mpr isClosed_closure, ?_⟩
simp only [closure_compl, disjoint_compl_left_iff_subset]
rw [← closure_eq_iff_isClosed.mpr t_cl] at clt_sub_g'
exact subset_closure.trans <| (clt_sub_g' n).trans <| (g'_open n).subset_interior_closure | theorem | Topology | [
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation.Regular",
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Separation/GDelta.lean | Disjoint.hasSeparatingCover_closed_gdelta_right | Lemma that allows the easy conclusion that perfectly normal spaces are completely normal. |
IsClosed.isGδ [PerfectlyNormalSpace X] {s : Set X} (hs : IsClosed s) : IsGδ s :=
PerfectlyNormalSpace.closed_gdelta hs | theorem | Topology | [
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation.Regular",
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Separation/GDelta.lean | IsClosed.isGδ | In a perfectly normal space, all closed sets are Gδ. |
T6Space (X : Type u) [TopologicalSpace X] : Prop extends T0Space X, PerfectlyNormalSpace X | class | Topology | [
"Mathlib.Topology.Compactness.Lindelof",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation.Regular",
"Mathlib.Topology.GDelta.Basic"
] | Mathlib/Topology/Separation/GDelta.lean | T6Space | A T₆ space is a perfectly normal T₀ space. |
@[mk_iff]
T2Space (X : Type u) [TopologicalSpace X] : Prop where
/-- Every two points in a Hausdorff space admit disjoint open neighbourhoods. -/
t2 : Pairwise fun x y => ∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v | class | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | T2Space | A T₂ space, also known as a Hausdorff space, is one in which for every
`x ≠ y` there exists disjoint open sets around `x` and `y`. This is
the most widely used of the separation axioms. |
t2_separation [T2Space X] {x y : X} (h : x ≠ y) :
∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v :=
T2Space.t2 h | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | t2_separation | Two different points can be separated by open sets. |
t2Space_iff_disjoint_nhds : T2Space X ↔ Pairwise fun x y : X => Disjoint (𝓝 x) (𝓝 y) := by
refine (t2Space_iff X).trans (forall₃_congr fun x y _ => ?_)
simp only [(nhds_basis_opens x).disjoint_iff (nhds_basis_opens y), ← exists_and_left,
and_assoc, and_comm, and_left_comm]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | t2Space_iff_disjoint_nhds | null |
disjoint_nhds_nhds [T2Space X] {x y : X} : Disjoint (𝓝 x) (𝓝 y) ↔ x ≠ y :=
⟨fun hd he => by simp [he, nhds_neBot.ne] at hd, (t2Space_iff_disjoint_nhds.mp ‹_› ·)⟩ | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | disjoint_nhds_nhds | null |
pairwise_disjoint_nhds [T2Space X] : Pairwise (Disjoint on (𝓝 : X → Filter X)) := fun _ _ =>
disjoint_nhds_nhds.2 | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | pairwise_disjoint_nhds | null |
protected Set.pairwiseDisjoint_nhds [T2Space X] (s : Set X) : s.PairwiseDisjoint 𝓝 :=
pairwise_disjoint_nhds.set_pairwise s | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Set.pairwiseDisjoint_nhds | null |
Set.Finite.t2_separation [T2Space X] {s : Set X} (hs : s.Finite) :
∃ U : X → Set X, (∀ x, x ∈ U x ∧ IsOpen (U x)) ∧ s.PairwiseDisjoint U :=
s.pairwiseDisjoint_nhds.exists_mem_filter_basis hs nhds_basis_opens | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Set.Finite.t2_separation | Points of a finite set can be separated by open sets from each other. |
SeparationQuotient.t2Space_iff : T2Space (SeparationQuotient X) ↔ R1Space X := by
simp only [t2Space_iff_disjoint_nhds, Pairwise, surjective_mk.forall₂, ne_eq, mk_eq_mk,
r1Space_iff_inseparable_or_disjoint_nhds, ← disjoint_comap_iff surjective_mk, comap_mk_nhds_mk,
← or_iff_not_imp_left] | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | SeparationQuotient.t2Space_iff | null |
SeparationQuotient.t2Space [R1Space X] : T2Space (SeparationQuotient X) :=
t2Space_iff.2 ‹_› | instance | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | SeparationQuotient.t2Space | null |
R1Space.t2Space_iff_t0Space [R1Space X] : T2Space X ↔ T0Space X := by
constructor <;> intro <;> infer_instance | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | R1Space.t2Space_iff_t0Space | null |
t2_iff_nhds : T2Space X ↔ ∀ {x y : X}, NeBot (𝓝 x ⊓ 𝓝 y) → x = y := by
simp only [t2Space_iff_disjoint_nhds, disjoint_iff, neBot_iff, Ne, not_imp_comm, Pairwise] | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | t2_iff_nhds | A space is T₂ iff the neighbourhoods of distinct points generate the bottom filter. |
eq_of_nhds_neBot [T2Space X] {x y : X} (h : NeBot (𝓝 x ⊓ 𝓝 y)) : x = y :=
t2_iff_nhds.mp ‹_› h | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | eq_of_nhds_neBot | null |
t2Space_iff_nhds :
T2Space X ↔ Pairwise fun x y : X => ∃ U ∈ 𝓝 x, ∃ V ∈ 𝓝 y, Disjoint U V := by
simp only [t2Space_iff_disjoint_nhds, Filter.disjoint_iff, Pairwise] | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | t2Space_iff_nhds | null |
t2_separation_nhds [T2Space X] {x y : X} (h : x ≠ y) :
∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ Disjoint u v :=
let ⟨u, v, open_u, open_v, x_in, y_in, huv⟩ := t2_separation h
⟨u, v, open_u.mem_nhds x_in, open_v.mem_nhds y_in, huv⟩ | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | t2_separation_nhds | null |
t2_separation_compact_nhds [LocallyCompactSpace X] [T2Space X] {x y : X} (h : x ≠ y) :
∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ IsCompact u ∧ IsCompact v ∧ Disjoint u v := by
simpa only [exists_prop, ← exists_and_left, and_comm, and_assoc, and_left_comm] using
((compact_basis_nhds x).disjoint_iff (compact_basis_nhds y)).1 (disjoint_nhds_nhds.2 h) | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | t2_separation_compact_nhds | null |
t2_iff_ultrafilter :
T2Space X ↔ ∀ {x y : X} (f : Ultrafilter X), ↑f ≤ 𝓝 x → ↑f ≤ 𝓝 y → x = y :=
t2_iff_nhds.trans <| by simp only [← exists_ultrafilter_iff, and_imp, le_inf_iff, exists_imp] | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | t2_iff_ultrafilter | null |
t2_iff_isClosed_diagonal : T2Space X ↔ IsClosed (diagonal X) := by
simp only [t2Space_iff_disjoint_nhds, ← isOpen_compl_iff, isOpen_iff_mem_nhds, Prod.forall,
nhds_prod_eq, compl_diagonal_mem_prod, mem_compl_iff, mem_diagonal_iff, Pairwise] | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | t2_iff_isClosed_diagonal | null |
isClosed_diagonal [T2Space X] : IsClosed (diagonal X) :=
t2_iff_isClosed_diagonal.mp ‹_› | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | isClosed_diagonal | null |
tendsto_nhds_unique [T2Space X] {f : Y → X} {l : Filter Y} {a b : X} [NeBot l]
(ha : Tendsto f l (𝓝 a)) (hb : Tendsto f l (𝓝 b)) : a = b :=
(tendsto_nhds_unique_inseparable ha hb).eq | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | tendsto_nhds_unique | null |
tendsto_nhds_unique' [T2Space X] {f : Y → X} {l : Filter Y} {a b : X} (_ : NeBot l)
(ha : Tendsto f l (𝓝 a)) (hb : Tendsto f l (𝓝 b)) : a = b :=
tendsto_nhds_unique ha hb | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | tendsto_nhds_unique' | null |
tendsto_nhds_unique_of_eventuallyEq [T2Space X] {f g : Y → X} {l : Filter Y} {a b : X}
[NeBot l] (ha : Tendsto f l (𝓝 a)) (hb : Tendsto g l (𝓝 b)) (hfg : f =ᶠ[l] g) : a = b :=
tendsto_nhds_unique (ha.congr' hfg) hb | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | tendsto_nhds_unique_of_eventuallyEq | null |
tendsto_nhds_unique_of_frequently_eq [T2Space X] {f g : Y → X} {l : Filter Y} {a b : X}
(ha : Tendsto f l (𝓝 a)) (hb : Tendsto g l (𝓝 b)) (hfg : ∃ᶠ x in l, f x = g x) : a = b :=
have : ∃ᶠ z : X × X in 𝓝 (a, b), z.1 = z.2 := (ha.prodMk_nhds hb).frequently hfg
not_not.1 fun hne => this (isClosed_diagonal.isOpen_compl.mem_nhds hne) | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | tendsto_nhds_unique_of_frequently_eq | null |
IsCompact.nhdsSet_inter_eq [T2Space X] {s t : Set X} (hs : IsCompact s) (ht : IsCompact t) :
𝓝ˢ (s ∩ t) = 𝓝ˢ s ⊓ 𝓝ˢ t := by
refine le_antisymm (nhdsSet_inter_le _ _) ?_
simp_rw [hs.nhdsSet_inf_eq_biSup, ht.inf_nhdsSet_eq_biSup, nhdsSet, sSup_image]
refine iSup₂_le fun x hxs ↦ iSup₂_le fun y hyt ↦ ?_
rcases eq_or_ne x y with (rfl | hne)
· exact le_iSup₂_of_le x ⟨hxs, hyt⟩ (inf_idem _).le
· exact (disjoint_nhds_nhds.mpr hne).eq_bot ▸ bot_le | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | IsCompact.nhdsSet_inter_eq | If `s` and `t` are compact sets in a T₂ space, then the set neighborhoods filter of `s ∩ t`
is the infimum of set neighborhoods filters for `s` and `t`.
For general sets, only the `≤` inequality holds, see `nhdsSet_inter_le`. |
IsCompact.separation_of_notMem {X : Type u_1} [TopologicalSpace X] [T2Space X] {x : X}
{t : Set X} (H1 : IsCompact t) (H2 : x ∉ t) :
∃ (U : Set X), ∃ (V : Set X), IsOpen U ∧ IsOpen V ∧ t ⊆ U ∧ x ∈ V ∧ Disjoint U V := by
simpa [SeparatedNhds] using SeparatedNhds.of_isCompact_isCompact_isClosed H1 isCompact_singleton
isClosed_singleton <| disjoint_singleton_right.mpr H2
@[deprecated (since := "2025-05-23")]
alias IsCompact.separation_of_not_mem := IsCompact.separation_of_notMem | lemma | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | IsCompact.separation_of_notMem | In a `T2Space X`, for a compact set `t` and a point `x` outside `t`, there are open sets `U`,
`V` that separate `t` and `x`. |
IsCompact.disjoint_nhdsSet_nhds {X : Type u_1} [TopologicalSpace X] [T2Space X] {x : X}
{t : Set X} (H1 : IsCompact t) (H2 : x ∉ t) :
Disjoint (𝓝ˢ t) (𝓝 x) := by
simpa using SeparatedNhds.disjoint_nhdsSet <| .of_isCompact_isCompact_isClosed H1
isCompact_singleton isClosed_singleton <| disjoint_singleton_right.mpr H2 | lemma | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | IsCompact.disjoint_nhdsSet_nhds | In a `T2Space X`, for a compact set `t` and a point `x` outside `t`, `𝓝ˢ t` and `𝓝 x` are
disjoint. |
Set.InjOn.exists_mem_nhdsSet {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
[T2Space Y] {f : X → Y} {s : Set X} (inj : InjOn f s) (sc : IsCompact s)
(fc : ∀ x ∈ s, ContinuousAt f x) (loc : ∀ x ∈ s, ∃ u ∈ 𝓝 x, InjOn f u) :
∃ t ∈ 𝓝ˢ s, InjOn f t := by
have : ∀ x ∈ s ×ˢ s, ∀ᶠ y in 𝓝 x, f y.1 = f y.2 → y.1 = y.2 := fun (x, y) ⟨hx, hy⟩ ↦ by
rcases eq_or_ne x y with rfl | hne
· rcases loc x hx with ⟨u, hu, hf⟩
exact Filter.mem_of_superset (prod_mem_nhds hu hu) <| forall_prod_set.2 hf
· suffices ∀ᶠ z in 𝓝 (x, y), f z.1 ≠ f z.2 from this.mono fun _ hne h ↦ absurd h hne
refine (fc x hx).prodMap' (fc y hy) <| isClosed_diagonal.isOpen_compl.mem_nhds ?_
exact inj.ne hx hy hne
rw [← eventually_nhdsSet_iff_forall, sc.nhdsSet_prod_eq sc] at this
exact eventually_prod_self_iff.1 this | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Set.InjOn.exists_mem_nhdsSet | If a function `f` is
- injective on a compact set `s`;
- continuous at every point of this set;
- injective on a neighborhood of each point of this set,
then it is injective on a neighborhood of this set. |
Set.InjOn.exists_isOpen_superset {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
[T2Space Y] {f : X → Y} {s : Set X} (inj : InjOn f s) (sc : IsCompact s)
(fc : ∀ x ∈ s, ContinuousAt f x) (loc : ∀ x ∈ s, ∃ u ∈ 𝓝 x, InjOn f u) :
∃ t, IsOpen t ∧ s ⊆ t ∧ InjOn f t :=
let ⟨_t, hst, ht⟩ := inj.exists_mem_nhdsSet sc fc loc
let ⟨u, huo, hsu, hut⟩ := mem_nhdsSet_iff_exists.1 hst
⟨u, huo, hsu, ht.mono hut⟩ | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Set.InjOn.exists_isOpen_superset | If a function `f` is
- injective on a compact set `s`;
- continuous at every point of this set;
- injective on a neighborhood of each point of this set,
then it is injective on an open neighborhood of this set. |
lim_eq {x : X} [NeBot f] (h : f ≤ 𝓝 x) : @lim _ _ ⟨x⟩ f = x :=
tendsto_nhds_unique (le_nhds_lim ⟨x, h⟩) h | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | lim_eq | null |
lim_eq_iff [NeBot f] (h : ∃ x : X, f ≤ 𝓝 x) {x} : @lim _ _ ⟨x⟩ f = x ↔ f ≤ 𝓝 x :=
⟨fun c => c ▸ le_nhds_lim h, lim_eq⟩ | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | lim_eq_iff | null |
Ultrafilter.lim_eq_iff_le_nhds [CompactSpace X] {x : X} {F : Ultrafilter X} :
F.lim = x ↔ ↑F ≤ 𝓝 x :=
⟨fun h => h ▸ F.le_nhds_lim, lim_eq⟩ | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Ultrafilter.lim_eq_iff_le_nhds | null |
isOpen_iff_ultrafilter' [CompactSpace X] (U : Set X) :
IsOpen U ↔ ∀ F : Ultrafilter X, F.lim ∈ U → U ∈ F.1 := by
rw [isOpen_iff_ultrafilter]
refine ⟨fun h F hF => h F.lim hF F F.le_nhds_lim, ?_⟩
intro cond x hx f h
rw [← Ultrafilter.lim_eq_iff_le_nhds.2 h] at hx
exact cond _ hx | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | isOpen_iff_ultrafilter' | null |
Filter.Tendsto.limUnder_eq {x : X} {f : Filter Y} [NeBot f] {g : Y → X}
(h : Tendsto g f (𝓝 x)) : @limUnder _ _ _ ⟨x⟩ f g = x :=
lim_eq h | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Filter.Tendsto.limUnder_eq | null |
Filter.limUnder_eq_iff {f : Filter Y} [NeBot f] {g : Y → X} (h : ∃ x, Tendsto g f (𝓝 x))
{x} : @limUnder _ _ _ ⟨x⟩ f g = x ↔ Tendsto g f (𝓝 x) :=
⟨fun c => c ▸ tendsto_nhds_limUnder h, Filter.Tendsto.limUnder_eq⟩ | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Filter.limUnder_eq_iff | null |
Continuous.limUnder_eq [TopologicalSpace Y] {f : Y → X} (h : Continuous f) (y : Y) :
@limUnder _ _ _ ⟨f y⟩ (𝓝 y) f = f y :=
(h.tendsto y).limUnder_eq
@[simp] | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Continuous.limUnder_eq | null |
lim_nhds (x : X) : @lim _ _ ⟨x⟩ (𝓝 x) = x :=
lim_eq le_rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | lim_nhds | null |
limUnder_nhds_id (x : X) : @limUnder _ _ _ ⟨x⟩ (𝓝 x) id = x :=
lim_nhds x
@[simp] | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | limUnder_nhds_id | null |
lim_nhdsWithin {x : X} {s : Set X} (h : x ∈ closure s) : @lim _ _ ⟨x⟩ (𝓝[s] x) = x :=
haveI : NeBot (𝓝[s] x) := mem_closure_iff_clusterPt.1 h
lim_eq inf_le_left
@[simp] | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | lim_nhdsWithin | null |
limUnder_nhdsWithin_id {x : X} {s : Set X} (h : x ∈ closure s) :
@limUnder _ _ _ ⟨x⟩ (𝓝[s] x) id = x :=
lim_nhdsWithin h | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | limUnder_nhdsWithin_id | null |
separated_by_continuous [TopologicalSpace Y] [T2Space Y]
{f : X → Y} (hf : Continuous f) {x y : X} (h : f x ≠ f y) :
∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v :=
let ⟨u, v, uo, vo, xu, yv, uv⟩ := t2_separation h
⟨f ⁻¹' u, f ⁻¹' v, uo.preimage hf, vo.preimage hf, xu, yv, uv.preimage _⟩ | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | separated_by_continuous | null |
separated_by_isOpenEmbedding [TopologicalSpace Y] [T2Space X]
{f : X → Y} (hf : IsOpenEmbedding f) {x y : X} (h : x ≠ y) :
∃ u v : Set Y, IsOpen u ∧ IsOpen v ∧ f x ∈ u ∧ f y ∈ v ∧ Disjoint u v :=
let ⟨u, v, uo, vo, xu, yv, uv⟩ := t2_separation h
⟨f '' u, f '' v, hf.isOpenMap _ uo, hf.isOpenMap _ vo, mem_image_of_mem _ xu,
mem_image_of_mem _ yv, disjoint_image_of_injective hf.injective uv⟩ | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | separated_by_isOpenEmbedding | null |
Prod.t2Space [T2Space X] [TopologicalSpace Y] [T2Space Y] : T2Space (X × Y) :=
inferInstance | instance | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Prod.t2Space | null |
T2Space.of_injective_continuous [TopologicalSpace Y] [T2Space Y] {f : X → Y}
(hinj : Injective f) (hc : Continuous f) : T2Space X :=
⟨fun _ _ h => separated_by_continuous hc (hinj.ne h)⟩ | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | T2Space.of_injective_continuous | If the codomain of an injective continuous function is a Hausdorff space, then so is its
domain. |
Topology.IsEmbedding.t2Space [TopologicalSpace Y] [T2Space Y] {f : X → Y}
(hf : IsEmbedding f) : T2Space X :=
.of_injective_continuous hf.injective hf.continuous | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Topology.IsEmbedding.t2Space | If the codomain of a topological embedding is a Hausdorff space, then so is its domain.
See also `T2Space.of_continuous_injective`. |
protected Homeomorph.t2Space [TopologicalSpace Y] [T2Space X] (h : X ≃ₜ Y) : T2Space Y :=
h.symm.isEmbedding.t2Space | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Homeomorph.t2Space | null |
ULift.instT2Space [T2Space X] : T2Space (ULift X) :=
IsEmbedding.uliftDown.t2Space | instance | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | ULift.instT2Space | null |
Pi.t2Space {Y : X → Type v} [∀ a, TopologicalSpace (Y a)]
[∀ a, T2Space (Y a)] : T2Space (∀ a, Y a) :=
inferInstance | instance | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Pi.t2Space | null |
Sigma.t2Space {ι} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ a, T2Space (X a)] :
T2Space (Σ i, X i) := by
constructor
rintro ⟨i, x⟩ ⟨j, y⟩ neq
rcases eq_or_ne i j with (rfl | h)
· replace neq : x ≠ y := ne_of_apply_ne _ neq
exact separated_by_isOpenEmbedding .sigmaMk neq
· let _ := (⊥ : TopologicalSpace ι); have : DiscreteTopology ι := ⟨rfl⟩
exact separated_by_continuous (continuous_def.2 fun u _ => isOpen_sigma_fst_preimage u) h | instance | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Sigma.t2Space | null |
t2Setoid : Setoid X := sInf {s | T2Space (Quotient s)} | def | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | t2Setoid | The smallest equivalence relation on a topological space giving a T2 quotient. |
T2Quotient := Quotient (t2Setoid X)
@[deprecated (since := "2025-05-15")] alias t2Quotient := T2Quotient | def | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | T2Quotient | The largest T2 quotient of a topological space. This construction is left-adjoint to the
inclusion of T2 spaces into all topological spaces. |
mk : X → T2Quotient X := Quotient.mk (t2Setoid X) | def | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | mk | The map from a topological space to its largest T2 quotient. |
mk_eq {x y : X} : mk x = mk y ↔ ∀ s : Setoid X, T2Space (Quotient s) → s x y :=
Setoid.quotient_mk_sInf_eq
variable (X) | lemma | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | mk_eq | null |
surjective_mk : Surjective (mk : X → T2Quotient X) := Quotient.mk_surjective | lemma | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | surjective_mk | null |
continuous_mk : Continuous (mk : X → T2Quotient X) :=
continuous_quotient_mk'
variable {X}
@[elab_as_elim] | lemma | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | continuous_mk | null |
protected inductionOn {motive : T2Quotient X → Prop} (q : T2Quotient X)
(h : ∀ x, motive (T2Quotient.mk x)) : motive q := Quotient.inductionOn q h
@[elab_as_elim] | lemma | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | inductionOn | null |
protected inductionOn₂ [TopologicalSpace Y] {motive : T2Quotient X → T2Quotient Y → Prop}
(q : T2Quotient X) (q' : T2Quotient Y) (h : ∀ x y, motive (mk x) (mk y)) : motive q q' :=
Quotient.inductionOn₂ q q' h | lemma | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | inductionOn₂ | null |
lift {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y]
{f : X → Y} (hf : Continuous f) : T2Quotient X → Y :=
Quotient.lift f (T2Quotient.compatible hf) | def | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | lift | The largest T2 quotient of a topological space is indeed T2. -/
instance : T2Space (T2Quotient X) := by
rw [t2Space_iff]
rintro ⟨x⟩ ⟨y⟩ (h : ¬ T2Quotient.mk x = T2Quotient.mk y)
obtain ⟨s, hs, hsxy⟩ : ∃ s, T2Space (Quotient s) ∧ Quotient.mk s x ≠ Quotient.mk s y := by
simpa [T2Quotient.mk_eq] using h
exact separated_by_continuous (continuous_map_sInf (by exact hs)) hsxy
lemma compatible {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y]
{f : X → Y} (hf : Continuous f) : letI _ := t2Setoid X
∀ (a b : X), a ≈ b → f a = f b := by
change t2Setoid X ≤ Setoid.ker f
exact sInf_le <| .of_injective_continuous
(Setoid.ker_lift_injective _) (hf.quotient_lift fun _ _ ↦ id)
/-- The universal property of the largest T2 quotient of a topological space `X`: any continuous
map from `X` to a T2 space `Y` uniquely factors through `T2Quotient X`. This declaration builds the
factored map. Its continuity is `T2Quotient.continuous_lift`, the fact that it indeed factors the
original map is `T2Quotient.lift_mk` and uniqueness is `T2Quotient.unique_lift`. |
continuous_lift {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y]
{f : X → Y} (hf : Continuous f) : Continuous (T2Quotient.lift hf) :=
continuous_coinduced_dom.mpr hf
@[simp] | lemma | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | continuous_lift | null |
lift_mk {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y]
{f : X → Y} (hf : Continuous f) (x : X) : lift hf (mk x) = f x :=
Quotient.lift_mk (s := t2Setoid X) f (T2Quotient.compatible hf) x | lemma | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | lift_mk | null |
unique_lift {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y]
{f : X → Y} (hf : Continuous f) {g : T2Quotient X → Y} (hfg : g ∘ mk = f) :
g = lift hf := by
apply surjective_mk X |>.right_cancellable |>.mp <| funext _
simp [← hfg] | lemma | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | unique_lift | null |
isClosed_eq [T2Space X] {f g : Y → X} (hf : Continuous f) (hg : Continuous g) :
IsClosed { y : Y | f y = g y } :=
continuous_iff_isClosed.mp (hf.prodMk hg) _ isClosed_diagonal | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | isClosed_eq | null |
protected IsClosed.isClosed_eq [T2Space Y] {f g : X → Y} {s : Set X} (hs : IsClosed s)
(hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed {x ∈ s | f x = g x} :=
(hf.prodMk hg).preimage_isClosed_of_isClosed hs isClosed_diagonal | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | IsClosed.isClosed_eq | If functions `f` and `g` are continuous on a closed set `s`,
then the set of points `x ∈ s` such that `f x = g x` is a closed set. |
isOpen_ne_fun [T2Space X] {f g : Y → X} (hf : Continuous f) (hg : Continuous g) :
IsOpen { y : Y | f y ≠ g y } :=
isOpen_compl_iff.mpr <| isClosed_eq hf hg | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | isOpen_ne_fun | null |
protected Set.EqOn.closure [T2Space X] {s : Set Y} {f g : Y → X} (h : EqOn f g s)
(hf : Continuous f) (hg : Continuous g) : EqOn f g (closure s) :=
closure_minimal h (isClosed_eq hf hg) | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Set.EqOn.closure | If two continuous maps are equal on `s`, then they are equal on the closure of `s`. See also
`Set.EqOn.of_subset_closure` for a more general version. |
Continuous.ext_on [T2Space X] {s : Set Y} (hs : Dense s) {f g : Y → X} (hf : Continuous f)
(hg : Continuous g) (h : EqOn f g s) : f = g :=
funext fun x => h.closure hf hg (hs x) | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | Continuous.ext_on | If two continuous functions are equal on a dense set, then they are equal. |
eqOn_closure₂' [T2Space Z] {s : Set X} {t : Set Y} {f g : X → Y → Z}
(h : ∀ x ∈ s, ∀ y ∈ t, f x y = g x y) (hf₁ : ∀ x, Continuous (f x))
(hf₂ : ∀ y, Continuous fun x => f x y) (hg₁ : ∀ x, Continuous (g x))
(hg₂ : ∀ y, Continuous fun x => g x y) : ∀ x ∈ closure s, ∀ y ∈ closure t, f x y = g x y :=
suffices closure s ⊆ ⋂ y ∈ closure t, { x | f x y = g x y } by simpa only [subset_def, mem_iInter]
(closure_minimal fun x hx => mem_iInter₂.2 <| Set.EqOn.closure (h x hx) (hf₁ _) (hg₁ _)) <|
isClosed_biInter fun _ _ => isClosed_eq (hf₂ _) (hg₂ _) | theorem | Topology | [
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Separation.Basic"
] | Mathlib/Topology/Separation/Hausdorff.lean | eqOn_closure₂' | null |
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