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 ⌀ |
|---|---|---|---|---|---|---|
protected Topology.IsEmbedding.t1Space [TopologicalSpace Y] [T1Space Y] {f : X → Y}
(hf : IsEmbedding f) : T1Space X :=
t1Space_of_injective_of_continuous hf.injective hf.continuous | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | Topology.IsEmbedding.t1Space | null |
protected Homeomorph.t1Space [TopologicalSpace Y] [T1Space X] (h : X ≃ₜ Y) : T1Space Y :=
h.symm.isEmbedding.t1Space | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | Homeomorph.t1Space | null |
Subtype.t1Space {X : Type u} [TopologicalSpace X] [T1Space X] {p : X → Prop} :
T1Space (Subtype p) :=
IsEmbedding.subtypeVal.t1Space | instance | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | Subtype.t1Space | null |
ULift.instT1Space [T1Space X] : T1Space (ULift X) :=
IsEmbedding.uliftDown.t1Space | instance | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | ULift.instT1Space | null |
@[simp]
compl_singleton_mem_nhds_iff [T1Space X] {x y : X} : {x}ᶜ ∈ 𝓝 y ↔ y ≠ x :=
isOpen_compl_singleton.mem_nhds_iff | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | compl_singleton_mem_nhds_iff | null |
compl_singleton_mem_nhds [T1Space X] {x y : X} (h : y ≠ x) : {x}ᶜ ∈ 𝓝 y :=
compl_singleton_mem_nhds_iff.mpr h | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | compl_singleton_mem_nhds | null |
closure_singleton [T1Space X] {x : X} : closure ({x} : Set X) = {x} :=
isClosed_singleton.closure_eq | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | closure_singleton | null |
Set.Subsingleton.isClosed [T1Space X] {s : Set X} (hs : s.Subsingleton) : IsClosed s := by
rcases hs.eq_empty_or_singleton with rfl | ⟨x, rfl⟩
· exact isClosed_empty
· exact isClosed_singleton | lemma | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | Set.Subsingleton.isClosed | null |
Set.Subsingleton.closure_eq [T1Space X] {s : Set X} (hs : s.Subsingleton) :
closure s = s :=
hs.isClosed.closure_eq | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | Set.Subsingleton.closure_eq | null |
Set.Subsingleton.closure [T1Space X] {s : Set X} (hs : s.Subsingleton) :
(closure s).Subsingleton := by
rwa [hs.closure_eq]
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | Set.Subsingleton.closure | null |
subsingleton_closure [T1Space X] {s : Set X} : (closure s).Subsingleton ↔ s.Subsingleton :=
⟨fun h => h.anti subset_closure, fun h => h.closure⟩ | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | subsingleton_closure | null |
isClosedMap_const {X Y} [TopologicalSpace X] [TopologicalSpace Y] [T1Space Y] {y : Y} :
IsClosedMap (Function.const X y) :=
IsClosedMap.of_nonempty fun s _ h2s => by simp_rw [const, h2s.image_const, isClosed_singleton] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | isClosedMap_const | null |
isClosedMap_prodMk_left [TopologicalSpace Y] [T1Space X] (x : X) :
IsClosedMap (fun y : Y ↦ Prod.mk x y) :=
fun _K hK ↦ Set.singleton_prod ▸ isClosed_singleton.prod hK | lemma | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | isClosedMap_prodMk_left | null |
isClosedMap_prodMk_right [TopologicalSpace Y] [T1Space Y] (y : Y) :
IsClosedMap (fun x : X ↦ Prod.mk x y) :=
fun _K hK ↦ Set.prod_singleton ▸ hK.prod isClosed_singleton | lemma | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | isClosedMap_prodMk_right | null |
nhdsWithin_insert_of_ne [T1Space X] {x y : X} {s : Set X} (hxy : x ≠ y) :
𝓝[insert y s] x = 𝓝[s] x := by
refine le_antisymm (Filter.le_def.2 fun t ht => ?_) (nhdsWithin_mono x <| subset_insert y s)
obtain ⟨o, ho, hxo, host⟩ := mem_nhdsWithin.mp ht
refine mem_nhdsWithin.mpr ⟨o \ {y}, ho.sdiff isClosed_singleton, ⟨hxo, hxy⟩, ?_⟩
rw [inter_insert_of_notMem <| notMem_diff_of_mem (mem_singleton y)]
exact (inter_subset_inter diff_subset Subset.rfl).trans host | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | nhdsWithin_insert_of_ne | null |
insert_mem_nhdsWithin_of_subset_insert [T1Space X] {x y : X} {s t : Set X}
(hu : t ⊆ insert y s) : insert x s ∈ 𝓝[t] x := by
rcases eq_or_ne x y with (rfl | h)
· exact mem_of_superset self_mem_nhdsWithin hu
refine nhdsWithin_mono x hu ?_
rw [nhdsWithin_insert_of_ne h]
exact mem_of_superset self_mem_nhdsWithin (subset_insert x s) | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | insert_mem_nhdsWithin_of_subset_insert | If `t` is a subset of `s`, except for one point,
then `insert x s` is a neighborhood of `x` within `t`. |
eventuallyEq_insert [T1Space X] {s t : Set X} {x y : X} (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
(insert x s : Set X) =ᶠ[𝓝 x] (insert x t : Set X) := by
simp_rw [eventuallyEq_set] at h ⊢
simp_rw [← union_singleton, ← nhdsWithin_univ, ← compl_union_self {x},
nhdsWithin_union, eventually_sup, nhdsWithin_singleton,
eventually_pure, union_singleton, mem_insert_iff, true_or, and_true]
filter_upwards [nhdsWithin_compl_singleton_le x y h] with y using or_congr (Iff.rfl)
@[simp] | lemma | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | eventuallyEq_insert | null |
ker_nhds [T1Space X] (x : X) : (𝓝 x).ker = {x} := by
simp [ker_nhds_eq_specializes] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | ker_nhds | null |
biInter_basis_nhds [T1Space X] {ι : Sort*} {p : ι → Prop} {s : ι → Set X} {x : X}
(h : (𝓝 x).HasBasis p s) : ⋂ (i) (_ : p i), s i = {x} := by
rw [← h.ker, ker_nhds]
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | biInter_basis_nhds | null |
compl_singleton_mem_nhdsSet_iff [T1Space X] {x : X} {s : Set X} : {x}ᶜ ∈ 𝓝ˢ s ↔ x ∉ s := by
rw [isOpen_compl_singleton.mem_nhdsSet, subset_compl_singleton_iff]
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | compl_singleton_mem_nhdsSet_iff | null |
nhdsSet_le_iff [T1Space X] {s t : Set X} : 𝓝ˢ s ≤ 𝓝ˢ t ↔ s ⊆ t := by
refine ⟨?_, fun h => monotone_nhdsSet h⟩
simp_rw [Filter.le_def]; intro h x hx
specialize h {x}ᶜ
simp_rw [compl_singleton_mem_nhdsSet_iff] at h
by_contra hxt
exact h hxt hx
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | nhdsSet_le_iff | null |
nhdsSet_inj_iff [T1Space X] {s t : Set X} : 𝓝ˢ s = 𝓝ˢ t ↔ s = t := by
simp_rw [le_antisymm_iff]
exact and_congr nhdsSet_le_iff nhdsSet_le_iff | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | nhdsSet_inj_iff | null |
injective_nhdsSet [T1Space X] : Function.Injective (𝓝ˢ : Set X → Filter X) := fun _ _ hst =>
nhdsSet_inj_iff.mp hst | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | injective_nhdsSet | null |
strictMono_nhdsSet [T1Space X] : StrictMono (𝓝ˢ : Set X → Filter X) :=
monotone_nhdsSet.strictMono_of_injective injective_nhdsSet
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | strictMono_nhdsSet | null |
nhds_le_nhdsSet_iff [T1Space X] {s : Set X} {x : X} : 𝓝 x ≤ 𝓝ˢ s ↔ x ∈ s := by
rw [← nhdsSet_singleton, nhdsSet_le_iff, singleton_subset_iff] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | nhds_le_nhdsSet_iff | null |
Dense.diff_singleton [T1Space X] {s : Set X} (hs : Dense s) (x : X) [NeBot (𝓝[≠] x)] :
Dense (s \ {x}) :=
hs.inter_of_isOpen_right (dense_compl_singleton x) isOpen_compl_singleton | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | Dense.diff_singleton | Removing a non-isolated point from a dense set, one still obtains a dense set. |
Dense.diff_finset [T1Space X] [∀ x : X, NeBot (𝓝[≠] x)] {s : Set X} (hs : Dense s)
(t : Finset X) : Dense (s \ t) := by
classical
induction t using Finset.induction_on with
| empty => simpa using hs
| insert _ _ _ ih =>
rw [Finset.coe_insert, ← union_singleton, ← diff_diff]
exact ih.diff_singleton _ | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | Dense.diff_finset | Removing a finset from a dense set in a space without isolated points, one still
obtains a dense set. |
Dense.diff_finite [T1Space X] [∀ x : X, NeBot (𝓝[≠] x)] {s : Set X} (hs : Dense s)
{t : Set X} (ht : t.Finite) : Dense (s \ t) := by
convert hs.diff_finset ht.toFinset
exact (Finite.coe_toFinset _).symm | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | Dense.diff_finite | Removing a finite set from a dense set in a space without isolated points, one still
obtains a dense set. |
eq_of_tendsto_nhds [TopologicalSpace Y] [T1Space Y] {f : X → Y} {x : X} {y : Y}
(h : Tendsto f (𝓝 x) (𝓝 y)) : f x = y :=
by_contra fun hfa : f x ≠ y =>
have fact₁ : {f x}ᶜ ∈ 𝓝 y := compl_singleton_mem_nhds hfa.symm
have fact₂ : Tendsto f (pure x) (𝓝 y) := h.comp (tendsto_id'.2 <| pure_le_nhds x)
fact₂ fact₁ (Eq.refl <| f x) | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | eq_of_tendsto_nhds | If a function to a `T1Space` tends to some limit `y` at some point `x`, then necessarily
`y = f x`. |
Filter.Tendsto.eventually_ne {X} [TopologicalSpace Y] [T1Space Y] {g : X → Y}
{l : Filter X} {b₁ b₂ : Y} (hg : Tendsto g l (𝓝 b₁)) (hb : b₁ ≠ b₂) : ∀ᶠ z in l, g z ≠ b₂ :=
hg.eventually (isOpen_compl_singleton.eventually_mem hb) | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | Filter.Tendsto.eventually_ne | null |
ContinuousAt.eventually_ne [TopologicalSpace Y] [T1Space Y] {g : X → Y} {x : X} {y : Y}
(hg1 : ContinuousAt g x) (hg2 : g x ≠ y) : ∀ᶠ z in 𝓝 x, g z ≠ y :=
hg1.tendsto.eventually_ne hg2 | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | ContinuousAt.eventually_ne | null |
eventually_ne_nhds [T1Space X] {a b : X} (h : a ≠ b) : ∀ᶠ x in 𝓝 a, x ≠ b :=
IsOpen.eventually_mem isOpen_ne h | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | eventually_ne_nhds | null |
eventually_ne_nhdsWithin [T1Space X] {a b : X} {s : Set X} (h : a ≠ b) :
∀ᶠ x in 𝓝[s] a, x ≠ b :=
Filter.Eventually.filter_mono nhdsWithin_le_nhds <| eventually_ne_nhds h | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | eventually_ne_nhdsWithin | null |
continuousWithinAt_insert [TopologicalSpace Y] [T1Space X]
{x y : X} {s : Set X} {f : X → Y} :
ContinuousWithinAt f (insert y s) x ↔ ContinuousWithinAt f s x := by
rcases eq_or_ne x y with (rfl | h)
· exact continuousWithinAt_insert_self
simp_rw [ContinuousWithinAt, nhdsWithin_insert_of_ne h]
alias ⟨ContinuousWithinAt.of_insert, ContinuousWithinAt.insert'⟩ := continuousWithinAt_insert | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | continuousWithinAt_insert | null |
continuousWithinAt_diff_singleton [TopologicalSpace Y] [T1Space X]
{x y : X} {s : Set X} {f : X → Y} :
ContinuousWithinAt f (s \ {y}) x ↔ ContinuousWithinAt f s x := by
rw [← continuousWithinAt_insert, insert_diff_singleton, continuousWithinAt_insert] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | continuousWithinAt_diff_singleton | See also `continuousWithinAt_diff_self` for the case `y = x` but not requiring `T1Space`. |
continuousWithinAt_congr_set' [TopologicalSpace Y] [T1Space X]
{x : X} {s t : Set X} {f : X → Y} (y : X) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
ContinuousWithinAt f s x ↔ ContinuousWithinAt f t x := by
rw [← continuousWithinAt_insert_self (s := s), ← continuousWithinAt_insert_self (s := t)]
exact continuousWithinAt_congr_set (eventuallyEq_insert h) | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | continuousWithinAt_congr_set' | If two sets coincide locally around `x`, except maybe at `y`, then it is equivalent to be
continuous at `x` within one set or the other. |
ContinuousWithinAt.eq_const_of_mem_closure [TopologicalSpace Y] [T1Space Y]
{f : X → Y} {s : Set X} {x : X} {c : Y} (h : ContinuousWithinAt f s x) (hx : x ∈ closure s)
(ht : ∀ y ∈ s, f y = c) : f x = c := by
rw [← Set.mem_singleton_iff, ← closure_singleton]
exact h.mem_closure hx ht
@[deprecated (since := "2025-08-22")] alias ContinousWithinAt.eq_const_of_mem_closure :=
ContinuousWithinAt.eq_const_of_mem_closure | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | ContinuousWithinAt.eq_const_of_mem_closure | null |
ContinuousWithinAt.eqOn_const_closure [TopologicalSpace Y] [T1Space Y]
{f : X → Y} {s : Set X} {c : Y} (h : ∀ x ∈ closure s, ContinuousWithinAt f s x)
(ht : s.EqOn f (fun _ ↦ c)) : (closure s).EqOn f (fun _ ↦ c) := by
intro x hx
apply ContinuousWithinAt.eq_const_of_mem_closure (h x hx) hx ht | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | ContinuousWithinAt.eqOn_const_closure | null |
continuousAt_of_tendsto_nhds [TopologicalSpace Y] [T1Space Y] {f : X → Y} {x : X} {y : Y}
(h : Tendsto f (𝓝 x) (𝓝 y)) : ContinuousAt f x := by
rwa [ContinuousAt, eq_of_tendsto_nhds h]
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | continuousAt_of_tendsto_nhds | To prove a function to a `T1Space` is continuous at some point `x`, it suffices to prove that
`f` admits *some* limit at `x`. |
tendsto_const_nhds_iff [T1Space X] {l : Filter Y} [NeBot l] {c d : X} :
Tendsto (fun _ => c) l (𝓝 d) ↔ c = d := by simp_rw [Tendsto, Filter.map_const, pure_le_nhds_iff] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | tendsto_const_nhds_iff | null |
isOpen_singleton_of_finite_mem_nhds [T1Space X] (x : X)
{s : Set X} (hs : s ∈ 𝓝 x) (hsf : s.Finite) : IsOpen ({x} : Set X) := by
have A : {x} ⊆ s := by simp only [singleton_subset_iff, mem_of_mem_nhds hs]
have B : IsClosed (s \ {x}) := (hsf.subset diff_subset).isClosed
have C : (s \ {x})ᶜ ∈ 𝓝 x := B.isOpen_compl.mem_nhds fun h => h.2 rfl
have D : {x} ∈ 𝓝 x := by simpa only [← diff_eq, diff_diff_cancel_left A] using inter_mem hs C
rwa [← mem_interior_iff_mem_nhds, ← singleton_subset_iff, subset_interior_iff_isOpen] at D | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | isOpen_singleton_of_finite_mem_nhds | A point with a finite neighborhood has to be isolated. |
infinite_of_mem_nhds {X} [TopologicalSpace X] [T1Space X] (x : X) [hx : NeBot (𝓝[≠] x)]
{s : Set X} (hs : s ∈ 𝓝 x) : Set.Infinite s := by
refine fun hsf => hx.1 ?_
rw [← isOpen_singleton_iff_punctured_nhds]
exact isOpen_singleton_of_finite_mem_nhds x hs hsf | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | infinite_of_mem_nhds | If the punctured neighborhoods of a point form a nontrivial filter, then any neighborhood is
infinite. |
Finite.instDiscreteTopology [T1Space X] [Finite X] : DiscreteTopology X :=
discreteTopology_iff_forall_isClosed.mpr (· |>.toFinite.isClosed) | instance | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | Finite.instDiscreteTopology | null |
Set.Finite.continuousOn [T1Space X] [TopologicalSpace Y] {s : Set X} (hs : s.Finite)
(f : X → Y) : ContinuousOn f s := by
rw [continuousOn_iff_continuous_restrict]
have : Finite s := hs
fun_prop | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | Set.Finite.continuousOn | null |
SeparationQuotient.t1Space_iff : T1Space (SeparationQuotient X) ↔ R0Space X := by
rw [r0Space_iff, ((t1Space_TFAE (SeparationQuotient X)).out 0 9 :)]
constructor
· intro h x y xspecy
rw [← IsInducing.specializes_iff isInducing_mk, h xspecy] at *
· -- TODO is there are better way to do this,
rintro h ⟨x⟩ ⟨y⟩ sxspecsy
change mk _ = mk _
have xspecy : x ⤳ y := isInducing_mk.specializes_iff.mp sxspecsy
have yspecx : y ⤳ x := h xspecy
rw [mk_eq_mk, inseparable_iff_specializes_and]
exact ⟨xspecy, yspecx⟩ | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | SeparationQuotient.t1Space_iff | null |
isClosed_inter_singleton [T1Space X] {A : Set X} {a : X} : IsClosed (A ∩ {a}) :=
Subsingleton.inter_singleton.isClosed | lemma | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | isClosed_inter_singleton | null |
isClosed_singleton_inter [T1Space X] {A : Set X} {a : X} : IsClosed ({a} ∩ A) :=
Subsingleton.singleton_inter.isClosed | lemma | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | isClosed_singleton_inter | null |
singleton_mem_nhdsWithin_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X}
(hx : x ∈ s) : {x} ∈ 𝓝[s] x := by
have : ({⟨x, hx⟩} : Set s) ∈ 𝓝 (⟨x, hx⟩ : s) := by simp [nhds_discrete]
simpa only [nhdsWithin_eq_map_subtype_coe hx, image_singleton] using
@image_mem_map _ _ _ ((↑) : s → X) _ this | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | singleton_mem_nhdsWithin_of_mem_discrete | null |
nhdsWithin_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X} (hx : x ∈ s) :
𝓝[s] x = pure x :=
le_antisymm (le_pure_iff.2 <| singleton_mem_nhdsWithin_of_mem_discrete hx) (pure_le_nhdsWithin hx) | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | nhdsWithin_of_mem_discrete | The neighbourhoods filter of `x` within `s`, under the discrete topology, is equal to
the pure `x` filter (which is the principal filter at the singleton `{x}`.) |
Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete {ι : Type*} {p : ι → Prop}
{t : ι → Set X} {s : Set X} [DiscreteTopology s] {x : X} (hb : (𝓝 x).HasBasis p t)
(hx : x ∈ s) : ∃ i, p i ∧ t i ∩ s = {x} := by
rcases (nhdsWithin_hasBasis hb s).mem_iff.1 (singleton_mem_nhdsWithin_of_mem_discrete hx) with
⟨i, hi, hix⟩
exact ⟨i, hi, hix.antisymm <| singleton_subset_iff.2 ⟨mem_of_mem_nhds <| hb.mem_of_mem hi, hx⟩⟩ | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete | null |
nhds_inter_eq_singleton_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X}
(hx : x ∈ s) : ∃ U ∈ 𝓝 x, U ∩ s = {x} := by
simpa using (𝓝 x).basis_sets.exists_inter_eq_singleton_of_mem_discrete hx | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | nhds_inter_eq_singleton_of_mem_discrete | A point `x` in a discrete subset `s` of a topological space admits a neighbourhood
that only meets `s` at `x`. |
isOpen_inter_eq_singleton_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X}
(hx : x ∈ s) : ∃ U : Set X, IsOpen U ∧ U ∩ s = {x} := by
obtain ⟨U, hU_nhds, hU_inter⟩ := nhds_inter_eq_singleton_of_mem_discrete hx
obtain ⟨t, ht_sub, ht_open, ht_x⟩ := mem_nhds_iff.mp hU_nhds
refine ⟨t, ht_open, Set.Subset.antisymm ?_ ?_⟩
· exact hU_inter ▸ Set.inter_subset_inter_left s ht_sub
· rw [Set.subset_inter_iff, Set.singleton_subset_iff, Set.singleton_subset_iff]
exact ⟨ht_x, hx⟩ | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | isOpen_inter_eq_singleton_of_mem_discrete | Let `x` be a point in a discrete subset `s` of a topological space, then there exists an open
set that only meets `s` at `x`. |
disjoint_nhdsWithin_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X} (hx : x ∈ s) :
∃ U ∈ 𝓝[≠] x, Disjoint U s :=
let ⟨V, h, h'⟩ := nhds_inter_eq_singleton_of_mem_discrete hx
⟨{x}ᶜ ∩ V, inter_mem_nhdsWithin _ h,
disjoint_iff_inter_eq_empty.mpr (by rw [inter_assoc, h', compl_inter_self])⟩ | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | disjoint_nhdsWithin_of_mem_discrete | For point `x` in a discrete subset `s` of a topological space, there is a set `U`
such that
1. `U` is a punctured neighborhood of `x` (i.e. `U ∪ {x}` is a neighbourhood of `x`),
2. `U` is disjoint from `s`. |
isClosedEmbedding_update {ι : Type*} {β : ι → Type*}
[DecidableEq ι] [(i : ι) → TopologicalSpace (β i)]
(x : (i : ι) → β i) (i : ι) [(i : ι) → T1Space (β i)] :
IsClosedEmbedding (update x i) := by
refine .of_continuous_injective_isClosedMap (continuous_const.update i continuous_id)
(update_injective x i) fun s hs ↦ ?_
rw [update_image]
apply isClosed_set_pi
simp [forall_update_iff, hs] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | isClosedEmbedding_update | null |
nhdsNE_le_cofinite {α : Type*} [TopologicalSpace α] [T1Space α] (a : α) :
𝓝[≠] a ≤ cofinite := by
refine le_cofinite_iff_compl_singleton_mem.mpr fun x ↦ ?_
rcases eq_or_ne a x with rfl | hx
exacts [self_mem_nhdsWithin, eventually_ne_nhdsWithin hx] | lemma | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | nhdsNE_le_cofinite | null |
Function.update_eventuallyEq_nhdsNE
{α β : Type*} [TopologicalSpace α] [T1Space α] [DecidableEq α] (f : α → β) (a a' : α) (b : β) :
Function.update f a b =ᶠ[𝓝[≠] a'] f :=
(Function.update_eventuallyEq_cofinite f a b).filter_mono (nhdsNE_le_cofinite a')
/-! ### R₁ (preregular) spaces -/ | lemma | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | Function.update_eventuallyEq_nhdsNE | null |
@[mk_iff r1Space_iff_specializes_or_disjoint_nhds]
R1Space (X : Type*) [TopologicalSpace X] : Prop where
specializes_or_disjoint_nhds (x y : X) : Specializes x y ∨ Disjoint (𝓝 x) (𝓝 y)
export R1Space (specializes_or_disjoint_nhds)
variable [R1Space X] {x y : X} | class | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | R1Space | A topological space is called a *preregular* (a.k.a. R₁) space,
if any two topologically distinguishable points have disjoint neighbourhoods. |
disjoint_nhds_nhds_iff_not_specializes : Disjoint (𝓝 x) (𝓝 y) ↔ ¬x ⤳ y :=
⟨fun hd hspec ↦ hspec.not_disjoint hd, (specializes_or_disjoint_nhds _ _).resolve_left⟩ | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | disjoint_nhds_nhds_iff_not_specializes | null |
specializes_iff_not_disjoint : x ⤳ y ↔ ¬Disjoint (𝓝 x) (𝓝 y) :=
disjoint_nhds_nhds_iff_not_specializes.not_left.symm | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | specializes_iff_not_disjoint | null |
disjoint_nhds_nhds_iff_not_inseparable : Disjoint (𝓝 x) (𝓝 y) ↔ ¬Inseparable x y := by
rw [disjoint_nhds_nhds_iff_not_specializes, specializes_iff_inseparable] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | disjoint_nhds_nhds_iff_not_inseparable | null |
r1Space_iff_inseparable_or_disjoint_nhds {X : Type*} [TopologicalSpace X] :
R1Space X ↔ ∀ x y : X, Inseparable x y ∨ Disjoint (𝓝 x) (𝓝 y) :=
⟨fun _h x y ↦ (specializes_or_disjoint_nhds x y).imp_left Specializes.inseparable, fun h ↦
⟨fun x y ↦ (h x y).imp_left Inseparable.specializes⟩⟩ | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | r1Space_iff_inseparable_or_disjoint_nhds | null |
Inseparable.of_nhds_neBot {x y : X} (h : NeBot (𝓝 x ⊓ 𝓝 y)) :
Inseparable x y :=
(r1Space_iff_inseparable_or_disjoint_nhds.mp ‹_› _ _).resolve_right fun h' => h.ne h'.eq_bot | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | Inseparable.of_nhds_neBot | null |
r1_separation {x y : X} (h : ¬Inseparable x y) :
∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v := by
rw [← disjoint_nhds_nhds_iff_not_inseparable,
(nhds_basis_opens x).disjoint_iff (nhds_basis_opens y)] at h
obtain ⟨u, ⟨hxu, hu⟩, v, ⟨hyv, hv⟩, huv⟩ := h
exact ⟨u, v, hu, hv, hxu, hyv, huv⟩ | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | r1_separation | null |
tendsto_nhds_unique_inseparable {f : Y → X} {l : Filter Y} {a b : X} [NeBot l]
(ha : Tendsto f l (𝓝 a)) (hb : Tendsto f l (𝓝 b)) : Inseparable a b :=
.of_nhds_neBot <| neBot_of_le <| le_inf ha hb | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | tendsto_nhds_unique_inseparable | Limits are unique up to separability.
A weaker version of `tendsto_nhds_unique` for `R1Space`. |
isClosed_setOf_specializes : IsClosed { p : X × X | p.1 ⤳ p.2 } := by
simp only [← isOpen_compl_iff, compl_setOf, ← disjoint_nhds_nhds_iff_not_specializes,
isOpen_setOf_disjoint_nhds_nhds] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | isClosed_setOf_specializes | null |
isClosed_setOf_inseparable : IsClosed { p : X × X | Inseparable p.1 p.2 } := by
simp only [← specializes_iff_inseparable, isClosed_setOf_specializes] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | isClosed_setOf_inseparable | null |
IsCompact.mem_closure_iff_exists_inseparable {K : Set X} (hK : IsCompact K) :
y ∈ closure K ↔ ∃ x ∈ K, Inseparable x y := by
refine ⟨fun hy ↦ ?_, fun ⟨x, hxK, hxy⟩ ↦
(hxy.mem_closed_iff isClosed_closure).1 <| subset_closure hxK⟩
contrapose! hy
have : Disjoint (𝓝 y) (𝓝ˢ K) := hK.disjoint_nhdsSet_right.2 fun x hx ↦
(disjoint_nhds_nhds_iff_not_inseparable.2 (hy x hx)).symm
simpa only [disjoint_iff, notMem_closure_iff_nhdsWithin_eq_bot]
using this.mono_right principal_le_nhdsSet | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | IsCompact.mem_closure_iff_exists_inseparable | In an R₁ space, a point belongs to the closure of a compact set `K`
if and only if it is topologically inseparable from some point of `K`. |
IsCompact.closure_eq_biUnion_inseparable {K : Set X} (hK : IsCompact K) :
closure K = ⋃ x ∈ K, {y | Inseparable x y} := by
ext; simp [hK.mem_closure_iff_exists_inseparable] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | IsCompact.closure_eq_biUnion_inseparable | null |
IsCompact.closure_eq_biUnion_closure_singleton {K : Set X} (hK : IsCompact K) :
closure K = ⋃ x ∈ K, closure {x} := by
simp only [hK.closure_eq_biUnion_inseparable, ← specializes_iff_inseparable,
specializes_iff_mem_closure, setOf_mem_eq] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | IsCompact.closure_eq_biUnion_closure_singleton | In an R₁ space, the closure of a compact set is the union of the closures of its points. |
IsCompact.closure_subset_of_isOpen {K : Set X} (hK : IsCompact K)
{U : Set X} (hU : IsOpen U) (hKU : K ⊆ U) : closure K ⊆ U := by
rw [hK.closure_eq_biUnion_inseparable, iUnion₂_subset_iff]
exact fun x hx y hxy ↦ (hxy.mem_open_iff hU).1 (hKU hx) | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | IsCompact.closure_subset_of_isOpen | In an R₁ space, if a compact set `K` is contained in an open set `U`,
then its closure is also contained in `U`. |
protected IsCompact.closure {K : Set X} (hK : IsCompact K) : IsCompact (closure K) := by
refine isCompact_of_finite_subcover fun U hUo hKU ↦ ?_
rcases hK.elim_finite_subcover U hUo (subset_closure.trans hKU) with ⟨t, ht⟩
exact ⟨t, hK.closure_subset_of_isOpen (isOpen_biUnion fun _ _ ↦ hUo _) ht⟩ | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | IsCompact.closure | The closure of a compact set in an R₁ space is a compact set. |
IsCompact.closure_of_subset {s K : Set X} (hK : IsCompact K) (h : s ⊆ K) :
IsCompact (closure s) :=
hK.closure.of_isClosed_subset isClosed_closure (closure_mono h)
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | IsCompact.closure_of_subset | null |
exists_isCompact_superset_iff {s : Set X} :
(∃ K, IsCompact K ∧ s ⊆ K) ↔ IsCompact (closure s) :=
⟨fun ⟨_K, hK, hsK⟩ => hK.closure_of_subset hsK, fun h => ⟨closure s, h, subset_closure⟩⟩ | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | exists_isCompact_superset_iff | null |
SeparatedNhds.of_isCompact_isCompact_isClosed {K L : Set X} (hK : IsCompact K)
(hL : IsCompact L) (h'L : IsClosed L) (hd : Disjoint K L) : SeparatedNhds K L := by
simp_rw [separatedNhds_iff_disjoint, hK.disjoint_nhdsSet_left, hL.disjoint_nhdsSet_right,
disjoint_nhds_nhds_iff_not_inseparable]
intro x hx y hy h
exact absurd ((h.mem_closed_iff h'L).2 hy) <| disjoint_left.1 hd hx | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | SeparatedNhds.of_isCompact_isCompact_isClosed | If `K` and `L` are disjoint compact sets in an R₁ topological space
and `L` is also closed, then `K` and `L` have disjoint neighborhoods. |
IsCompact.binary_compact_cover {K U V : Set X}
(hK : IsCompact K) (hU : IsOpen U) (hV : IsOpen V) (h2K : K ⊆ U ∪ V) :
∃ K₁ K₂ : Set X, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂ := by
have hK' : IsCompact (closure K) := hK.closure
have : SeparatedNhds (closure K \ U) (closure K \ V) := by
apply SeparatedNhds.of_isCompact_isCompact_isClosed (hK'.diff hU) (hK'.diff hV)
(isClosed_closure.sdiff hV)
rw [disjoint_iff_inter_eq_empty, diff_inter_diff, diff_eq_empty]
exact hK.closure_subset_of_isOpen (hU.union hV) h2K
have : SeparatedNhds (K \ U) (K \ V) :=
this.mono (diff_subset_diff_left (subset_closure)) (diff_subset_diff_left (subset_closure))
rcases this with ⟨O₁, O₂, h1O₁, h1O₂, h2O₁, h2O₂, hO⟩
exact ⟨K \ O₁, K \ O₂, hK.diff h1O₁, hK.diff h1O₂, diff_subset_comm.mp h2O₁,
diff_subset_comm.mp h2O₂, by rw [← diff_inter, hO.inter_eq, diff_empty]⟩ | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | IsCompact.binary_compact_cover | If a compact set is covered by two open sets, then we can cover it by two compact subsets. |
IsCompact.finite_compact_cover {s : Set X} (hs : IsCompact s) {ι : Type*}
(t : Finset ι) (U : ι → Set X) (hU : ∀ i ∈ t, IsOpen (U i)) (hsC : s ⊆ ⋃ i ∈ t, U i) :
∃ K : ι → Set X, (∀ i, IsCompact (K i)) ∧ (∀ i, K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i := by
classical
induction t using Finset.induction generalizing U s with
| empty =>
refine ⟨fun _ => ∅, fun _ => isCompact_empty, fun i => empty_subset _, ?_⟩
simpa only [subset_empty_iff, Finset.notMem_empty, iUnion_false, iUnion_empty] using hsC
| insert x t hx ih =>
simp only [Finset.set_biUnion_insert] at hsC
simp only [Finset.forall_mem_insert] at hU
have hU' : ∀ i ∈ t, IsOpen (U i) := fun i hi => hU.2 i hi
rcases hs.binary_compact_cover hU.1 (isOpen_biUnion hU') hsC with
⟨K₁, K₂, h1K₁, h1K₂, h2K₁, h2K₂, hK⟩
rcases ih h1K₂ U hU' h2K₂ with ⟨K, h1K, h2K, h3K⟩
refine ⟨update K x K₁, ?_, ?_, ?_⟩
· intro i
rcases eq_or_ne i x with rfl | hi
· simp only [update_self, h1K₁]
· simp only [update_of_ne hi, h1K]
· intro i
rcases eq_or_ne i x with rfl | hi
· simp only [update_self, h2K₁]
· simp only [update_of_ne hi, h2K]
· simp only [Finset.set_biUnion_insert_update _ hx, hK, h3K] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | IsCompact.finite_compact_cover | For every finite open cover `Uᵢ` of a compact set, there exists a compact cover `Kᵢ ⊆ Uᵢ`. |
R1Space.of_continuous_specializes_imp [TopologicalSpace Y] {f : Y → X} (hc : Continuous f)
(hspec : ∀ x y, f x ⤳ f y → x ⤳ y) : R1Space Y where
specializes_or_disjoint_nhds x y := (specializes_or_disjoint_nhds (f x) (f y)).imp (hspec x y) <|
((hc.tendsto _).disjoint · (hc.tendsto _)) | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | R1Space.of_continuous_specializes_imp | null |
Topology.IsInducing.r1Space [TopologicalSpace Y] {f : Y → X} (hf : IsInducing f) :
R1Space Y := .of_continuous_specializes_imp hf.continuous fun _ _ ↦ hf.specializes_iff.1 | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | Topology.IsInducing.r1Space | null |
protected R1Space.induced (f : Y → X) : @R1Space Y (.induced f ‹_›) :=
@IsInducing.r1Space _ _ _ _ (.induced f _) f (.induced f) | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | R1Space.induced | null |
protected R1Space.sInf {X : Type*} {T : Set (TopologicalSpace X)}
(hT : ∀ t ∈ T, @R1Space X t) : @R1Space X (sInf T) := by
let _ := sInf T
refine ⟨fun x y ↦ ?_⟩
simp only [Specializes, nhds_sInf]
rcases em (∃ t ∈ T, Disjoint (@nhds X t x) (@nhds X t y)) with ⟨t, htT, htd⟩ | hTd
· exact .inr <| htd.mono (iInf₂_le t htT) (iInf₂_le t htT)
· push_neg at hTd
exact .inl <| iInf₂_mono fun t ht ↦ ((hT t ht).1 x y).resolve_right (hTd t ht) | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | R1Space.sInf | null |
protected R1Space.iInf {ι X : Type*} {t : ι → TopologicalSpace X}
(ht : ∀ i, @R1Space X (t i)) : @R1Space X (iInf t) :=
.sInf <| forall_mem_range.2 ht | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | R1Space.iInf | null |
protected R1Space.inf {X : Type*} {t₁ t₂ : TopologicalSpace X}
(h₁ : @R1Space X t₁) (h₂ : @R1Space X t₂) : @R1Space X (t₁ ⊓ t₂) := by
rw [inf_eq_iInf]
apply R1Space.iInf
simp [*] | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | R1Space.inf | null |
exists_mem_nhds_isCompact_mapsTo_of_isCompact_mem_nhds
{X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [R1Space Y] {f : X → Y} {x : X}
{K : Set X} {s : Set Y} (hf : Continuous f) (hs : s ∈ 𝓝 (f x)) (hKc : IsCompact K)
(hKx : K ∈ 𝓝 x) : ∃ L ∈ 𝓝 x, IsCompact L ∧ MapsTo f L s := by
have hc : IsCompact (f '' K \ interior s) := (hKc.image hf).diff isOpen_interior
obtain ⟨U, V, Uo, Vo, hxU, hV, hd⟩ : SeparatedNhds {f x} (f '' K \ interior s) := by
simp_rw [separatedNhds_iff_disjoint, nhdsSet_singleton, hc.disjoint_nhdsSet_right,
disjoint_nhds_nhds_iff_not_inseparable]
rintro y ⟨-, hys⟩ hxy
refine hys <| (hxy.mem_open_iff isOpen_interior).1 ?_
rwa [mem_interior_iff_mem_nhds]
refine ⟨K \ f ⁻¹' V, diff_mem hKx ?_, hKc.diff <| Vo.preimage hf, fun y hy ↦ ?_⟩
· filter_upwards [hf.continuousAt <| Uo.mem_nhds (hxU rfl)] with x hx
using Set.disjoint_left.1 hd hx
· by_contra hys
exact hy.2 (hV ⟨mem_image_of_mem _ hy.1, notMem_subset interior_subset hys⟩) | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | exists_mem_nhds_isCompact_mapsTo_of_isCompact_mem_nhds | null |
IsCompact.isCompact_isClosed_basis_nhds {x : X} {L : Set X} (hLc : IsCompact L)
(hxL : L ∈ 𝓝 x) : (𝓝 x).HasBasis (fun K ↦ K ∈ 𝓝 x ∧ IsCompact K ∧ IsClosed K) (·) :=
hasBasis_self.2 fun _U hU ↦
let ⟨K, hKx, hKc, hKU⟩ := exists_mem_nhds_isCompact_mapsTo_of_isCompact_mem_nhds
continuous_id (interior_mem_nhds.2 hU) hLc hxL
⟨closure K, mem_of_superset hKx subset_closure, ⟨hKc.closure, isClosed_closure⟩,
(hKc.closure_subset_of_isOpen isOpen_interior hKU).trans interior_subset⟩ | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | IsCompact.isCompact_isClosed_basis_nhds | If a point in an R₁ space has a compact neighborhood,
then it has a basis of compact closed neighborhoods. |
@[simp]
Filter.coclosedCompact_eq_cocompact : coclosedCompact X = cocompact X := by
refine le_antisymm ?_ cocompact_le_coclosedCompact
rw [hasBasis_coclosedCompact.le_basis_iff hasBasis_cocompact]
exact fun K hK ↦ ⟨closure K, ⟨isClosed_closure, hK.closure⟩, compl_subset_compl.2 subset_closure⟩ | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | Filter.coclosedCompact_eq_cocompact | In an R₁ space, the filters `coclosedCompact` and `cocompact` are equal. |
@[simp]
Bornology.relativelyCompact_eq_inCompact :
Bornology.relativelyCompact X = Bornology.inCompact X :=
Bornology.ext _ _ Filter.coclosedCompact_eq_cocompact
/-! | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | Bornology.relativelyCompact_eq_inCompact | In an R₁ space, the bornologies `relativelyCompact` and `inCompact` are equal. |
isCompact_isClosed_basis_nhds (x : X) :
(𝓝 x).HasBasis (fun K => K ∈ 𝓝 x ∧ IsCompact K ∧ IsClosed K) (·) :=
let ⟨_L, hLc, hLx⟩ := exists_compact_mem_nhds x
hLc.isCompact_isClosed_basis_nhds hLx | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | isCompact_isClosed_basis_nhds | In a (weakly) locally compact R₁ space, compact closed neighborhoods of a point `x`
form a basis of neighborhoods of `x`. |
exists_mem_nhds_isCompact_isClosed (x : X) : ∃ K ∈ 𝓝 x, IsCompact K ∧ IsClosed K :=
(isCompact_isClosed_basis_nhds x).ex_mem | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | exists_mem_nhds_isCompact_isClosed | In a (weakly) locally compact R₁ space, each point admits a compact closed neighborhood. |
exists_isOpen_superset_and_isCompact_closure {K : Set X} (hK : IsCompact K) :
∃ V, IsOpen V ∧ K ⊆ V ∧ IsCompact (closure V) := by
rcases exists_compact_superset hK with ⟨K', hK', hKK'⟩
exact ⟨interior K', isOpen_interior, hKK', hK'.closure_of_subset interior_subset⟩ | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | exists_isOpen_superset_and_isCompact_closure | A weakly locally compact R₁ space is locally compact. -/
instance (priority := 80) WeaklyLocallyCompactSpace.locallyCompactSpace : LocallyCompactSpace X :=
.of_hasBasis isCompact_isClosed_basis_nhds fun _ _ ⟨_, h, _⟩ ↦ h
/-- In a weakly locally compact R₁ space,
every compact set has an open neighborhood with compact closure. |
exists_isOpen_mem_isCompact_closure (x : X) :
∃ U : Set X, IsOpen U ∧ x ∈ U ∧ IsCompact (closure U) := by
simpa only [singleton_subset_iff]
using exists_isOpen_superset_and_isCompact_closure isCompact_singleton | theorem | Topology | [
"Mathlib.Algebra.Notation.Support",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Piecewise",
"Mathlib.Topology.Separation.SeparatedNhds",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Bases",
"Mathlib.Tactic.StacksAttribute"
] | Mathlib/Topology/Separation/Basic.lean | exists_isOpen_mem_isCompact_closure | In a weakly locally compact R₁ space,
every point has an open neighborhood with compact closure. |
@[mk_iff]
CompletelyRegularSpace (X : Type u) [TopologicalSpace X] : Prop where
completely_regular : ∀ (x : X), ∀ K : Set X, IsClosed K → x ∉ K →
∃ f : X → I, Continuous f ∧ f x = 0 ∧ EqOn f 1 K | class | 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 | A space is completely regular if points can be separated from closed sets via
continuous functions to the unit interval. |
completelyRegularSpace_iff_isOpen : CompletelyRegularSpace X ↔
∀ (x : X), ∀ K : Set X, IsOpen K → x ∈ K →
∃ f : X → I, Continuous f ∧ f x = 0 ∧ EqOn f 1 Kᶜ := by
conv_lhs => tactic =>
simp_rw +singlePass [completelyRegularSpace_iff, compl_surjective.forall, isClosed_compl_iff,
mem_compl_iff, not_not] | lemma | 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_isOpen | null |
CompletelyRegularSpace.completely_regular_isOpen [CompletelyRegularSpace X] :
∀ (x : X), ∀ K : Set X, IsOpen K → x ∈ K →
∃ f : X → I, Continuous f ∧ f x = 0 ∧ EqOn f 1 Kᶜ :=
completelyRegularSpace_iff_isOpen.mp inferInstance | lemma | 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.completely_regular_isOpen | null |
CompletelyRegularSpace.instRegularSpace [CompletelyRegularSpace X] :
RegularSpace X := by
rw [regularSpace_iff]
intro s a hs ha
obtain ⟨f, cf, hf, hhf⟩ := CompletelyRegularSpace.completely_regular a s hs ha
apply disjoint_of_map (f := f)
apply Disjoint.mono (cf.tendsto_nhdsSet_nhds hhf) cf.continuousAt
exact disjoint_nhds_nhds.mpr (hf.symm ▸ zero_ne_one).symm | instance | 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.instRegularSpace | null |
NormalSpace.instCompletelyRegularSpace [NormalSpace X] [R0Space X] :
CompletelyRegularSpace X := by
rw [completelyRegularSpace_iff]
intro x K hK hx
have cx : IsClosed (closure {x}) := isClosed_closure
have d : Disjoint (closure {x}) K := by
rw [Set.disjoint_iff]
intro a ⟨hax, haK⟩
exact hx ((specializes_iff_mem_closure.mpr hax).symm.mem_closed hK haK)
let ⟨⟨f, cf⟩, hfx, hfK, hficc⟩ := exists_continuous_zero_one_of_isClosed cx hK d
let g : X → I := fun x => ⟨f x, hficc x⟩
have cg : Continuous g := cf.subtype_mk hficc
have hgx : g x = 0 := Subtype.ext (hfx (subset_closure (mem_singleton x)))
have hgK : EqOn g 1 K := fun k hk => Subtype.ext (hfK hk)
exact ⟨g, cg, hgx, hgK⟩ | 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 | NormalSpace.instCompletelyRegularSpace | null |
Topology.IsInducing.completelyRegularSpace
{Y : Type v} [TopologicalSpace Y] [CompletelyRegularSpace Y]
{f : X → Y} (hf : IsInducing f) : CompletelyRegularSpace X where
completely_regular x K hK hxK := by
rw [hf.isClosed_iff] at hK
obtain ⟨K, hK, rfl⟩ := hK
rw [mem_preimage] at hxK
obtain ⟨g, hcf, egfx, hgK⟩ := CompletelyRegularSpace.completely_regular _ _ hK hxK
refine ⟨g ∘ f, hcf.comp hf.continuous, egfx, ?_⟩
conv => arg 2; equals (1 : Y → ↥I) ∘ f => rfl
exact hgK.comp_right <| mapsTo_preimage _ _ | 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.IsInducing.completelyRegularSpace | null |
completelyRegularSpace_induced
{X Y : Type*} {t : TopologicalSpace Y} (ht : @CompletelyRegularSpace Y t)
(f : X → Y) : @CompletelyRegularSpace X (t.induced f) :=
@IsInducing.completelyRegularSpace _ (t.induced f) _ t _ _ (IsInducing.induced f) | lemma | 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_induced | null |
completelyRegularSpace_iInf {ι X : Type*} {t : ι → TopologicalSpace X}
(ht : ∀ i, @CompletelyRegularSpace X (t i)) : @CompletelyRegularSpace X (⨅ i, t i) := by
letI := (⨅ i, t i) -- register this as default topological space to reduce `@`s
rw [completelyRegularSpace_iff_isOpen]
intro x K hK hxK
simp_rw [← hK.mem_nhds_iff, nhds_iInf, mem_iInf, exists_finite_iff_finset,
Finset.coe_sort_coe] at hxK; clear hK
obtain ⟨I', V, hV, rfl⟩ := hxK
simp only [mem_nhds_iff] at hV
choose U hUV hU hxU using hV
replace hU := fun (i : ↥I') =>
@CompletelyRegularSpace.completely_regular_isOpen _ (t i) (ht i) x (U i) (hU i) (hxU i)
clear hxU
choose fs hfs hxfs hfsU using hU
use I'.attach.sup fs
constructorm* _ ∧ _
· solve_by_elim [Continuous.finset_sup, continuous_iInf_dom]
· simpa [show (0 : ↥I) = ⊥ from rfl] using hxfs
· simp only [EqOn, Pi.one_apply, show (1 : ↥I) = ⊤ from rfl] at hfsU ⊢
conv => equals ∀ x i, x ∈ (V i)ᶜ → ∃ b, fs b x = ⊤ => simp [Finset.sup_eq_top_iff]
intro x i hxi
specialize hfsU i (by tauto_set)
exists i | 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_iInf | null |
completelyRegularSpace_inf {X : Type*} {t₁ t₂ : TopologicalSpace X}
(ht₁ : @CompletelyRegularSpace X t₁) (ht₂ : @CompletelyRegularSpace X t₂) :
@CompletelyRegularSpace X (t₁ ⊓ t₂) := by
rw [inf_eq_iInf]; apply completelyRegularSpace_iInf; simp [*] | lemma | 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_inf | null |
isInducing_stoneCechUnit [CompletelyRegularSpace X] :
IsInducing (stoneCechUnit : X → StoneCech X) := by
rw [isInducing_iff_nhds]
intro x
apply le_antisymm
· rw [← map_le_iff_le_comap]; exact continuous_stoneCechUnit.continuousAt
· simp_rw [le_nhds_iff, ((nhds_basis_opens _).comap _).mem_iff, and_assoc]
intro U hxU hU
obtain ⟨f, hf, efx, hfU⟩ :=
CompletelyRegularSpace.completely_regular_isOpen x U hU hxU
conv at hfU => equals Uᶜ ⊆ f ⁻¹' {1} => simp [EqOn, subset_def]
rw [← compl_subset_comm, ← preimage_compl, ← stoneCechExtend_extends hf, preimage_comp] at hfU
refine ⟨stoneCechExtend hf ⁻¹' {1}ᶜ, ?_,
isOpen_compl_singleton.preimage (continuous_stoneCechExtend hf), hfU⟩
rw [mem_preimage, stoneCechExtend_stoneCechUnit, efx, mem_compl_iff, mem_singleton_iff]
simp | 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 | isInducing_stoneCechUnit | null |
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