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 ⌀ |
|---|---|---|---|---|---|---|
nhdsSet_eq_principal_upperClosure (s : Set α) : 𝓝ˢ s = 𝓟 ↑(upperClosure s) := by
rw [← principal_nhdsKer, nhdsKer_eq_upperClosure] | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | nhdsSet_eq_principal_upperClosure | null |
protected _root_.Topology.isUpperSet_iff_nhds {α : Type*} [TopologicalSpace α] [Preorder α] :
Topology.IsUpperSet α ↔ (∀ a : α, 𝓝 a = 𝓟 (Ici a)) where
mp _ a := nhds_eq_principal_Ici a
mpr hα := ⟨by simp [TopologicalSpace.ext_iff_nhds, hα, nhds_eq_principal_Ici]⟩ | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | _root_.Topology.isUpperSet_iff_nhds | null |
protected monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f := by
constructor
· intro hf
simp_rw [continuous_def, isOpen_iff_isUpperSet]
exact fun _ hs ↦ IsUpperSet.preimage hs hf
· intro hf a b hab
rw [← mem_Iic, ← closure_singleton] at hab ⊢
apply Continuous.closure_preimage_subset hf {f b}
apply mem_of_mem_of_subset hab
apply closure_mono
rw [singleton_subset_iff, mem_preimage, mem_singleton_iff] | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | monotone_iff_continuous | null |
monotone_to_upperTopology_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [IsUpper β] {f : α → β} (hf : Monotone f) : Continuous f := by
simp_rw [continuous_def, isOpen_iff_isUpperSet]
intro s hs
exact (IsUpper.isUpperSet_of_isOpen hs).preimage hf | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | monotone_to_upperTopology_continuous | null |
upperSet_le_upper {t₁ t₂ : TopologicalSpace α} [@Topology.IsUpperSet α t₁ _]
[@Topology.IsUpper α t₂ _] : t₁ ≤ t₂ := fun s hs => by
rw [@isOpen_iff_isUpperSet α _ t₁]
exact IsUpper.isUpperSet_of_isOpen hs | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | upperSet_le_upper | null |
topology_eq : ‹_› = lowerSet α := topology_eq_lowerSetTopology
variable {α} | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | topology_eq | null |
_root_.OrderDual.instIsUpperSet [Preorder α] [TopologicalSpace α] [Topology.IsLowerSet α] :
Topology.IsUpperSet αᵒᵈ where
topology_eq_upperSetTopology := by ext; rw [IsLowerSet.topology_eq α] | instance | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | _root_.OrderDual.instIsUpperSet | null |
WithLowerSetHomeomorph : WithLowerSet α ≃ₜ α :=
WithLowerSet.ofLowerSet.toHomeomorphOfIsInducing ⟨topology_eq α ▸ induced_id.symm⟩ | def | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | WithLowerSetHomeomorph | If `α` is equipped with the lower set topology, then it is homeomorphic to `WithLowerSet α`. |
isOpen_iff_isLowerSet : IsOpen s ↔ IsLowerSet s := by rw [topology_eq α]; rfl | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | isOpen_iff_isLowerSet | null |
toAlexandrovDiscrete : AlexandrovDiscrete α := IsUpperSet.toAlexandrovDiscrete (α := αᵒᵈ) | instance | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | toAlexandrovDiscrete | null |
isClosed_iff_isUpper : IsClosed s ↔ IsUpperSet s := by
rw [← isOpen_compl_iff, isOpen_iff_isLowerSet, isUpperSet_compl.symm, compl_compl] | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | isClosed_iff_isUpper | null |
closure_eq_upperClosure {s : Set α} : closure s = upperClosure s :=
IsUpperSet.closure_eq_lowerClosure (α := αᵒᵈ) | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | closure_eq_upperClosure | null |
@[simp] closure_singleton {a : α} : closure {a} = Ici a := by
rw [closure_eq_upperClosure, upperClosure_singleton]
rfl | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | closure_singleton | The closure of a singleton `{a}` in the lower set topology is the right-closed left-infinite
interval (-∞,a]. |
specializes_iff_le {a b : α} : a ⤳ b ↔ a ≤ b := by
simp only [specializes_iff_closure_subset, closure_singleton, Ici_subset_Ici] | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | specializes_iff_le | null |
nhdsKer_eq_lowerClosure (s : Set α) : nhdsKer s = ↑(lowerClosure s) := by
ext; simp [mem_nhdsKer_iff_specializes, specializes_iff_le]
@[simp] lemma nhdsKer_singleton (a : α) : nhdsKer {a} = Iic a := by
rw [nhdsKer_eq_lowerClosure, lowerClosure_singleton, LowerSet.coe_Iic] | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | nhdsKer_eq_lowerClosure | null |
nhds_eq_principal_Iic (a : α) : 𝓝 a = 𝓟 (Iic a) := by
rw [← principal_nhdsKer_singleton, nhdsKer_singleton] | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | nhds_eq_principal_Iic | null |
nhdsSet_eq_principal_lowerClosure (s : Set α) : 𝓝ˢ s = 𝓟 ↑(lowerClosure s) := by
rw [← principal_nhdsKer, nhdsKer_eq_lowerClosure] | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | nhdsSet_eq_principal_lowerClosure | null |
protected _root_.Topology.isLowerSet_iff_nhds {α : Type*} [TopologicalSpace α] [Preorder α] :
Topology.IsLowerSet α ↔ (∀ a : α, 𝓝 a = 𝓟 (Iic a)) where
mp _ a := nhds_eq_principal_Iic a
mpr hα := ⟨by simp [TopologicalSpace.ext_iff_nhds, hα, nhds_eq_principal_Iic]⟩ | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | _root_.Topology.isLowerSet_iff_nhds | null |
protected monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsLowerSet α] [Topology.IsLowerSet β] {f : α → β} : Monotone f ↔ Continuous f := by
rw [← monotone_dual_iff]
exact IsUpperSet.monotone_iff_continuous (α := αᵒᵈ) (β := βᵒᵈ)
(f := (toDual ∘ f ∘ ofDual : αᵒᵈ → βᵒᵈ)) | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | monotone_iff_continuous | null |
monotone_to_lowerTopology_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsLowerSet α] [IsLower β] {f : α → β} (hf : Monotone f) : Continuous f :=
IsUpperSet.monotone_to_upperTopology_continuous (α := αᵒᵈ) (β := βᵒᵈ) hf.dual | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | monotone_to_lowerTopology_continuous | null |
lowerSet_le_lower {t₁ t₂ : TopologicalSpace α} [@Topology.IsLowerSet α t₁ _]
[@IsLower α t₂ _] : t₁ ≤ t₂ := fun s hs => by
rw [@isOpen_iff_isLowerSet α _ t₁]
exact IsLower.isLowerSet_of_isOpen hs | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | lowerSet_le_lower | null |
isUpperSet_orderDual [Preorder α] [TopologicalSpace α] :
Topology.IsUpperSet αᵒᵈ ↔ Topology.IsLowerSet α := by
constructor
· apply OrderDual.instIsLowerSet
· apply OrderDual.instIsUpperSet | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | isUpperSet_orderDual | null |
isLowerSet_orderDual [Preorder α] [TopologicalSpace α] :
Topology.IsLowerSet αᵒᵈ ↔ Topology.IsUpperSet α := isUpperSet_orderDual.symm | lemma | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | isLowerSet_orderDual | null |
map (f : α →o β) : C(WithUpperSet α, WithUpperSet β) where
toFun := toUpperSet ∘ f ∘ ofUpperSet
continuous_toFun := continuous_def.2 fun _s hs ↦ IsUpperSet.preimage hs f.monotone
@[simp] lemma map_id : map (OrderHom.id : α →o α) = ContinuousMap.id _ := rfl
@[simp] lemma map_comp (g : β →o γ) (f : α →o β) : map (g.comp f) = (map g).comp (map f) := rfl
@[simp] lemma toUpperSet_specializes_toUpperSet {a b : α} :
toUpperSet a ⤳ toUpperSet b ↔ b ≤ a := by
simp_rw [specializes_iff_closure_subset, IsUpperSet.closure_singleton, Iic_subset_Iic,
toUpperSet_le_iff]
@[simp] lemma ofUpperSet_le_ofUpperSet {a b : WithUpperSet α} :
ofUpperSet a ≤ ofUpperSet b ↔ b ⤳ a := toUpperSet_specializes_toUpperSet.symm
@[simp] lemma isUpperSet_toUpperSet_preimage {s : Set (WithUpperSet α)} :
IsUpperSet (toUpperSet ⁻¹' s) ↔ IsOpen s := Iff.rfl
@[simp] lemma isOpen_ofUpperSet_preimage {s : Set α} :
IsOpen (ofUpperSet ⁻¹' s) ↔ IsUpperSet s := isUpperSet_toUpperSet_preimage.symm | def | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | map | A monotone map between preorders spaces induces a continuous map between themselves considered
with the upper set topology. |
map (f : α →o β) : C(WithLowerSet α, WithLowerSet β) where
toFun := toLowerSet ∘ f ∘ ofLowerSet
continuous_toFun := continuous_def.2 fun _s hs ↦ IsLowerSet.preimage hs f.monotone
@[simp] lemma map_id : map (OrderHom.id : α →o α) = ContinuousMap.id _ := rfl
@[simp] lemma map_comp (g : β →o γ) (f : α →o β) : map (g.comp f) = (map g).comp (map f) := rfl
@[simp] lemma toLowerSet_specializes_toLowerSet {a b : α} :
toLowerSet a ⤳ toLowerSet b ↔ a ≤ b := by
simp_rw [specializes_iff_closure_subset, IsLowerSet.closure_singleton, Ici_subset_Ici,
toLowerSet_le_iff]
@[simp] lemma ofLowerSet_le_ofLowerSet {a b : WithLowerSet α} :
ofLowerSet a ≤ ofLowerSet b ↔ a ⤳ b := toLowerSet_specializes_toLowerSet.symm
@[simp] lemma isLowerSet_toLowerSet_preimage {s : Set (WithLowerSet α)} :
IsLowerSet (toLowerSet ⁻¹' s) ↔ IsOpen s := Iff.rfl
@[simp] lemma isOpen_ofLowerSet_preimage {s : Set α} :
IsOpen (ofLowerSet ⁻¹' s) ↔ IsLowerSet s := isLowerSet_toLowerSet_preimage.symm | def | Topology | [
"Mathlib.Logic.Lemmas",
"Mathlib.Topology.AlexandrovDiscrete",
"Mathlib.Topology.ContinuousMap.Basic",
"Mathlib.Topology.Order.LowerUpperTopology"
] | Mathlib/Topology/Order/UpperLowerSetTopology.lean | map | A monotone map between preorders spaces induces a continuous map between themselves considered
with the lower set topology. |
@[simp]
nhdsKer_eq_of_t1Space [T1Space X] (s : Set X) : nhdsKer s = s := by
ext; simp [mem_nhdsKer_iff_specializes] | lemma | Topology | [
"Mathlib.Topology.Separation.Basic",
"Mathlib.Topology.AlexandrovDiscrete"
] | Mathlib/Topology/Separation/AlexandrovDiscrete.lean | nhdsKer_eq_of_t1Space | null |
@[stacks 004X "(2)"]
T0Space (X : Type u) [TopologicalSpace X] : Prop where
/-- Two inseparable points in a T₀ space are equal. -/
t0 : ∀ ⦃x y : X⦄, Inseparable x y → x = y | 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 | T0Space | A T₀ space, also known as a Kolmogorov space, is a topological space such that for every pair
`x ≠ y`, there is an open set containing one but not the other. We formulate the definition in terms
of the `Inseparable` relation. |
t0Space_iff_inseparable (X : Type u) [TopologicalSpace X] :
T0Space X ↔ ∀ x y : X, Inseparable x y → x = y :=
⟨fun ⟨h⟩ => h, fun h => ⟨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 | t0Space_iff_inseparable | null |
t0Space_iff_not_inseparable (X : Type u) [TopologicalSpace X] :
T0Space X ↔ Pairwise fun x y : X => ¬Inseparable x y := by
simp only [t0Space_iff_inseparable, Ne, not_imp_not, Pairwise] | 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 | t0Space_iff_not_inseparable | null |
Inseparable.eq [T0Space X] {x y : X} (h : Inseparable x y) : x = y :=
T0Space.t0 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 | Inseparable.eq | null |
protected Topology.IsInducing.injective [TopologicalSpace Y] [T0Space X] {f : X → Y}
(hf : IsInducing f) : Injective f := fun _ _ h =>
(hf.inseparable_iff.1 <| .of_eq h).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 | Topology.IsInducing.injective | A topology inducing map from a T₀ space is injective. |
protected Topology.IsInducing.isEmbedding [TopologicalSpace Y] [T0Space X] {f : X → Y}
(hf : IsInducing f) : IsEmbedding f :=
⟨hf, hf.injective⟩ | 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.isEmbedding | A topology inducing map from a T₀ space is a topological embedding. |
isEmbedding_iff_isInducing [TopologicalSpace Y] [T0Space X] {f : X → Y} :
IsEmbedding f ↔ IsInducing f :=
⟨IsEmbedding.isInducing, IsInducing.isEmbedding⟩ | 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 | isEmbedding_iff_isInducing | null |
t0Space_iff_nhds_injective (X : Type u) [TopologicalSpace X] :
T0Space X ↔ Injective (𝓝 : X → Filter X) :=
t0Space_iff_inseparable 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 | t0Space_iff_nhds_injective | null |
nhds_injective [T0Space X] : Injective (𝓝 : X → Filter X) :=
(t0Space_iff_nhds_injective X).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 | nhds_injective | null |
inseparable_iff_eq [T0Space X] {x y : X} : Inseparable x y ↔ x = y :=
nhds_injective.eq_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 | inseparable_iff_eq | null |
nhds_eq_nhds_iff [T0Space X] {a b : X} : 𝓝 a = 𝓝 b ↔ a = b :=
nhds_injective.eq_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 | nhds_eq_nhds_iff | null |
inseparable_eq_eq [T0Space X] : Inseparable = @Eq X :=
funext₂ fun _ _ => propext inseparable_iff_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 | inseparable_eq_eq | null |
TopologicalSpace.IsTopologicalBasis.inseparable_iff {b : Set (Set X)}
(hb : IsTopologicalBasis b) {x y : X} : Inseparable x y ↔ ∀ s ∈ b, (x ∈ s ↔ y ∈ s) :=
⟨fun h _ hs ↦ inseparable_iff_forall_isOpen.1 h _ (hb.isOpen hs),
fun h ↦ hb.nhds_hasBasis.eq_of_same_basis <| by
convert hb.nhds_hasBasis using 2
exact and_congr_right (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 | TopologicalSpace.IsTopologicalBasis.inseparable_iff | null |
TopologicalSpace.IsTopologicalBasis.eq_iff [T0Space X] {b : Set (Set X)}
(hb : IsTopologicalBasis b) {x y : X} : x = y ↔ ∀ s ∈ b, (x ∈ s ↔ y ∈ s) :=
inseparable_iff_eq.symm.trans hb.inseparable_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 | TopologicalSpace.IsTopologicalBasis.eq_iff | null |
t0Space_iff_exists_isOpen_xor'_mem (X : Type u) [TopologicalSpace X] :
T0Space X ↔ Pairwise fun x y => ∃ U : Set X, IsOpen U ∧ Xor' (x ∈ U) (y ∈ U) := by
simp only [t0Space_iff_not_inseparable, xor_iff_not_iff, not_forall, exists_prop,
inseparable_iff_forall_isOpen, Pairwise] | 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 | t0Space_iff_exists_isOpen_xor'_mem | null |
exists_isOpen_xor'_mem [T0Space X] {x y : X} (h : x ≠ y) :
∃ U : Set X, IsOpen U ∧ Xor' (x ∈ U) (y ∈ U) :=
(t0Space_iff_exists_isOpen_xor'_mem X).1 ‹_› 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 | exists_isOpen_xor'_mem | null |
specializationOrder (X) [TopologicalSpace X] [T0Space X] : PartialOrder X :=
{ specializationPreorder X, PartialOrder.lift (OrderDual.toDual ∘ 𝓝) nhds_injective with } | def | 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 | specializationOrder | Specialization forms a partial order on a t0 topological space. |
SeparationQuotient.instT0Space : T0Space (SeparationQuotient X) :=
⟨fun x y => Quotient.inductionOn₂' x y fun _ _ h =>
SeparationQuotient.mk_eq_mk.2 <| SeparationQuotient.isInducing_mk.inseparable_iff.1 h⟩ | 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 | SeparationQuotient.instT0Space | null |
minimal_nonempty_closed_subsingleton [T0Space X] {s : Set X} (hs : IsClosed s)
(hmin : ∀ t, t ⊆ s → t.Nonempty → IsClosed t → t = s) : s.Subsingleton := by
refine fun x hx y hy => of_not_not fun hxy => ?_
rcases exists_isOpen_xor'_mem hxy with ⟨U, hUo, hU⟩
wlog h : x ∈ U ∧ y ∉ U
· refine this hs hmin y hy x hx (Ne.symm hxy) U hUo hU.symm (hU.resolve_left h)
obtain ⟨hxU, hyU⟩ := h
have : s \ U = s := hmin (s \ U) diff_subset ⟨y, hy, hyU⟩ (hs.sdiff hUo)
exact (this.symm.subset hx).2 hxU | 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 | minimal_nonempty_closed_subsingleton | null |
minimal_nonempty_closed_eq_singleton [T0Space X] {s : Set X} (hs : IsClosed s)
(hne : s.Nonempty) (hmin : ∀ t, t ⊆ s → t.Nonempty → IsClosed t → t = s) : ∃ x, s = {x} :=
exists_eq_singleton_iff_nonempty_subsingleton.2
⟨hne, minimal_nonempty_closed_subsingleton hs hmin⟩ | 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 | minimal_nonempty_closed_eq_singleton | null |
IsClosed.exists_closed_singleton [T0Space X] [CompactSpace X] {S : Set X}
(hS : IsClosed S) (hne : S.Nonempty) : ∃ x : X, x ∈ S ∧ IsClosed ({x} : Set X) := by
obtain ⟨V, Vsub, Vne, Vcls, hV⟩ := hS.exists_minimal_nonempty_closed_subset hne
rcases minimal_nonempty_closed_eq_singleton Vcls Vne hV with ⟨x, rfl⟩
exact ⟨x, Vsub (mem_singleton x), Vcls⟩ | 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.exists_closed_singleton | Given a closed set `S` in a compact T₀ space, there is some `x ∈ S` such that `{x}` is
closed. |
minimal_nonempty_open_subsingleton [T0Space X] {s : Set X} (hs : IsOpen s)
(hmin : ∀ t, t ⊆ s → t.Nonempty → IsOpen t → t = s) : s.Subsingleton := by
refine fun x hx y hy => of_not_not fun hxy => ?_
rcases exists_isOpen_xor'_mem hxy with ⟨U, hUo, hU⟩
wlog h : x ∈ U ∧ y ∉ U
· exact this hs hmin y hy x hx (Ne.symm hxy) U hUo hU.symm (hU.resolve_left h)
obtain ⟨hxU, hyU⟩ := h
have : s ∩ U = s := hmin (s ∩ U) inter_subset_left ⟨x, hx, hxU⟩ (hs.inter hUo)
exact hyU (this.symm.subset hy).2 | 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 | minimal_nonempty_open_subsingleton | null |
minimal_nonempty_open_eq_singleton [T0Space X] {s : Set X} (hs : IsOpen s)
(hne : s.Nonempty) (hmin : ∀ t, t ⊆ s → t.Nonempty → IsOpen t → t = s) : ∃ x, s = {x} :=
exists_eq_singleton_iff_nonempty_subsingleton.2 ⟨hne, minimal_nonempty_open_subsingleton hs hmin⟩ | 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 | minimal_nonempty_open_eq_singleton | null |
exists_isOpen_singleton_of_isOpen_finite [T0Space X] {s : Set X} (hfin : s.Finite)
(hne : s.Nonempty) (ho : IsOpen s) : ∃ x ∈ s, IsOpen ({x} : Set X) := by
lift s to Finset X using hfin
induction s using Finset.strongInductionOn
rename_i s ihs
rcases em (∃ t, t ⊂ s ∧ t.Nonempty ∧ IsOpen (t : Set X)) with (⟨t, hts, htne, hto⟩ | ht)
· rcases ihs t hts htne hto with ⟨x, hxt, hxo⟩
exact ⟨x, hts.1 hxt, hxo⟩
· -- Porting note: was `rcases minimal_nonempty_open_eq_singleton ho hne _ with ⟨x, hx⟩`
rsuffices ⟨x, hx⟩ : ∃ x, s.toSet = {x}
· exact ⟨x, hx.symm ▸ rfl, hx ▸ ho⟩
refine minimal_nonempty_open_eq_singleton ho hne ?_
refine fun t hts htne hto => of_not_not fun hts' => ht ?_
lift t to Finset X using s.finite_toSet.subset hts
exact ⟨t, ssubset_iff_subset_ne.2 ⟨hts, mt Finset.coe_inj.2 hts'⟩, htne, hto⟩ | 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_singleton_of_isOpen_finite | Given an open finite set `S` in a T₀ space, there is some `x ∈ S` such that `{x}` is open. |
exists_open_singleton_of_finite [T0Space X] [Finite X] [Nonempty X] :
∃ x : X, IsOpen ({x} : Set X) :=
let ⟨x, _, h⟩ := exists_isOpen_singleton_of_isOpen_finite (Set.toFinite _)
univ_nonempty isOpen_univ
⟨x, 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 | exists_open_singleton_of_finite | null |
t0Space_of_injective_of_continuous [TopologicalSpace Y] {f : X → Y}
(hf : Function.Injective f) (hf' : Continuous f) [T0Space Y] : T0Space X :=
⟨fun _ _ h => hf <| (h.map hf').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 | t0Space_of_injective_of_continuous | null |
protected Topology.IsEmbedding.t0Space [TopologicalSpace Y] [T0Space Y] {f : X → Y}
(hf : IsEmbedding f) : T0Space X :=
t0Space_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.t0Space | null |
protected Homeomorph.t0Space [TopologicalSpace Y] [T0Space X] (h : X ≃ₜ Y) : T0Space Y :=
h.symm.isEmbedding.t0Space
@[stacks 0B31 "part 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 | Homeomorph.t0Space | null |
Subtype.t0Space [T0Space X] {p : X → Prop} : T0Space (Subtype p) :=
IsEmbedding.subtypeVal.t0Space | 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.t0Space | null |
t0Space_iff_or_notMem_closure (X : Type u) [TopologicalSpace X] :
T0Space X ↔ Pairwise fun a b : X => a ∉ closure ({b} : Set X) ∨ b ∉ closure ({a} : Set X) := by
simp only [t0Space_iff_not_inseparable, inseparable_iff_mem_closure, not_and_or]
@[deprecated (since := "2025-05-23")]
alias t0Space_iff_or_not_mem_closure := t0Space_iff_or_notMem_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 | t0Space_iff_or_notMem_closure | null |
Prod.instT0Space [TopologicalSpace Y] [T0Space X] [T0Space Y] : T0Space (X × Y) :=
⟨fun _ _ h => Prod.ext (h.map continuous_fst).eq (h.map continuous_snd).eq⟩ | 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 | Prod.instT0Space | null |
Pi.instT0Space {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)]
[∀ i, T0Space (X i)] :
T0Space (∀ i, X i) :=
⟨fun _ _ h => funext fun i => (h.map (continuous_apply i)).eq⟩ | 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 | Pi.instT0Space | null |
ULift.instT0Space [T0Space X] : T0Space (ULift X) := IsEmbedding.uliftDown.t0Space | 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.instT0Space | null |
T0Space.of_cover (h : ∀ x y, Inseparable x y → ∃ s : Set X, x ∈ s ∧ y ∈ s ∧ T0Space s) :
T0Space X := by
refine ⟨fun x y hxy => ?_⟩
rcases h x y hxy with ⟨s, hxs, hys, hs⟩
lift x to s using hxs; lift y to s using hys
rw [← subtype_inseparable_iff] at hxy
exact congr_arg Subtype.val hxy.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 | T0Space.of_cover | null |
T0Space.of_open_cover (h : ∀ x, ∃ s : Set X, x ∈ s ∧ IsOpen s ∧ T0Space s) : T0Space X :=
T0Space.of_cover fun x _ hxy =>
let ⟨s, hxs, hso, hs⟩ := h x
⟨s, hxs, (hxy.mem_open_iff hso).1 hxs, 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 | T0Space.of_open_cover | null |
@[mk_iff]
R0Space (X : Type u) [TopologicalSpace X] : Prop where
/-- In an R₀ space, the `Specializes` relation is symmetric. -/
specializes_symmetric : Symmetric (Specializes : X → X → Prop)
export R0Space (specializes_symmetric) | 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 | R0Space | A topological space is called an R₀ space, if `Specializes` relation is symmetric.
In other words, given two points `x y : X`,
if every neighborhood of `y` contains `x`, then every neighborhood of `x` contains `y`. |
Specializes.symm (h : x ⤳ y) : y ⤳ x := specializes_symmetric 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 | Specializes.symm | In an R₀ space, the `Specializes` relation is symmetric, dot notation version. |
specializes_comm : x ⤳ y ↔ y ⤳ x := ⟨Specializes.symm, Specializes.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_comm | In an R₀ space, the `Specializes` relation is symmetric, `Iff` version. |
specializes_iff_inseparable : x ⤳ y ↔ Inseparable x y :=
⟨fun h ↦ h.antisymm h.symm, 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 | specializes_iff_inseparable | In an R₀ space, `Specializes` is equivalent to `Inseparable`. |
isCompact_closure_singleton : IsCompact (closure {x}) := by
refine isCompact_of_finite_subcover fun U hUo hxU ↦ ?_
obtain ⟨i, hi⟩ : ∃ i, x ∈ U i := mem_iUnion.1 <| hxU <| subset_closure rfl
refine ⟨{i}, fun y hy ↦ ?_⟩
rw [← specializes_iff_mem_closure, specializes_comm] at hy
simpa using hy.mem_open (hUo i) hi | 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_singleton | In an R₀ space, `Specializes` implies `Inseparable`. -/
alias ⟨Specializes.inseparable, _⟩ := specializes_iff_inseparable
theorem Topology.IsInducing.r0Space [TopologicalSpace Y] {f : Y → X} (hf : IsInducing f) :
R0Space Y where
specializes_symmetric a b := by
simpa only [← hf.specializes_iff] using Specializes.symm
instance {p : X → Prop} : R0Space {x // p x} := IsInducing.subtypeVal.r0Space
instance [TopologicalSpace Y] [R0Space Y] : R0Space (X × Y) where
specializes_symmetric _ _ h := h.fst.symm.prod h.snd.symm
instance {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i, R0Space (X i)] :
R0Space (∀ i, X i) where
specializes_symmetric _ _ h := specializes_pi.2 fun i ↦ (specializes_pi.1 h i).symm
/-- In an R₀ space, the closure of a singleton is a compact set. |
Filter.coclosedCompact_le_cofinite : coclosedCompact X ≤ cofinite :=
le_cofinite_iff_compl_singleton_mem.2 fun _ ↦
compl_mem_coclosedCompact.2 isCompact_closure_singleton
variable (X) in | 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_le_cofinite | null |
Bornology.relativelyCompact : Bornology X where
cobounded := Filter.coclosedCompact X
le_cofinite := Filter.coclosedCompact_le_cofinite | def | 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 | In an R₀ space, relatively compact sets form a bornology.
Its cobounded filter is `Filter.coclosedCompact`.
See also `Bornology.inCompact` the bornology of sets contained in a compact set. |
Bornology.relativelyCompact.isBounded_iff {s : Set X} :
@Bornology.IsBounded _ (Bornology.relativelyCompact X) s ↔ IsCompact (closure s) :=
compl_mem_coclosedCompact | 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.isBounded_iff | null |
Set.Finite.isCompact_closure {s : Set X} (hs : s.Finite) : IsCompact (closure s) :=
let _ : Bornology X := .relativelyCompact X
Bornology.relativelyCompact.isBounded_iff.1 hs.isBounded | 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.isCompact_closure | In an R₀ space, the closure of a finite set is a compact set. |
T1Space (X : Type u) [TopologicalSpace X] : Prop where
/-- A singleton in a T₁ space is a closed set. -/
t1 : ∀ x, IsClosed ({x} : Set 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 | T1Space | A T₁ space, also known as a Fréchet space, is a topological space
where every singleton set is closed. Equivalently, for every pair
`x ≠ y`, there is an open set containing `x` and not `y`. |
isClosed_singleton [T1Space X] {x : X} : IsClosed ({x} : Set X) :=
T1Space.t1 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 | isClosed_singleton | null |
isOpen_compl_singleton [T1Space X] {x : X} : IsOpen ({x}ᶜ : Set X) :=
isClosed_singleton.isOpen_compl | 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_compl_singleton | null |
isOpen_ne [T1Space X] {x : X} : IsOpen { y | y ≠ x } :=
isOpen_compl_singleton
@[to_additive] | 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_ne | null |
Continuous.isOpen_mulSupport [T1Space X] [One X] [TopologicalSpace Y] {f : Y → X}
(hf : Continuous f) : IsOpen (mulSupport f) :=
isOpen_ne.preimage hf | 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 | Continuous.isOpen_mulSupport | null |
Ne.nhdsWithin_compl_singleton [T1Space X] {x y : X} (h : x ≠ y) : 𝓝[{y}ᶜ] x = 𝓝 x :=
isOpen_ne.nhdsWithin_eq 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 | Ne.nhdsWithin_compl_singleton | null |
Ne.nhdsWithin_diff_singleton [T1Space X] {x y : X} (h : x ≠ y) (s : Set X) :
𝓝[s \ {y}] x = 𝓝[s] x := by
rw [diff_eq, inter_comm, nhdsWithin_inter_of_mem]
exact mem_nhdsWithin_of_mem_nhds (isOpen_ne.mem_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 | Ne.nhdsWithin_diff_singleton | null |
nhdsWithin_compl_singleton_le [T1Space X] (x y : X) : 𝓝[{x}ᶜ] x ≤ 𝓝[{y}ᶜ] x := by
rcases eq_or_ne x y with rfl | hy
· exact Eq.le rfl
· rw [Ne.nhdsWithin_compl_singleton hy]
exact nhdsWithin_le_nhds | 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 | nhdsWithin_compl_singleton_le | null |
isOpen_setOf_eventually_nhdsWithin [T1Space X] {p : X → Prop} :
IsOpen { x | ∀ᶠ y in 𝓝[≠] x, p y } := by
refine isOpen_iff_mem_nhds.mpr fun a ha => ?_
filter_upwards [eventually_nhds_nhdsWithin.mpr ha] with b hb
rcases eq_or_ne a b with rfl | h
· exact hb
· rw [h.symm.nhdsWithin_compl_singleton] at hb
exact hb.filter_mono nhdsWithin_le_nhds
@[simp] protected lemma Set.Finite.isClosed [T1Space X] {s : Set X} (hs : s.Finite) :
IsClosed s := by
rw [← biUnion_of_singleton s]
exact hs.isClosed_biUnion fun i _ => 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 | isOpen_setOf_eventually_nhdsWithin | null |
TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne [T1Space X] {b : Set (Set X)}
(hb : IsTopologicalBasis b) {x y : X} (h : x ≠ y) : ∃ a ∈ b, x ∈ a ∧ y ∉ a := by
rcases hb.isOpen_iff.1 isOpen_ne x h with ⟨a, ab, xa, ha⟩
exact ⟨a, ab, xa, fun h => ha h rfl⟩ | 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 | TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne | null |
protected Finset.isClosed [T1Space X] (s : Finset X) : IsClosed (s : Set X) :=
s.finite_toSet.isClosed | 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 | Finset.isClosed | null |
t1Space_TFAE (X : Type u) [TopologicalSpace X] :
List.TFAE [T1Space X,
∀ x, IsClosed ({ x } : Set X),
∀ x, IsOpen ({ x }ᶜ : Set X),
Continuous (@CofiniteTopology.of X),
∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x,
∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
∀ ⦃x y : X⦄, x ≠ y → ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ y ∉ U,
∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y),
∀ ⦃x y : X⦄, x ⤳ y → x = y,
T0Space X ∧ R0Space X] := by
tfae_have 1 ↔ 2 := ⟨fun h => h.1, fun h => ⟨h⟩⟩
tfae_have 2 ↔ 3 := by
simp only [isOpen_compl_iff]
tfae_have 5 ↔ 3 := by
refine forall_swap.trans ?_
simp only [isOpen_iff_mem_nhds, mem_compl_iff, mem_singleton_iff]
tfae_have 5 ↔ 6 := by
simp only [← subset_compl_singleton_iff, exists_mem_subset_iff]
tfae_have 5 ↔ 7 := by
simp only [(nhds_basis_opens _).mem_iff, subset_compl_singleton_iff, and_assoc,
and_left_comm]
tfae_have 5 ↔ 8 := by
simp only [← principal_singleton, disjoint_principal_right]
tfae_have 8 ↔ 9 := forall_swap.trans (by simp only [disjoint_comm, ne_comm])
tfae_have 1 → 4 := by
simp only [continuous_def, CofiniteTopology.isOpen_iff']
rintro H s (rfl | hs)
exacts [isOpen_empty, compl_compl s ▸ (@Set.Finite.isClosed _ _ H _ hs).isOpen_compl]
tfae_have 4 → 2 :=
fun h x => (CofiniteTopology.isClosed_iff.2 <| Or.inr (finite_singleton _)).preimage h
tfae_have 2 ↔ 10 := by
simp only [← closure_subset_iff_isClosed, specializes_iff_mem_closure, subset_def,
mem_singleton_iff, eq_comm]
tfae_have 10 ↔ 11 :=
⟨fun h => ⟨⟨fun _ _ h₂ => h h₂.specializes⟩, ⟨fun _ _ h₂ => specializes_of_eq (h h₂).symm⟩⟩,
fun ⟨_, _⟩ _ _ h => (h.antisymm h.symm).eq⟩
tfae_finish | 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 | t1Space_TFAE | null |
t1Space_iff_continuous_cofinite_of : T1Space X ↔ Continuous (@CofiniteTopology.of X) :=
(t1Space_TFAE X).out 0 3 | 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 | t1Space_iff_continuous_cofinite_of | null |
CofiniteTopology.continuous_of [T1Space X] : Continuous (@CofiniteTopology.of X) :=
t1Space_iff_continuous_cofinite_of.mp ‹_› | 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 | CofiniteTopology.continuous_of | null |
t1Space_iff_exists_open :
T1Space X ↔ Pairwise fun x y => ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ y ∉ U :=
(t1Space_TFAE X).out 0 6 | 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 | t1Space_iff_exists_open | null |
t1Space_iff_disjoint_pure_nhds : T1Space X ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y) :=
(t1Space_TFAE X).out 0 8 | 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 | t1Space_iff_disjoint_pure_nhds | null |
t1Space_iff_disjoint_nhds_pure : T1Space X ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y) :=
(t1Space_TFAE X).out 0 7 | 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 | t1Space_iff_disjoint_nhds_pure | null |
t1Space_iff_specializes_imp_eq : T1Space X ↔ ∀ ⦃x y : X⦄, x ⤳ y → x = y :=
(t1Space_TFAE X).out 0 9 | 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 | t1Space_iff_specializes_imp_eq | null |
t1Space_iff_t0Space_and_r0Space : T1Space X ↔ T0Space X ∧ R0Space X :=
(t1Space_TFAE X).out 0 10 | 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 | t1Space_iff_t0Space_and_r0Space | null |
disjoint_pure_nhds [T1Space X] {x y : X} (h : x ≠ y) : Disjoint (pure x) (𝓝 y) :=
t1Space_iff_disjoint_pure_nhds.mp ‹_› 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 | disjoint_pure_nhds | null |
disjoint_nhds_pure [T1Space X] {x y : X} (h : x ≠ y) : Disjoint (𝓝 x) (pure y) :=
t1Space_iff_disjoint_nhds_pure.mp ‹_› 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 | disjoint_nhds_pure | null |
Specializes.eq [T1Space X] {x y : X} (h : x ⤳ y) : x = y :=
t1Space_iff_specializes_imp_eq.1 ‹_› 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 | Specializes.eq | null |
specializes_iff_eq [T1Space X] {x y : X} : x ⤳ y ↔ x = y :=
⟨Specializes.eq, fun h => h ▸ specializes_rfl⟩
@[simp] theorem specializes_eq_eq [T1Space X] : (· ⤳ ·) = @Eq X :=
funext₂ fun _ _ => propext specializes_iff_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 | specializes_iff_eq | null |
pure_le_nhds_iff [T1Space X] {a b : X} : pure a ≤ 𝓝 b ↔ a = b :=
specializes_iff_pure.symm.trans specializes_iff_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 | pure_le_nhds_iff | null |
nhds_le_nhds_iff [T1Space X] {a b : X} : 𝓝 a ≤ 𝓝 b ↔ a = b :=
specializes_iff_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 | nhds_le_nhds_iff | null |
t1Space_antitone {X} : Antitone (@T1Space X) := fun a _ h _ =>
@T1Space.mk _ a fun x => (T1Space.t1 x).mono 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 | t1Space_antitone | null |
continuousWithinAt_update_of_ne [T1Space X] [DecidableEq X] [TopologicalSpace Y] {f : X → Y}
{s : Set X} {x x' : X} {y : Y} (hne : x' ≠ x) :
ContinuousWithinAt (Function.update f x y) s x' ↔ ContinuousWithinAt f s x' :=
EventuallyEq.congr_continuousWithinAt
(mem_nhdsWithin_of_mem_nhds <| mem_of_superset (isOpen_ne.mem_nhds hne) fun _y' hy' =>
Function.update_of_ne hy' _ _)
(Function.update_of_ne hne ..) | 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_update_of_ne | null |
continuousAt_update_of_ne [T1Space X] [DecidableEq X] [TopologicalSpace Y]
{f : X → Y} {x x' : X} {y : Y} (hne : x' ≠ x) :
ContinuousAt (Function.update f x y) x' ↔ ContinuousAt f x' := by
simp only [← continuousWithinAt_univ, continuousWithinAt_update_of_ne hne] | 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_update_of_ne | null |
continuousOn_update_iff [T1Space X] [DecidableEq X] [TopologicalSpace Y] {f : X → Y}
{s : Set X} {x : X} {y : Y} :
ContinuousOn (Function.update f x y) s ↔
ContinuousOn f (s \ {x}) ∧ (x ∈ s → Tendsto f (𝓝[s \ {x}] x) (𝓝 y)) := by
rw [ContinuousOn, ← and_forall_ne x, and_comm]
refine and_congr ⟨fun H z hz => ?_, fun H z hzx hzs => ?_⟩ (forall_congr' fun _ => ?_)
· specialize H z hz.2 hz.1
rw [continuousWithinAt_update_of_ne hz.2] at H
exact H.mono diff_subset
· rw [continuousWithinAt_update_of_ne hzx]
refine (H z ⟨hzs, hzx⟩).mono_of_mem_nhdsWithin (inter_mem_nhdsWithin _ ?_)
exact isOpen_ne.mem_nhds hzx
· exact continuousWithinAt_update_same | 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 | continuousOn_update_iff | null |
t1Space_of_injective_of_continuous [TopologicalSpace Y] {f : X → Y}
(hf : Function.Injective f) (hf' : Continuous f) [T1Space Y] : T1Space X :=
t1Space_iff_specializes_imp_eq.2 fun _ _ h => hf (h.map hf').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 | t1Space_of_injective_of_continuous | null |
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