Inseparable points in a topological space

In this file we define

• Specializes (notation: x ⤳ y) : a relation saying that 𝓝 x ≤ 𝓝 y;

• Inseparable: a relation saying that two points in a topological space have the same neighbourhoods; equivalently, they can't be separated by an open set;

• InseparableSetoid X: same relation, as a Setoid;

• SeparationQuotient X: the quotient of X by its InseparableSetoid.

We also prove various basic properties of the relation Inseparable.

Notations

• x ⤳ y: notation for Specializes x y;
• x ~ᵢ y is used as a local notation for Inseparable x y;
• 𝓝 x is the neighbourhoods filter nhds x of a point x, defined elsewhere.

Tags

topological space, separation setoid

Specializes relation

def Specializes {X : Type u_1} [TopologicalSpace X] (x : X) (y : X) : Prop

x specializes to y (notation: x ⤳ y) if either of the following equivalent properties hold:

• 𝓝 x ≤ 𝓝 y; this property is used as the definition;
• pure x ≤ 𝓝 y; in other words, any neighbourhood of y contains x;
• y ∈ closure {x};
• closure {y} ⊆ closure {x};
• for any closed set s we have x ∈ s → y ∈ s;
• for any open set s we have y ∈ s → x ∈ s;
• y is a cluster point of the filter pure x = 𝓟 {x}.

This relation defines a Preorder on X. If X is a T₀ space, then this preorder is a partial order. If X is a T₁ space, then this partial order is trivial : x ⤳ y ↔ x = y.

Equations

• x ⤳ y = (nhds x ≤ nhds y)

def «term_⤳_» : Lean.TrailingParserDescr

x specializes to y (notation: x ⤳ y) if either of the following equivalent properties hold:

• 𝓝 x ≤ 𝓝 y; this property is used as the definition;
• pure x ≤ 𝓝 y; in other words, any neighbourhood of y contains x;
• y ∈ closure {x};
• closure {y} ⊆ closure {x};
• for any closed set s we have x ∈ s → y ∈ s;
• for any open set s we have y ∈ s → x ∈ s;
• y is a cluster point of the filter pure x = 𝓟 {x}.

This relation defines a Preorder on X. If X is a T₀ space, then this preorder is a partial order. If X is a T₁ space, then this partial order is trivial : x ⤳ y ↔ x = y.

Equations

• «term_⤳_» = Lean.ParserDescr.trailingNode `term_⤳_ 300 300 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ⤳ ") (Lean.ParserDescr.cat `term 301))

theorem specializes_TFAE {X : Type u_1} [TopologicalSpace X] (x : X) (y : X) : List.TFAE [x ⤳ y, pure x ≤ nhds y, ∀ (s : Set X), IsOpen s → y ∈ s → x ∈ s, ∀ (s : Set X), IsClosed s → x ∈ s → y ∈ s, y ∈ closure {x}, closure {y} ⊆ closure {x}, ClusterPt y (pure x)]

A collection of equivalent definitions of x ⤳ y. The public API is given by iff lemmas below.

theorem specializes_iff_nhds {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : x ⤳ y ↔ nhds x ≤ nhds y

theorem Specializes.not_disjoint {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} (h : x ⤳ y) : ¬Disjoint (nhds x) (nhds y)

theorem specializes_iff_pure {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : x ⤳ y ↔ pure x ≤ nhds y

theorem Specializes.nhds_le_nhds {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : x ⤳ y → nhds x ≤ nhds y

Alias of the forward direction of specializes_iff_nhds.

theorem Specializes.pure_le_nhds {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : x ⤳ y → pure x ≤ nhds y

Alias of the forward direction of specializes_iff_pure.

theorem ker_nhds_eq_specializes {X : Type u_1} [TopologicalSpace X] {x : X} : Filter.ker (nhds x) = {y : X | y ⤳ x}

theorem specializes_iff_forall_open {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : x ⤳ y ↔ ∀ (s : Set X), IsOpen s → y ∈ s → x ∈ s

theorem Specializes.mem_open {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} {s : Set X} (h : x ⤳ y) (hs : IsOpen s) (hy : y ∈ s) : x ∈ s

theorem IsOpen.not_specializes {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} {s : Set X} (hs : IsOpen s) (hx : x ∉ s) (hy : y ∈ s) : ¬x ⤳ y

theorem specializes_iff_forall_closed {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : x ⤳ y ↔ ∀ (s : Set X), IsClosed s → x ∈ s → y ∈ s

theorem Specializes.mem_closed {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} {s : Set X} (h : x ⤳ y) (hs : IsClosed s) (hx : x ∈ s) : y ∈ s

theorem IsClosed.not_specializes {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} {s : Set X} (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ⤳ y

theorem specializes_iff_mem_closure {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : x ⤳ y ↔ y ∈ closure {x}

theorem Specializes.mem_closure {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : x ⤳ y → y ∈ closure {x}

Alias of the forward direction of specializes_iff_mem_closure.

theorem specializes_iff_closure_subset {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : x ⤳ y ↔ closure {y} ⊆ closure {x}

theorem Specializes.closure_subset {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : x ⤳ y → closure {y} ⊆ closure {x}

Alias of the forward direction of specializes_iff_closure_subset.

theorem specializes_iff_clusterPt {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : x ⤳ y ↔ ClusterPt y (pure x)

theorem Filter.HasBasis.specializes_iff {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} {ι : Sort u_7} {p : ι → Prop} {s : ι → Set X} (h : Filter.HasBasis (nhds y) p s) : x ⤳ y ↔ ∀ (i : ι), p i → x ∈ s i

theorem specializes_rfl {X : Type u_1} [TopologicalSpace X] {x : X} : x ⤳ x

theorem specializes_refl {X : Type u_1} [TopologicalSpace X] (x : X) : x ⤳ x

theorem Specializes.trans {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} {z : X} : x ⤳ y → y ⤳ z → x ⤳ z

theorem specializes_of_eq {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} (e : x = y) : x ⤳ y

theorem specializes_of_nhdsWithin {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} {s : Set X} (h₁ : nhdsWithin x s ≤ nhdsWithin y s) (h₂ : x ∈ s) : x ⤳ y

theorem Specializes.map_of_continuousAt {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x : X} {y : X} {f : X → Y} (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y

theorem Specializes.map {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x : X} {y : X} {f : X → Y} (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y

theorem Inducing.specializes_iff {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x : X} {y : X} {f : X → Y} (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y

theorem subtype_specializes_iff {X : Type u_1} [TopologicalSpace X] {p : X → Prop} (x : Subtype p) (y : Subtype p) : x ⤳ y ↔ ↑x ⤳ ↑y

@[simp]

theorem specializes_prod {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x₁ : X} {x₂ : X} {y₁ : Y} {y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂

theorem Specializes.prod {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x₁ : X} {x₂ : X} {y₁ : Y} {y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) : (x₁, y₁) ⤳ (x₂, y₂)

@[simp]

theorem specializes_pi {ι : Type u_5} {π : ι → Type u_6} [(i : ι) → TopologicalSpace (π i)] {f : (i : ι) → π i} {g : (i : ι) → π i} : f ⤳ g ↔ ∀ (i : ι), f i ⤳ g i

theorem not_specializes_iff_exists_open {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : ¬x ⤳ y ↔ ∃ (S : Set X), IsOpen S ∧ y ∈ S ∧ x ∉ S

theorem not_specializes_iff_exists_closed {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : ¬x ⤳ y ↔ ∃ (S : Set X), IsClosed S ∧ x ∈ S ∧ y ∉ S

theorem IsOpen.continuous_piecewise_of_specializes {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} {g : X → Y} [DecidablePred fun (x : X) => x ∈ s] (hs : IsOpen s) (hf : Continuous f) (hg : Continuous g) (hspec : ∀ (x : X), f x ⤳ g x) : Continuous (Set.piecewise s f g)

theorem IsClosed.continuous_piecewise_of_specializes {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : X → Y} {g : X → Y} [DecidablePred fun (x : X) => x ∈ s] (hs : IsClosed s) (hf : Continuous f) (hg : Continuous g) (hspec : ∀ (x : X), g x ⤳ f x) : Continuous (Set.piecewise s f g)

def specializationPreorder (X : Type u_1) [TopologicalSpace X] : Preorder X

Specialization forms a preorder on the topological space.

Equations

• specializationPreorder X = let src := Preorder.lift (⇑OrderDual.toDual ∘ nhds); Preorder.mk (_ : ∀ (a : X), a ≤ a) (_ : ∀ (a b c : X), a ≤ b → b ≤ c → a ≤ c)

theorem Continuous.specialization_monotone {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) : Monotone f

A continuous function is monotone with respect to the specialization preorders on the domain and the codomain.

Inseparable relation

def Inseparable {X : Type u_1} [TopologicalSpace X] (x : X) (y : X) : Prop

Two points x and y in a topological space are Inseparable if any of the following equivalent properties hold:

• 𝓝 x = 𝓝 y; we use this property as the definition;
• for any open set s, x ∈ s ↔ y ∈ s, see inseparable_iff_open;
• for any closed set s, x ∈ s ↔ y ∈ s, see inseparable_iff_closed;
• x ∈ closure {y} and y ∈ closure {x}, see inseparable_iff_mem_closure;
• closure {x} = closure {y}, see inseparable_iff_closure_eq.

Equations

• Inseparable x y = (nhds x = nhds y)

theorem inseparable_def {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : Inseparable x y ↔ nhds x = nhds y

theorem inseparable_iff_specializes_and {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : Inseparable x y ↔ x ⤳ y ∧ y ⤳ x

theorem Inseparable.specializes {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} (h : Inseparable x y) : x ⤳ y

theorem Inseparable.specializes' {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} (h : Inseparable x y) : y ⤳ x

theorem Specializes.antisymm {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} (h₁ : x ⤳ y) (h₂ : y ⤳ x) : Inseparable x y

theorem inseparable_iff_forall_open {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : Inseparable x y ↔ ∀ (s : Set X), IsOpen s → (x ∈ s ↔ y ∈ s)

theorem not_inseparable_iff_exists_open {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : ¬Inseparable x y ↔ ∃ (s : Set X), IsOpen s ∧ Xor' (x ∈ s) (y ∈ s)

theorem inseparable_iff_forall_closed {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : Inseparable x y ↔ ∀ (s : Set X), IsClosed s → (x ∈ s ↔ y ∈ s)

theorem inseparable_iff_mem_closure {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : Inseparable x y ↔ x ∈ closure {y} ∧ y ∈ closure {x}

theorem inseparable_iff_closure_eq {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : Inseparable x y ↔ closure {x} = closure {y}

theorem inseparable_of_nhdsWithin_eq {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} {s : Set X} (hx : x ∈ s) (hy : y ∈ s) (h : nhdsWithin x s = nhdsWithin y s) : Inseparable x y

theorem Inducing.inseparable_iff {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x : X} {y : X} {f : X → Y} (hf : Inducing f) : Inseparable (f x) (f y) ↔ Inseparable x y

theorem subtype_inseparable_iff {X : Type u_1} [TopologicalSpace X] {p : X → Prop} (x : Subtype p) (y : Subtype p) : Inseparable x y ↔ Inseparable ↑x ↑y

@[simp]

theorem inseparable_prod {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x₁ : X} {x₂ : X} {y₁ : Y} {y₂ : Y} : Inseparable (x₁, y₁) (x₂, y₂) ↔ Inseparable x₁ x₂ ∧ Inseparable y₁ y₂

theorem Inseparable.prod {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x₁ : X} {x₂ : X} {y₁ : Y} {y₂ : Y} (hx : Inseparable x₁ x₂) (hy : Inseparable y₁ y₂) : Inseparable (x₁, y₁) (x₂, y₂)

@[simp]

theorem inseparable_pi {ι : Type u_5} {π : ι → Type u_6} [(i : ι) → TopologicalSpace (π i)] {f : (i : ι) → π i} {g : (i : ι) → π i} : Inseparable f g ↔ ∀ (i : ι), Inseparable (f i) (g i)

theorem Inseparable.refl {X : Type u_1} [TopologicalSpace X] (x : X) : Inseparable x x

theorem Inseparable.rfl {X : Type u_1} [TopologicalSpace X] {x : X} : Inseparable x x

theorem Inseparable.of_eq {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} (e : x = y) : Inseparable x y

theorem Inseparable.symm {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} (h : Inseparable x y) : Inseparable y x

theorem Inseparable.trans {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} {z : X} (h₁ : Inseparable x y) (h₂ : Inseparable y z) : Inseparable x z

theorem Inseparable.nhds_eq {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} (h : Inseparable x y) : nhds x = nhds y

theorem Inseparable.mem_open_iff {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} {s : Set X} (h : Inseparable x y) (hs : IsOpen s) : x ∈ s ↔ y ∈ s

theorem Inseparable.mem_closed_iff {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} {s : Set X} (h : Inseparable x y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s

theorem Inseparable.map_of_continuousAt {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x : X} {y : X} {f : X → Y} (h : Inseparable x y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) : Inseparable (f x) (f y)

theorem Inseparable.map {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x : X} {y : X} {f : X → Y} (h : Inseparable x y) (hf : Continuous f) : Inseparable (f x) (f y)

theorem IsClosed.not_inseparable {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} {s : Set X} (hs : IsClosed s) (hx : x ∈ s) (hy : y ∉ s) : ¬Inseparable x y

theorem IsOpen.not_inseparable {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} {s : Set X} (hs : IsOpen s) (hx : x ∈ s) (hy : y ∉ s) : ¬Inseparable x y

Separation quotient

In this section we define the quotient of a topological space by the Inseparable relation.

def inseparableSetoid (X : Type u_1) [TopologicalSpace X] : Setoid X

A setoid version of Inseparable, used to define the SeparationQuotient.

Equations

• inseparableSetoid X = let src := Setoid.comap nhds ⊥; { r := Inseparable, iseqv := (_ : Equivalence Setoid.r) }

def SeparationQuotient (X : Type u_1) [TopologicalSpace X] : Type u_1

The quotient of a topological space by its inseparableSetoid. This quotient is guaranteed to be a T₀ space.

Equations

• SeparationQuotient X = Quotient (inseparableSetoid X)

instance instTopologicalSpaceSeparationQuotient (X : Type u_1) [TopologicalSpace X] : TopologicalSpace (SeparationQuotient X)

Equations

• instTopologicalSpaceSeparationQuotient X = instTopologicalSpaceQuotient

def SeparationQuotient.mk {X : Type u_1} [TopologicalSpace X] : X → SeparationQuotient X

The natural map from a topological space to its separation quotient.

Equations

• SeparationQuotient.mk = Quotient.mk''

theorem SeparationQuotient.quotientMap_mk {X : Type u_1} [TopologicalSpace X] : QuotientMap SeparationQuotient.mk

theorem SeparationQuotient.continuous_mk {X : Type u_1} [TopologicalSpace X] : Continuous SeparationQuotient.mk

@[simp]

theorem SeparationQuotient.mk_eq_mk {X : Type u_1} [TopologicalSpace X] {x : X} {y : X} : SeparationQuotient.mk x = SeparationQuotient.mk y ↔ Inseparable x y

theorem SeparationQuotient.surjective_mk {X : Type u_1} [TopologicalSpace X] : Function.Surjective SeparationQuotient.mk

@[simp]

theorem SeparationQuotient.range_mk {X : Type u_1} [TopologicalSpace X] : Set.range SeparationQuotient.mk = Set.univ

instance SeparationQuotient.instNonemptySeparationQuotient {X : Type u_1} [TopologicalSpace X] [Nonempty X] : Nonempty (SeparationQuotient X)

Equations

• (_ : Nonempty (SeparationQuotient X)) = (_ : Nonempty (SeparationQuotient X))

instance SeparationQuotient.instInhabitedSeparationQuotient {X : Type u_1} [TopologicalSpace X] [Inhabited X] : Inhabited (SeparationQuotient X)

Equations

• SeparationQuotient.instInhabitedSeparationQuotient = { default := SeparationQuotient.mk default }

instance SeparationQuotient.instSubsingletonSeparationQuotient {X : Type u_1} [TopologicalSpace X] [Subsingleton X] : Subsingleton (SeparationQuotient X)

Equations

• (_ : Subsingleton (SeparationQuotient X)) = (_ : Subsingleton (SeparationQuotient X))

theorem SeparationQuotient.preimage_image_mk_open {X : Type u_1} [TopologicalSpace X] {s : Set X} (hs : IsOpen s) : SeparationQuotient.mk ⁻¹' (SeparationQuotient.mk '' s) = s

theorem SeparationQuotient.isOpenMap_mk {X : Type u_1} [TopologicalSpace X] : IsOpenMap SeparationQuotient.mk

theorem SeparationQuotient.preimage_image_mk_closed {X : Type u_1} [TopologicalSpace X] {s : Set X} (hs : IsClosed s) : SeparationQuotient.mk ⁻¹' (SeparationQuotient.mk '' s) = s

theorem SeparationQuotient.inducing_mk {X : Type u_1} [TopologicalSpace X] : Inducing SeparationQuotient.mk

theorem SeparationQuotient.isClosedMap_mk {X : Type u_1} [TopologicalSpace X] : IsClosedMap SeparationQuotient.mk

@[simp]

theorem SeparationQuotient.comap_mk_nhds_mk {X : Type u_1} [TopologicalSpace X] {x : X} : Filter.comap SeparationQuotient.mk (nhds (SeparationQuotient.mk x)) = nhds x

@[simp]

theorem SeparationQuotient.comap_mk_nhdsSet_image {X : Type u_1} [TopologicalSpace X] {s : Set X} : Filter.comap SeparationQuotient.mk (nhdsSet (SeparationQuotient.mk '' s)) = nhdsSet s

theorem SeparationQuotient.map_mk_nhds {X : Type u_1} [TopologicalSpace X] {x : X} : Filter.map SeparationQuotient.mk (nhds x) = nhds (SeparationQuotient.mk x)

theorem SeparationQuotient.map_mk_nhdsSet {X : Type u_1} [TopologicalSpace X] {s : Set X} : Filter.map SeparationQuotient.mk (nhdsSet s) = nhdsSet (SeparationQuotient.mk '' s)

theorem SeparationQuotient.comap_mk_nhdsSet {X : Type u_1} [TopologicalSpace X] {t : Set (SeparationQuotient X)} : Filter.comap SeparationQuotient.mk (nhdsSet t) = nhdsSet (SeparationQuotient.mk ⁻¹' t)

theorem SeparationQuotient.preimage_mk_closure {X : Type u_1} [TopologicalSpace X] {t : Set (SeparationQuotient X)} : SeparationQuotient.mk ⁻¹' closure t = closure (SeparationQuotient.mk ⁻¹' t)

theorem SeparationQuotient.preimage_mk_interior {X : Type u_1} [TopologicalSpace X] {t : Set (SeparationQuotient X)} : SeparationQuotient.mk ⁻¹' interior t = interior (SeparationQuotient.mk ⁻¹' t)

theorem SeparationQuotient.preimage_mk_frontier {X : Type u_1} [TopologicalSpace X] {t : Set (SeparationQuotient X)} : SeparationQuotient.mk ⁻¹' frontier t = frontier (SeparationQuotient.mk ⁻¹' t)

theorem SeparationQuotient.image_mk_closure {X : Type u_1} [TopologicalSpace X] {s : Set X} : SeparationQuotient.mk '' closure s = closure (SeparationQuotient.mk '' s)

theorem SeparationQuotient.map_prod_map_mk_nhds {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (x : X) (y : Y) : Filter.map (Prod.map SeparationQuotient.mk SeparationQuotient.mk) (nhds (x, y)) = nhds (SeparationQuotient.mk x, SeparationQuotient.mk y)

theorem SeparationQuotient.map_mk_nhdsWithin_preimage {X : Type u_1} [TopologicalSpace X] (s : Set (SeparationQuotient X)) (x : X) : Filter.map SeparationQuotient.mk (nhdsWithin x (SeparationQuotient.mk ⁻¹' s)) = nhdsWithin (SeparationQuotient.mk x) s

def SeparationQuotient.lift {X : Type u_1} {α : Type u_4} [TopologicalSpace X] (f : X → α) (hf : ∀ (x y : X), Inseparable x y → f x = f y) : SeparationQuotient X → α

Lift a map f : X → α such that Inseparable x y → f x = f y to a map SeparationQuotient X → α.

Equations

• SeparationQuotient.lift f hf x = Quotient.liftOn' x f hf

@[simp]

theorem SeparationQuotient.lift_mk {X : Type u_1} {α : Type u_4} [TopologicalSpace X] {f : X → α} (hf : ∀ (x y : X), Inseparable x y → f x = f y) (x : X) : SeparationQuotient.lift f hf (SeparationQuotient.mk x) = f x

@[simp]

theorem SeparationQuotient.lift_comp_mk {X : Type u_1} {α : Type u_4} [TopologicalSpace X] {f : X → α} (hf : ∀ (x y : X), Inseparable x y → f x = f y) : SeparationQuotient.lift f hf ∘ SeparationQuotient.mk = f

@[simp]

theorem SeparationQuotient.tendsto_lift_nhds_mk {X : Type u_1} {α : Type u_4} [TopologicalSpace X] {x : X} {f : X → α} {hf : ∀ (x y : X), Inseparable x y → f x = f y} {l : Filter α} : Filter.Tendsto (SeparationQuotient.lift f hf) (nhds (SeparationQuotient.mk x)) l ↔ Filter.Tendsto f (nhds x) l

@[simp]

theorem SeparationQuotient.tendsto_lift_nhdsWithin_mk {X : Type u_1} {α : Type u_4} [TopologicalSpace X] {x : X} {f : X → α} {hf : ∀ (x y : X), Inseparable x y → f x = f y} {s : Set (SeparationQuotient X)} {l : Filter α} : Filter.Tendsto (SeparationQuotient.lift f hf) (nhdsWithin (SeparationQuotient.mk x) s) l ↔ Filter.Tendsto f (nhdsWithin x (SeparationQuotient.mk ⁻¹' s)) l

@[simp]

theorem SeparationQuotient.continuousAt_lift {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x : X} {f : X → Y} {hf : ∀ (x y : X), Inseparable x y → f x = f y} : ContinuousAt (SeparationQuotient.lift f hf) (SeparationQuotient.mk x) ↔ ContinuousAt f x

@[simp]

theorem SeparationQuotient.continuousWithinAt_lift {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x : X} {f : X → Y} {hf : ∀ (x y : X), Inseparable x y → f x = f y} {s : Set (SeparationQuotient X)} : ContinuousWithinAt (SeparationQuotient.lift f hf) s (SeparationQuotient.mk x) ↔ ContinuousWithinAt f (SeparationQuotient.mk ⁻¹' s) x

@[simp]

theorem SeparationQuotient.continuousOn_lift {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {hf : ∀ (x y : X), Inseparable x y → f x = f y} {s : Set (SeparationQuotient X)} : ContinuousOn (SeparationQuotient.lift f hf) s ↔ ContinuousOn f (SeparationQuotient.mk ⁻¹' s)

@[simp]

theorem SeparationQuotient.continuous_lift {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {hf : ∀ (x y : X), Inseparable x y → f x = f y} : Continuous (SeparationQuotient.lift f hf) ↔ Continuous f

def SeparationQuotient.lift₂ {X : Type u_1} {Y : Type u_2} {α : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y → α) (hf : ∀ (a : X) (b : Y) (c : X) (d : Y), Inseparable a c → Inseparable b d → f a b = f c d) : SeparationQuotient X → SeparationQuotient Y → α

Lift a map f : X → Y → α such that Inseparable a b → Inseparable c d → f a c = f b d to a map SeparationQuotient X → SeparationQuotient Y → α.

Equations

• SeparationQuotient.lift₂ f hf x y = Quotient.liftOn₂' x y f hf

@[simp]

theorem SeparationQuotient.lift₂_mk {X : Type u_1} {Y : Type u_2} {α : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y → α} (hf : ∀ (a : X) (b : Y) (c : X) (d : Y), Inseparable a c → Inseparable b d → f a b = f c d) (x : X) (y : Y) : SeparationQuotient.lift₂ f hf (SeparationQuotient.mk x) (SeparationQuotient.mk y) = f x y

@[simp]

theorem SeparationQuotient.tendsto_lift₂_nhds {X : Type u_1} {Y : Type u_2} {α : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y → α} {hf : ∀ (a : X) (b : Y) (c : X) (d : Y), Inseparable a c → Inseparable b d → f a b = f c d} {x : X} {y : Y} {l : Filter α} : Filter.Tendsto (Function.uncurry (SeparationQuotient.lift₂ f hf)) (nhds (SeparationQuotient.mk x, SeparationQuotient.mk y)) l ↔ Filter.Tendsto (Function.uncurry f) (nhds (x, y)) l

@[simp]

theorem SeparationQuotient.tendsto_lift₂_nhdsWithin {X : Type u_1} {Y : Type u_2} {α : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y → α} {hf : ∀ (a : X) (b : Y) (c : X) (d : Y), Inseparable a c → Inseparable b d → f a b = f c d} {x : X} {y : Y} {s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} : Filter.Tendsto (Function.uncurry (SeparationQuotient.lift₂ f hf)) (nhdsWithin (SeparationQuotient.mk x, SeparationQuotient.mk y) s) l ↔ Filter.Tendsto (Function.uncurry f) (nhdsWithin (x, y) (Prod.map SeparationQuotient.mk SeparationQuotient.mk ⁻¹' s)) l

@[simp]

theorem SeparationQuotient.continuousAt_lift₂ {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y → Z} {hf : ∀ (a : X) (b : Y) (c : X) (d : Y), Inseparable a c → Inseparable b d → f a b = f c d} {x : X} {y : Y} : ContinuousAt (Function.uncurry (SeparationQuotient.lift₂ f hf)) (SeparationQuotient.mk x, SeparationQuotient.mk y) ↔ ContinuousAt (Function.uncurry f) (x, y)

@[simp]

theorem SeparationQuotient.continuousWithinAt_lift₂ {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y → Z} {hf : ∀ (a : X) (b : Y) (c : X) (d : Y), Inseparable a c → Inseparable b d → f a b = f c d} {s : Set (SeparationQuotient X × SeparationQuotient Y)} {x : X} {y : Y} : ContinuousWithinAt (Function.uncurry (SeparationQuotient.lift₂ f hf)) s (SeparationQuotient.mk x, SeparationQuotient.mk y) ↔ ContinuousWithinAt (Function.uncurry f) (Prod.map SeparationQuotient.mk SeparationQuotient.mk ⁻¹' s) (x, y)

@[simp]

theorem SeparationQuotient.continuousOn_lift₂ {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y → Z} {hf : ∀ (a : X) (b : Y) (c : X) (d : Y), Inseparable a c → Inseparable b d → f a b = f c d} {s : Set (SeparationQuotient X × SeparationQuotient Y)} : ContinuousOn (Function.uncurry (SeparationQuotient.lift₂ f hf)) s ↔ ContinuousOn (Function.uncurry f) (Prod.map SeparationQuotient.mk SeparationQuotient.mk ⁻¹' s)

@[simp]

theorem SeparationQuotient.continuous_lift₂ {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y → Z} {hf : ∀ (a : X) (b : Y) (c : X) (d : Y), Inseparable a c → Inseparable b d → f a b = f c d} : Continuous (Function.uncurry (SeparationQuotient.lift₂ f hf)) ↔ Continuous (Function.uncurry f)

theorem continuous_congr_of_inseparable {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {g : X → Y} (h : ∀ (x : X), Inseparable (f x) (g x)) : Continuous f ↔ Continuous g