Uniform isomorphisms

This file defines uniform isomorphisms between two uniform spaces. They are bijections with both directions uniformly continuous. We denote uniform isomorphisms with the notation ≃ᵤ.

Main definitions

• UniformEquiv α β: The type of uniform isomorphisms from α to β. This type can be denoted using the following notation: α ≃ᵤ β.

structure UniformEquiv (α : Type u_4) (β : Type u_5) [UniformSpace α] [UniformSpace β] extends α ≃ β : Type (max u_4 u_5)

Uniform isomorphism between α and β

• toFun : α → β
• invFun : β → α
• left_inv : Function.LeftInverse self.invFun self.toFun
• right_inv : Function.RightInverse self.invFun self.toFun
• uniformContinuous_toFun : UniformContinuous self.toFun

  Uniform continuity of the function

• uniformContinuous_invFun : UniformContinuous self.invFun

  Uniform continuity of the inverse

def «term_≃ᵤ_» : Lean.TrailingParserDescr

Uniform isomorphism between α and β

theorem UniformEquiv.toEquiv_injective {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] : Function.Injective toEquiv

instance UniformEquiv.instEquivLike {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] : EquivLike (α ≃ᵤ β) α β

@[simp]

theorem UniformEquiv.uniformEquiv_mk_coe {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (a : α ≃ β) (b : UniformContinuous a.toFun) (c : UniformContinuous a.invFun) : ⇑{ toEquiv := a, uniformContinuous_toFun := b, uniformContinuous_invFun := c } = ⇑a

def UniformEquiv.symm {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : β ≃ᵤ α

Inverse of a uniform isomorphism.

def UniformEquiv.Simps.apply {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : α → β

See Note [custom simps projection]. We need to specify this projection explicitly in this case, because it is a composition of multiple projections.

def UniformEquiv.Simps.symm_apply {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : β → α

See Note [custom simps projection]

@[simp]

theorem UniformEquiv.coe_toEquiv {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : ⇑h.toEquiv = ⇑h

@[simp]

theorem UniformEquiv.coe_symm_toEquiv {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : ⇑h.symm = ⇑h.symm

theorem UniformEquiv.ext {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] {h h' : α ≃ᵤ β} (H : ∀ (x : α), h x = h' x) : h = h'

theorem UniformEquiv.ext_iff {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] {h h' : α ≃ᵤ β} : h = h' ↔ ∀ (x : α), h x = h' x

def UniformEquiv.refl (α : Type u_4) [UniformSpace α] : α ≃ᵤ α

Identity map as a uniform isomorphism.

@[simp]

theorem UniformEquiv.refl_apply (α : Type u_4) [UniformSpace α] : ⇑(UniformEquiv.refl α) = id

def UniformEquiv.trans {α : Type u} {β : Type u_1} {γ : Type u_2} [UniformSpace α] [UniformSpace β] [UniformSpace γ] (h₁ : α ≃ᵤ β) (h₂ : β ≃ᵤ γ) : α ≃ᵤ γ

Composition of two uniform isomorphisms.

@[simp]

theorem UniformEquiv.trans_apply {α : Type u} {β : Type u_1} {γ : Type u_2} [UniformSpace α] [UniformSpace β] [UniformSpace γ] (h₁ : α ≃ᵤ β) (h₂ : β ≃ᵤ γ) (a : α) : (h₁.trans h₂) a = h₂ (h₁ a)

@[simp]

theorem UniformEquiv.uniformEquiv_mk_coe_symm {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (a : α ≃ β) (b : UniformContinuous a.toFun) (c : UniformContinuous a.invFun) : ⇑{ toEquiv := a, uniformContinuous_toFun := b, uniformContinuous_invFun := c }.symm = ⇑a.symm

@[simp]

theorem UniformEquiv.refl_symm {α : Type u} [UniformSpace α] : (UniformEquiv.refl α).symm = UniformEquiv.refl α

theorem UniformEquiv.uniformContinuous {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : UniformContinuous ⇑h

theorem UniformEquiv.continuous {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : Continuous ⇑h

theorem UniformEquiv.uniformContinuous_symm {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : UniformContinuous ⇑h.symm

theorem UniformEquiv.continuous_symm {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : Continuous ⇑h.symm

def UniformEquiv.toHomeomorph {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (e : α ≃ᵤ β) : α ≃ₜ β

A uniform isomorphism as a homeomorphism.

theorem UniformEquiv.toHomeomorph_apply {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (e : α ≃ᵤ β) : ⇑e.toHomeomorph = ⇑e

theorem UniformEquiv.toHomeomorph_symm_apply {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (e : α ≃ᵤ β) : ⇑e.toHomeomorph.symm = ⇑e.symm

@[simp]

theorem UniformEquiv.apply_symm_apply {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) (x : β) : h (h.symm x) = x

@[simp]

theorem UniformEquiv.symm_apply_apply {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) (x : α) : h.symm (h x) = x

theorem UniformEquiv.bijective {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : Function.Bijective ⇑h

theorem UniformEquiv.injective {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : Function.Injective ⇑h

theorem UniformEquiv.surjective {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : Function.Surjective ⇑h

def UniformEquiv.changeInv {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (f : α ≃ᵤ β) (g : β → α) (hg : Function.RightInverse g ⇑f) : α ≃ᵤ β

Change the uniform equiv f to make the inverse function definitionally equal to g.

@[simp]

theorem UniformEquiv.symm_comp_self {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : ⇑h.symm ∘ ⇑h = id

@[simp]

theorem UniformEquiv.self_comp_symm {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : ⇑h ∘ ⇑h.symm = id

theorem UniformEquiv.range_coe {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : Set.range ⇑h = Set.univ

theorem UniformEquiv.image_symm {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : Set.image ⇑h.symm = Set.preimage ⇑h

theorem UniformEquiv.preimage_symm {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : Set.preimage ⇑h.symm = Set.image ⇑h

@[simp]

theorem UniformEquiv.image_preimage {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) (s : Set β) : ⇑h '' (⇑h ⁻¹' s) = s

@[simp]

theorem UniformEquiv.preimage_image {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) (s : Set α) : ⇑h ⁻¹' (⇑h '' s) = s

theorem UniformEquiv.isUniformInducing {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : IsUniformInducing ⇑h

theorem UniformEquiv.comap_eq {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : UniformSpace.comap (⇑h) inst✝ = inst✝¹

theorem UniformEquiv.isUniformEmbedding {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : IsUniformEmbedding ⇑h

theorem UniformEquiv.completeSpace_iff {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (h : α ≃ᵤ β) : CompleteSpace α ↔ CompleteSpace β

noncomputable def UniformEquiv.ofIsUniformEmbedding {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (f : α → β) (hf : IsUniformEmbedding f) : α ≃ᵤ ↑(Set.range f)

Uniform equiv given a uniform embedding.

def UniformEquiv.setCongr {α : Type u} [UniformSpace α] {s t : Set α} (h : s = t) : ↑s ≃ᵤ ↑t

If two sets are equal, then they are uniformly equivalent.

def UniformEquiv.prodCongr {α : Type u} {β : Type u_1} {γ : Type u_2} {δ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] [UniformSpace δ] (h₁ : α ≃ᵤ β) (h₂ : γ ≃ᵤ δ) : α × γ ≃ᵤ β × δ

Product of two uniform isomorphisms.

@[simp]

theorem UniformEquiv.prodCongr_symm {α : Type u} {β : Type u_1} {γ : Type u_2} {δ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] [UniformSpace δ] (h₁ : α ≃ᵤ β) (h₂ : γ ≃ᵤ δ) : (h₁.prodCongr h₂).symm = h₁.symm.prodCongr h₂.symm

@[simp]

theorem UniformEquiv.coe_prodCongr {α : Type u} {β : Type u_1} {γ : Type u_2} {δ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] [UniformSpace δ] (h₁ : α ≃ᵤ β) (h₂ : γ ≃ᵤ δ) : ⇑(h₁.prodCongr h₂) = Prod.map ⇑h₁ ⇑h₂

def UniformEquiv.prodComm (α : Type u) (β : Type u_1) [UniformSpace α] [UniformSpace β] : α × β ≃ᵤ β × α

α × β is uniformly isomorphic to β × α.

@[simp]

theorem UniformEquiv.prodComm_symm (α : Type u) (β : Type u_1) [UniformSpace α] [UniformSpace β] : (prodComm α β).symm = prodComm β α

@[simp]

theorem UniformEquiv.coe_prodComm (α : Type u) (β : Type u_1) [UniformSpace α] [UniformSpace β] : ⇑(prodComm α β) = Prod.swap

def UniformEquiv.prodAssoc (α : Type u) (β : Type u_1) (γ : Type u_2) [UniformSpace α] [UniformSpace β] [UniformSpace γ] : (α × β) × γ ≃ᵤ α × β × γ

(α × β) × γ is uniformly isomorphic to α × (β × γ).

def UniformEquiv.prodPunit (α : Type u) [UniformSpace α] : α × PUnit.{u_4 + 1} ≃ᵤ α

α × {*} is uniformly isomorphic to α.

@[simp]

theorem UniformEquiv.prodPunit_apply (α : Type u) [UniformSpace α] : ⇑(prodPunit α) = fun (p : α × PUnit.{u_4 + 1}) => p.1

def UniformEquiv.punitProd (α : Type u) [UniformSpace α] : PUnit.{u_4 + 1} × α ≃ᵤ α

{*} × α is uniformly isomorphic to α.

@[simp]

theorem UniformEquiv.coe_punitProd (α : Type u) [UniformSpace α] : ⇑(punitProd α) = Prod.snd

def UniformEquiv.piCongrLeft {ι : Type u_4} {ι' : Type u_5} {β : ι' → Type u_6} [(j : ι') → UniformSpace (β j)] (e : ι ≃ ι') : ((i : ι) → β (e i)) ≃ᵤ ((j : ι') → β j)

Equiv.piCongrLeft as a uniform isomorphism: this is the natural isomorphism Π i, β (e i) ≃ᵤ Π j, β j obtained from a bijection ι ≃ ι'.

@[simp]

theorem UniformEquiv.piCongrLeft_toEquiv {ι : Type u_4} {ι' : Type u_5} {β : ι' → Type u_6} [(j : ι') → UniformSpace (β j)] (e : ι ≃ ι') : (piCongrLeft e).toEquiv = Equiv.piCongrLeft β e

theorem UniformEquiv.piCongrLeft_apply {ι : Type u_4} {ι' : Type u_5} {β : ι' → Type u_6} [(j : ι') → UniformSpace (β j)] (e : ι ≃ ι') (a✝ : (b : ι) → β (e.symm.symm b)) (a : ι') : (piCongrLeft e) a✝ a = (Equiv.piCongrLeft' β e.symm).symm a✝ a

@[simp]

theorem UniformEquiv.piCongrLeft_refl {ι : Type u_4} {X : ι → Type u_5} [(i : ι) → UniformSpace (X i)] : piCongrLeft (Equiv.refl ι) = UniformEquiv.refl ((i : ι) → X i)

@[simp]

theorem UniformEquiv.piCongrLeft_symm_apply {ι : Type u_4} {ι' : Type u_5} {X : ι' → Type u_6} [(j : ι') → UniformSpace (X j)] (e : ι ≃ ι') : ⇑(piCongrLeft e).symm = fun (x1 : (j : ι') → X j) (x2 : ι) => x1 (e x2)

@[simp]

theorem UniformEquiv.piCongrLeft_apply_apply {ι : Type u_4} {ι' : Type u_5} {X : ι' → Type u_6} [(j : ι') → UniformSpace (X j)] (e : ι ≃ ι') (x : (i : ι) → X (e i)) (i : ι) : (piCongrLeft e) x (e i) = x i

def UniformEquiv.piCongrRight {ι : Type u_4} {β₁ : ι → Type u_5} {β₂ : ι → Type u_6} [(i : ι) → UniformSpace (β₁ i)] [(i : ι) → UniformSpace (β₂ i)] (F : (i : ι) → β₁ i ≃ᵤ β₂ i) : ((i : ι) → β₁ i) ≃ᵤ ((i : ι) → β₂ i)

Equiv.piCongrRight as a uniform isomorphism: this is the natural isomorphism Π i, β₁ i ≃ᵤ Π j, β₂ i obtained from uniform isomorphisms β₁ i ≃ᵤ β₂ i for each i.

@[simp]

theorem UniformEquiv.piCongrRight_apply {ι : Type u_4} {β₁ : ι → Type u_5} {β₂ : ι → Type u_6} [(i : ι) → UniformSpace (β₁ i)] [(i : ι) → UniformSpace (β₂ i)] (F : (i : ι) → β₁ i ≃ᵤ β₂ i) (a✝ : (i : ι) → β₁ i) (i : ι) : (piCongrRight F) a✝ i = (F i) (a✝ i)

@[simp]

theorem UniformEquiv.piCongrRight_toEquiv {ι : Type u_4} {β₁ : ι → Type u_5} {β₂ : ι → Type u_6} [(i : ι) → UniformSpace (β₁ i)] [(i : ι) → UniformSpace (β₂ i)] (F : (i : ι) → β₁ i ≃ᵤ β₂ i) : (piCongrRight F).toEquiv = Equiv.piCongrRight fun (i : ι) => (F i).toEquiv

@[simp]

theorem UniformEquiv.piCongrRight_symm {ι : Type u_4} {β₁ : ι → Type u_5} {β₂ : ι → Type u_6} [(i : ι) → UniformSpace (β₁ i)] [(i : ι) → UniformSpace (β₂ i)] (F : (i : ι) → β₁ i ≃ᵤ β₂ i) : (piCongrRight F).symm = piCongrRight fun (i : ι) => (F i).symm

@[simp]

theorem UniformEquiv.piCongrRight_refl {ι : Type u_4} {X : ι → Type u_5} [(i : ι) → UniformSpace (X i)] : (piCongrRight fun (i : ι) => UniformEquiv.refl (X i)) = UniformEquiv.refl ((i : ι) → X i)

def UniformEquiv.piCongr {ι₁ : Type u_4} {ι₂ : Type u_5} {β₁ : ι₁ → Type u_6} {β₂ : ι₂ → Type u_7} [(i₁ : ι₁) → UniformSpace (β₁ i₁)] [(i₂ : ι₂) → UniformSpace (β₂ i₂)] (e : ι₁ ≃ ι₂) (F : (i₁ : ι₁) → β₁ i₁ ≃ᵤ β₂ (e i₁)) : ((i₁ : ι₁) → β₁ i₁) ≃ᵤ ((i₂ : ι₂) → β₂ i₂)

Equiv.piCongr as a uniform isomorphism: this is the natural isomorphism Π i₁, β₁ i ≃ᵤ Π i₂, β₂ i₂ obtained from a bijection ι₁ ≃ ι₂ and isomorphisms β₁ i₁ ≃ᵤ β₂ (e i₁) for each i₁ : ι₁.

@[simp]

theorem UniformEquiv.piCongr_toEquiv {ι₁ : Type u_4} {ι₂ : Type u_5} {β₁ : ι₁ → Type u_6} {β₂ : ι₂ → Type u_7} [(i₁ : ι₁) → UniformSpace (β₁ i₁)] [(i₂ : ι₂) → UniformSpace (β₂ i₂)] (e : ι₁ ≃ ι₂) (F : (i₁ : ι₁) → β₁ i₁ ≃ᵤ β₂ (e i₁)) : (piCongr e F).toEquiv = (Equiv.piCongrRight fun (i : ι₁) => (F i).toEquiv).trans (Equiv.piCongrLeft β₂ e)

@[simp]

theorem UniformEquiv.piCongr_apply {ι₁ : Type u_4} {ι₂ : Type u_5} {β₁ : ι₁ → Type u_6} {β₂ : ι₂ → Type u_7} [(i₁ : ι₁) → UniformSpace (β₁ i₁)] [(i₂ : ι₂) → UniformSpace (β₂ i₂)] (e : ι₁ ≃ ι₂) (F : (i₁ : ι₁) → β₁ i₁ ≃ᵤ β₂ (e i₁)) (a✝ : (i : ι₁) → β₁ i) (i₂ : ι₂) : (piCongr e F) a✝ i₂ = (Equiv.piCongrLeft β₂ e) ((Equiv.piCongrRight fun (i : ι₁) => (F i).toEquiv) a✝) i₂

def UniformEquiv.ulift (α : Type u) [UniformSpace α] : ULift.{v, u} α ≃ᵤ α

Uniform equivalence between ULift α and α.

def UniformEquiv.funUnique (ι : Type u_4) (α : Type u_5) [Unique ι] [UniformSpace α] : (ι → α) ≃ᵤ α

If ι has a unique element, then ι → α is uniformly isomorphic to α.

@[simp]

theorem UniformEquiv.funUnique_symm_apply (ι : Type u_4) (α : Type u_5) [Unique ι] [UniformSpace α] : ⇑(funUnique ι α).symm = uniqueElim

@[simp]

theorem UniformEquiv.funUnique_apply (ι : Type u_4) (α : Type u_5) [Unique ι] [UniformSpace α] : ⇑(funUnique ι α) = fun (f : ι → α) => f default

def UniformEquiv.piFinTwo (α : Fin 2 → Type u) [(i : Fin 2) → UniformSpace (α i)] : ((i : Fin 2) → α i) ≃ᵤ α 0 × α 1

Uniform isomorphism between dependent functions Π i : Fin 2, α i and α 0 × α 1.

@[simp]

theorem UniformEquiv.piFinTwo_symm_apply (α : Fin 2 → Type u) [(i : Fin 2) → UniformSpace (α i)] : ⇑(piFinTwo α).symm = fun (p : α 0 × α 1) => Fin.cons p.1 (Fin.cons p.2 finZeroElim)

@[simp]

theorem UniformEquiv.piFinTwo_apply (α : Fin 2 → Type u) [(i : Fin 2) → UniformSpace (α i)] : ⇑(piFinTwo α) = fun (f : (i : Fin 2) → α i) => (f 0, f 1)

def UniformEquiv.finTwoArrow (α : Type u_4) [UniformSpace α] : (Fin 2 → α) ≃ᵤ α × α

Uniform isomorphism between α² = Fin 2 → α and α × α.

@[simp]

theorem UniformEquiv.finTwoArrow_apply (α : Type u_4) [UniformSpace α] : ⇑(finTwoArrow α) = fun (f : Fin 2 → α) => (f 0, f 1)

@[simp]

theorem UniformEquiv.finTwoArrow_symm_apply (α : Type u_4) [UniformSpace α] : ⇑(finTwoArrow α).symm = fun (x : α × α) => ![x.1, x.2]

def UniformEquiv.image {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (e : α ≃ᵤ β) (s : Set α) : ↑s ≃ᵤ ↑(⇑e '' s)

A subset of a uniform space is uniformly isomorphic to its image under a uniform isomorphism.

def Equiv.toUniformEquivOfIsUniformInducing {α : Type u} {β : Type u_1} [UniformSpace α] [UniformSpace β] (f : α ≃ β) (hf : IsUniformInducing ⇑f) : α ≃ᵤ β

A uniform inducing equiv between uniform spaces is a uniform isomorphism.