Abstract theory of Hausdorff completions of uniform spaces

This file characterizes Hausdorff completions of a uniform space α as complete Hausdorff spaces equipped with a map from α which has dense image and induce the original uniform structure on α. Assuming these properties we "extend" uniformly continuous maps from α to complete Hausdorff spaces to the completions of α. This is the universal property expected from a completion. It is then used to extend uniformly continuous maps from α to α' to maps between completions of α and α'.

This file does not construct any such completion, it only study consequences of their existence. The first advantage is that formal properties are clearly highlighted without interference from construction details. The second advantage is that this framework can then be used to compare different completion constructions. See Topology/UniformSpace/CompareReals for an example. Of course the comparison comes from the universal property as usual.

A general explicit construction of completions is done in UniformSpace/Completion, leading to a functor from uniform spaces to complete Hausdorff uniform spaces that is left adjoint to the inclusion, see UniformSpace/UniformSpaceCat for the category packaging.

Implementation notes

A tiny technical advantage of using a characteristic predicate such as the properties listed in AbstractCompletion instead of stating the universal property is that the universal property derived from the predicate is more universe polymorphic.

References

We don't know any traditional text discussing this. Real world mathematics simply silently identify the results of any two constructions that lead to something one could reasonably call a completion.

Tags

uniform spaces, completion, universal property

structure AbstractCompletion (α : Type u) [UniformSpace α] : Type (u + 1)

A completion of α is the data of a complete separated uniform space (from the same universe) and a map from α with dense range and inducing the original uniform structure on α.

• space : Type u

  The underlying space of the completion.

• coe : α → self.space

  A map from a space to its completion.

• uniformStruct : UniformSpace self.space

  The completion carries a uniform structure.

• complete : CompleteSpace self.space

  The completion is complete.

• separation : T0Space self.space

  The completion is a T₀ space.

• uniformInducing : UniformInducing self.coe

  The map into the completion is uniform-inducing.

• dense : DenseRange self.coe

  The map into the completion has dense range.

theorem AbstractCompletion.complete {α : Type u} [UniformSpace α] (self : AbstractCompletion α) : CompleteSpace self.space

The completion is complete.

theorem AbstractCompletion.separation {α : Type u} [UniformSpace α] (self : AbstractCompletion α) : T0Space self.space

The completion is a T₀ space.

theorem AbstractCompletion.uniformInducing {α : Type u} [UniformSpace α] (self : AbstractCompletion α) : UniformInducing self.coe

The map into the completion is uniform-inducing.

theorem AbstractCompletion.dense {α : Type u} [UniformSpace α] (self : AbstractCompletion α) : DenseRange self.coe

The map into the completion has dense range.

def AbstractCompletion.ofComplete {α : Type u_1} [UniformSpace α] [T0Space α] [CompleteSpace α] : AbstractCompletion α

If α is complete, then it is an abstract completion of itself.

Equations

• AbstractCompletion.ofComplete = { space := α, coe := id, uniformStruct := inferInstance, complete := ⋯, separation := ⋯, uniformInducing := ⋯, dense := ⋯ }

theorem AbstractCompletion.closure_range {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) : closure (Set.range pkg.coe) = Set.univ

theorem AbstractCompletion.denseInducing {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) : DenseInducing pkg.coe

theorem AbstractCompletion.uniformContinuous_coe {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) : UniformContinuous pkg.coe

theorem AbstractCompletion.continuous_coe {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) : Continuous pkg.coe

theorem AbstractCompletion.induction_on {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {p : pkg.space → Prop} (a : pkg.space) (hp : IsClosed {a : pkg.space | p a}) (ih : ∀ (a : α), p (pkg.coe a)) : p a

theorem AbstractCompletion.funext {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [TopologicalSpace β] [T2Space β] {f : pkg.space → β} {g : pkg.space → β} (hf : Continuous f) (hg : Continuous g) (h : ∀ (a : α), f (pkg.coe a) = g (pkg.coe a)) : f = g

def AbstractCompletion.extend {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (f : α → β) : pkg.space → β

Extension of maps to completions

Equations

• pkg.extend f = if UniformContinuous f then ⋯.extend f else fun (x : pkg.space) => f (⋯.some x)

theorem AbstractCompletion.extend_def {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] {f : α → β} (hf : UniformContinuous f) : pkg.extend f = ⋯.extend f

theorem AbstractCompletion.extend_coe {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] {f : α → β} [T2Space β] (hf : UniformContinuous f) (a : α) : pkg.extend f (pkg.coe a) = f a

theorem AbstractCompletion.uniformContinuous_extend {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] {f : α → β} [CompleteSpace β] : UniformContinuous (pkg.extend f)

theorem AbstractCompletion.continuous_extend {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] {f : α → β} [CompleteSpace β] : Continuous (pkg.extend f)

theorem AbstractCompletion.extend_unique {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] {f : α → β} [CompleteSpace β] [T0Space β] (hf : UniformContinuous f) {g : pkg.space → β} (hg : UniformContinuous g) (h : ∀ (a : α), f a = g (pkg.coe a)) : pkg.extend f = g

@[simp]

theorem AbstractCompletion.extend_comp_coe {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] [CompleteSpace β] [T0Space β] {f : pkg.space → β} (hf : UniformContinuous f) : pkg.extend (f ∘ pkg.coe) = f

def AbstractCompletion.map {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) (f : α → β) : pkg.space → pkg'.space

Lifting maps to completions

Equations

• pkg.map pkg' f = pkg.extend (pkg'.coe ∘ f)

theorem AbstractCompletion.uniformContinuous_map {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) (f : α → β) : UniformContinuous (pkg.map pkg' f)

theorem AbstractCompletion.continuous_map {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) (f : α → β) : Continuous (pkg.map pkg' f)

@[simp]

theorem AbstractCompletion.map_coe {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {f : α → β} (hf : UniformContinuous f) (a : α) : pkg.map pkg' f (pkg.coe a) = pkg'.coe (f a)

theorem AbstractCompletion.map_unique {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {f : α → β} {g : pkg.space → pkg'.space} (hg : UniformContinuous g) (h : ∀ (a : α), pkg'.coe (f a) = g (pkg.coe a)) : pkg.map pkg' f = g

@[simp]

theorem AbstractCompletion.map_id {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) : pkg.map pkg id = id

theorem AbstractCompletion.extend_map {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {γ : Type u_3} [UniformSpace γ] [CompleteSpace γ] [T0Space γ] {f : β → γ} {g : α → β} (hf : UniformContinuous f) (hg : UniformContinuous g) : pkg'.extend f ∘ pkg.map pkg' g = pkg.extend (f ∘ g)

theorem AbstractCompletion.map_comp {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {γ : Type u_3} [UniformSpace γ] (pkg'' : AbstractCompletion γ) {g : β → γ} {f : α → β} (hg : UniformContinuous g) (hf : UniformContinuous f) : pkg'.map pkg'' g ∘ pkg.map pkg' f = pkg.map pkg'' (g ∘ f)

def AbstractCompletion.compare {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) (pkg' : AbstractCompletion α) : pkg.space → pkg'.space

The comparison map between two completions of the same uniform space.

Equations

• pkg.compare pkg' = pkg.extend pkg'.coe

theorem AbstractCompletion.uniformContinuous_compare {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) (pkg' : AbstractCompletion α) : UniformContinuous (pkg.compare pkg')

theorem AbstractCompletion.compare_coe {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) (pkg' : AbstractCompletion α) (a : α) : pkg.compare pkg' (pkg.coe a) = pkg'.coe a

theorem AbstractCompletion.inverse_compare {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) (pkg' : AbstractCompletion α) : pkg.compare pkg' ∘ pkg'.compare pkg = id

def AbstractCompletion.compareEquiv {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) (pkg' : AbstractCompletion α) : pkg.space ≃ᵤ pkg'.space

The uniform bijection between two completions of the same uniform space.

Equations

• pkg.compareEquiv pkg' = { toFun := pkg.compare pkg', invFun := pkg'.compare pkg, left_inv := ⋯, right_inv := ⋯, uniformContinuous_toFun := ⋯, uniformContinuous_invFun := ⋯ }

theorem AbstractCompletion.uniformContinuous_compareEquiv {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) (pkg' : AbstractCompletion α) : UniformContinuous ⇑(pkg.compareEquiv pkg')

theorem AbstractCompletion.uniformContinuous_compareEquiv_symm {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) (pkg' : AbstractCompletion α) : UniformContinuous ⇑(pkg.compareEquiv pkg').symm

def AbstractCompletion.prod {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) : AbstractCompletion (α × β)

Products of completions

Equations

• One or more equations did not get rendered due to their size.

def AbstractCompletion.extend₂ {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {γ : Type u_3} [UniformSpace γ] (f : α → β → γ) : pkg.space → pkg'.space → γ

Extend two variable map to completions.

Equations

• pkg.extend₂ pkg' f = Function.curry ((pkg.prod pkg').extend (Function.uncurry f))

theorem AbstractCompletion.extension₂_coe_coe {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {γ : Type u_3} [UniformSpace γ] [T0Space γ] {f : α → β → γ} (hf : UniformContinuous (Function.uncurry f)) (a : α) (b : β) : pkg.extend₂ pkg' f (pkg.coe a) (pkg'.coe b) = f a b

theorem AbstractCompletion.uniformContinuous_extension₂ {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {γ : Type u_3} [UniformSpace γ] (f : α → β → γ) [CompleteSpace γ] : UniformContinuous₂ (pkg.extend₂ pkg' f)

def AbstractCompletion.map₂ {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {γ : Type u_3} [UniformSpace γ] (pkg'' : AbstractCompletion γ) (f : α → β → γ) : pkg.space → pkg'.space → pkg''.space

Lift two variable maps to completions.

Equations

• pkg.map₂ pkg' pkg'' f = pkg.extend₂ pkg' (Function.bicompr pkg''.coe f)

theorem AbstractCompletion.uniformContinuous_map₂ {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {γ : Type u_3} [UniformSpace γ] (pkg'' : AbstractCompletion γ) (f : α → β → γ) : UniformContinuous₂ (pkg.map₂ pkg' pkg'' f)

theorem AbstractCompletion.continuous_map₂ {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {γ : Type u_3} [UniformSpace γ] (pkg'' : AbstractCompletion γ) {δ : Type u_4} [TopologicalSpace δ] {f : α → β → γ} {a : δ → pkg.space} {b : δ → pkg'.space} (ha : Continuous a) (hb : Continuous b) : Continuous fun (d : δ) => pkg.map₂ pkg' pkg'' f (a d) (b d)

theorem AbstractCompletion.map₂_coe_coe {α : Type u_1} [UniformSpace α] (pkg : AbstractCompletion α) {β : Type u_2} [UniformSpace β] (pkg' : AbstractCompletion β) {γ : Type u_3} [UniformSpace γ] (pkg'' : AbstractCompletion γ) (a : α) (b : β) (f : α → β → γ) (hf : UniformContinuous₂ f) : pkg.map₂ pkg' pkg'' f (pkg.coe a) (pkg'.coe b) = pkg''.coe (f a b)