Hausdorff completions of uniform spaces

The goal is to construct a left-adjoint to the inclusion of complete Hausdorff uniform spaces into all uniform spaces. Any uniform space α gets a completion Completion α and a morphism (i.e. uniformly continuous map) (↑) : α → Completion α which solves the universal mapping problem of factorizing morphisms from α to any complete Hausdorff uniform space β. It means any uniformly continuous f : α → β gives rise to a unique morphism Completion.extension f : Completion α → β such that f = Completion.extension f ∘ (↑). Actually Completion.extension f is defined for all maps from α to β but it has the desired properties only if f is uniformly continuous.

Beware that (↑) is not injective if α is not Hausdorff. But its image is always dense. The adjoint functor acting on morphisms is then constructed by the usual abstract nonsense. For every uniform spaces α and β, it turns f : α → β into a morphism Completion.map f : Completion α → Completion β such that (↑) ∘ f = (Completion.map f) ∘ (↑) provided f is uniformly continuous. This construction is compatible with composition.

In this file we introduce the following concepts:

• CauchyFilter α the uniform completion of the uniform space α (using Cauchy filters). These are not minimal filters.

• Completion α := Quotient (separationSetoid (CauchyFilter α)) the Hausdorff completion.

References

This formalization is mostly based on N. Bourbaki: General Topology I. M. James: Topologies and Uniformities From a slightly different perspective in order to reuse material in Topology.UniformSpace.Basic.

def CauchyFilter (α : Type u) [UniformSpace α] : Type u

Space of Cauchy filters

This is essentially the completion of a uniform space. The embeddings are the neighbourhood filters. This space is not minimal, the separated uniform space (i.e. quotiented on the intersection of all entourages) is necessary for this.

instance CauchyFilter.instNeBotValFilterCauchy {α : Type u} [UniformSpace α] (f : CauchyFilter α) : (↑f).NeBot

def CauchyFilter.gen {α : Type u} [UniformSpace α] (s : SetRel α α) : SetRel (CauchyFilter α) (CauchyFilter α)

The pairs of Cauchy filters generated by a set.

theorem CauchyFilter.monotone_gen {α : Type u} [UniformSpace α] : Monotone gen

@[implicit_reducible]

instance CauchyFilter.instUniformSpace {α : Type u} [UniformSpace α] : UniformSpace (CauchyFilter α)

theorem CauchyFilter.mem_uniformity {α : Type u} [UniformSpace α] {s : Set (CauchyFilter α × CauchyFilter α)} : s ∈ uniformity (CauchyFilter α) ↔ ∃ t ∈ uniformity α, gen t ⊆ s

theorem CauchyFilter.basis_uniformity {α : Type u} [UniformSpace α] {ι : Sort u_1} {p : ι → Prop} {s : ι → SetRel α α} (h : (uniformity α).HasBasis p s) : (uniformity (CauchyFilter α)).HasBasis p (gen ∘ s)

theorem CauchyFilter.mem_uniformity' {α : Type u} [UniformSpace α] {s : Set (CauchyFilter α × CauchyFilter α)} : s ∈ uniformity (CauchyFilter α) ↔ ∃ t ∈ uniformity α, ∀ (f g : CauchyFilter α), t ∈ ↑f ×ˢ ↑g → (f, g) ∈ s

def CauchyFilter.pureCauchy {α : Type u} [UniformSpace α] (a : α) : CauchyFilter α

Embedding of α into its completion CauchyFilter α

theorem CauchyFilter.isUniformInducing_pureCauchy {α : Type u} [UniformSpace α] : IsUniformInducing pureCauchy

theorem CauchyFilter.isUniformEmbedding_pureCauchy {α : Type u} [UniformSpace α] : IsUniformEmbedding pureCauchy

theorem CauchyFilter.denseRange_pureCauchy {α : Type u} [UniformSpace α] : DenseRange pureCauchy

theorem CauchyFilter.isDenseInducing_pureCauchy {α : Type u} [UniformSpace α] : IsDenseInducing pureCauchy

theorem CauchyFilter.isDenseEmbedding_pureCauchy {α : Type u} [UniformSpace α] : IsDenseEmbedding pureCauchy

theorem CauchyFilter.nonempty_cauchyFilter_iff {α : Type u} [UniformSpace α] : Nonempty (CauchyFilter α) ↔ Nonempty α

instance CauchyFilter.instCompleteSpace {α : Type u} [UniformSpace α] : CompleteSpace (CauchyFilter α)

@[implicit_reducible]

instance CauchyFilter.instInhabited {α : Type u} [UniformSpace α] [Inhabited α] : Inhabited (CauchyFilter α)

instance CauchyFilter.instNonempty {α : Type u} [UniformSpace α] [h : Nonempty α] : Nonempty (CauchyFilter α)

noncomputable def CauchyFilter.extend {α : Type u} [UniformSpace α] {β : Type v} [UniformSpace β] (f : α → β) : CauchyFilter α → β

Extend a uniformly continuous function α → β to a function CauchyFilter α → β. Outputs junk when f is not uniformly continuous.

theorem CauchyFilter.extend_pureCauchy {α : Type u} [UniformSpace α] {β : Type v} [UniformSpace β] [T0Space β] {f : α → β} (hf : UniformContinuous f) (a : α) : extend f (pureCauchy a) = f a

theorem CauchyFilter.uniformContinuous_extend {α : Type u} [UniformSpace α] {β : Type v} [UniformSpace β] [CompleteSpace β] {f : α → β} : UniformContinuous (extend f)

theorem CauchyFilter.inseparable_iff {α : Type u} [UniformSpace α] {f g : CauchyFilter α} : Inseparable f g ↔ ↑f ×ˢ ↑g ≤ uniformity α

theorem CauchyFilter.inseparable_iff_of_le_nhds {α : Type u} [UniformSpace α] {f g : CauchyFilter α} {a b : α} (ha : ↑f ≤ nhds a) (hb : ↑g ≤ nhds b) : Inseparable a b ↔ Inseparable f g

theorem CauchyFilter.inseparable_lim_iff {α : Type u} [UniformSpace α] [CompleteSpace α] {f g : CauchyFilter α} : Inseparable (↑f).lim (↑g).lim ↔ Inseparable f g

theorem CauchyFilter.cauchyFilter_eq {α : Type u_1} [UniformSpace α] [CompleteSpace α] [T0Space α] {f g : CauchyFilter α} : (↑f).lim = (↑g).lim ↔ Inseparable f g

theorem CauchyFilter.separated_pureCauchy_injective {α : Type u_1} [UniformSpace α] [T0Space α] : Function.Injective fun (a : α) => SeparationQuotient.mk (pureCauchy a)

def UniformSpace.Completion (α : Type u_1) [UniformSpace α] : Type u_1

Hausdorff completion of α

@[implicit_reducible]

instance UniformSpace.Completion.inhabited (α : Type u_1) [UniformSpace α] [Inhabited α] : Inhabited (Completion α)

@[implicit_reducible]

instance UniformSpace.Completion.uniformSpace (α : Type u_1) [UniformSpace α] : UniformSpace (Completion α)

instance UniformSpace.Completion.completeSpace (α : Type u_1) [UniformSpace α] : CompleteSpace (Completion α)

instance UniformSpace.Completion.t0Space (α : Type u_1) [UniformSpace α] : T0Space (Completion α)

def UniformSpace.Completion.coe' {α : Type u_1} [UniformSpace α] : α → Completion α

The map from a uniform space to its completion.

@[implicit_reducible]

instance UniformSpace.Completion.instCoe (α : Type u_1) [UniformSpace α] : Coe α (Completion α)

Automatic coercion from α to its completion. Not always injective.

theorem UniformSpace.Completion.coe_eq (α : Type u_1) [UniformSpace α] : coe' = SeparationQuotient.mk ∘ CauchyFilter.pureCauchy

theorem UniformSpace.Completion.isUniformInducing_coe (α : Type u_1) [UniformSpace α] : IsUniformInducing coe'

theorem UniformSpace.Completion.comap_coe_eq_uniformity (α : Type u_1) [UniformSpace α] : Filter.comap (fun (p : α × α) => (↑p.1, ↑p.2)) (uniformity (Completion α)) = uniformity α

theorem UniformSpace.Completion.denseRange_coe {α : Type u_1} [UniformSpace α] : DenseRange coe'

def UniformSpace.Completion.cPkg {α : Type u_4} [UniformSpace α] : AbstractCompletion.{u_4, u_4} α

The Hausdorff completion as an abstract completion.

@[implicit_reducible]

instance UniformSpace.Completion.AbstractCompletion.inhabited (α : Type u_1) [UniformSpace α] : Inhabited (AbstractCompletion.{u_1, u_1} α)

theorem UniformSpace.Completion.nonempty_completion_iff (α : Type u_1) [UniformSpace α] : Nonempty (Completion α) ↔ Nonempty α

theorem UniformSpace.Completion.uniformContinuous_coe (α : Type u_1) [UniformSpace α] : UniformContinuous coe'

theorem UniformSpace.Completion.continuous_coe (α : Type u_1) [UniformSpace α] : Continuous coe'

theorem UniformSpace.Completion.isUniformEmbedding_coe (α : Type u_1) [UniformSpace α] [T0Space α] : IsUniformEmbedding coe'

theorem UniformSpace.Completion.coe_injective (α : Type u_1) [UniformSpace α] [T0Space α] : Function.Injective coe'

@[simp]

theorem UniformSpace.Completion.coe_inj {α : Type u_1} [UniformSpace α] [T0Space α] {a b : α} : ↑a = ↑b ↔ a = b

theorem UniformSpace.Completion.isDenseInducing_coe {α : Type u_1} [UniformSpace α] : IsDenseInducing coe'

noncomputable def UniformSpace.Completion.UniformCompletion.completeEquivSelf {α : Type u_1} [UniformSpace α] [CompleteSpace α] [T0Space α] : Completion α ≃ᵤ α

The uniform bijection between a complete space and its uniform completion.

instance UniformSpace.Completion.separableSpace_completion {α : Type u_1} [UniformSpace α] [TopologicalSpace.SeparableSpace α] : TopologicalSpace.SeparableSpace (Completion α)

theorem UniformSpace.Completion.isDenseEmbedding_coe {α : Type u_1} [UniformSpace α] [T0Space α] : IsDenseEmbedding coe'

theorem UniformSpace.Completion.denseRange_coe₂ {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] : DenseRange fun (x : α × β) => (↑x.1, ↑x.2)

theorem UniformSpace.Completion.denseRange_coe₃ {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {γ : Type u_3} [UniformSpace γ] : DenseRange fun (x : α × β × γ) => (↑x.1, ↑x.2.1, ↑x.2.2)

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

theorem UniformSpace.Completion.induction_on₂ {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {p : Completion α → Completion β → Prop} (a : Completion α) (b : Completion β) (hp : IsClosed {x : Completion α × Completion β | p x.1 x.2}) (ih : ∀ (a : α) (b : β), p ↑a ↑b) : p a b

theorem UniformSpace.Completion.induction_on₃ {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {γ : Type u_3} [UniformSpace γ] {p : Completion α → Completion β → Completion γ → Prop} (a : Completion α) (b : Completion β) (c : Completion γ) (hp : IsClosed {x : Completion α × Completion β × Completion γ | p x.1 x.2.1 x.2.2}) (ih : ∀ (a : α) (b : β) (c : γ), p ↑a ↑b ↑c) : p a b c

theorem UniformSpace.Completion.ext {α : Type u_1} [UniformSpace α] {Y : Type u_4} [TopologicalSpace Y] [T2Space Y] {f g : Completion α → Y} (hf : Continuous f) (hg : Continuous g) (h : ∀ (a : α), f ↑a = g ↑a) : f = g

theorem UniformSpace.Completion.ext' {α : Type u_1} [UniformSpace α] {Y : Type u_4} [TopologicalSpace Y] [T2Space Y] {f g : Completion α → Y} (hf : Continuous f) (hg : Continuous g) (h : ∀ (a : α), f ↑a = g ↑a) (a : Completion α) : f a = g a

noncomputable def UniformSpace.Completion.extension {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] (f : α → β) : Completion α → β

"Extension" to the completion. It is defined for any map f but returns an arbitrary constant value if f is not uniformly continuous

theorem UniformSpace.Completion.uniformContinuous_extension {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {f : α → β} [CompleteSpace β] : UniformContinuous (Completion.extension f)

theorem UniformSpace.Completion.continuous_extension {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {f : α → β} [CompleteSpace β] : Continuous (Completion.extension f)

theorem UniformSpace.Completion.extension_coe {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {f : α → β} [T0Space β] (hf : UniformContinuous f) (a : α) : Completion.extension f ↑a = f a

theorem UniformSpace.Completion.inseparable_extension_coe {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {f : α → β} (hf : UniformContinuous f) (x : α) : Inseparable (Completion.extension f ↑x) (f x)

theorem UniformSpace.Completion.isUniformInducing_extension {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {f : α → β} [CompleteSpace β] (h : IsUniformInducing f) : IsUniformInducing (Completion.extension f)

theorem UniformSpace.Completion.extension_unique {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {f : α → β} [T0Space β] [CompleteSpace β] (hf : UniformContinuous f) {g : Completion α → β} (hg : UniformContinuous g) (h : ∀ (a : α), f a = g ↑a) : Completion.extension f = g

@[simp]

theorem UniformSpace.Completion.extension_comp_coe {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] [T0Space β] [CompleteSpace β] {f : Completion α → β} (hf : UniformContinuous f) : Completion.extension (f ∘ coe') = f

noncomputable def UniformSpace.Completion.map {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] (f : α → β) : Completion α → Completion β

Completion functor acting on morphisms

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

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

theorem UniformSpace.Completion.map_coe {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {f : α → β} (hf : UniformContinuous f) (a : α) : Completion.map f ↑a = ↑(f a)

theorem UniformSpace.Completion.map_unique {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {f : α → β} {g : Completion α → Completion β} (hg : UniformContinuous g) (h : ∀ (a : α), ↑(f a) = g ↑a) : Completion.map f = g

@[simp]

theorem UniformSpace.Completion.map_id {α : Type u_1} [UniformSpace α] : Completion.map id = id

theorem UniformSpace.Completion.extension_map {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {γ : Type u_3} [UniformSpace γ] [CompleteSpace γ] [T0Space γ] {f : β → γ} {g : α → β} (hf : UniformContinuous f) (hg : UniformContinuous g) : Completion.extension f ∘ Completion.map g = Completion.extension (f ∘ g)

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

noncomputable def UniformSpace.Completion.mapEquiv {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] (e : α ≃ᵤ β) : Completion α ≃ᵤ Completion β

The uniform isomorphism between two completions of isomorphic uniform spaces.

@[simp]

theorem UniformSpace.Completion.mapEquiv_symm {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] (e : α ≃ᵤ β) : (mapEquiv e).symm = mapEquiv e.symm

@[simp]

theorem UniformSpace.Completion.mapEquiv_coe {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] (e : α ≃ᵤ β) (a : α) : (mapEquiv e) ↑a = ↑(e a)

noncomputable def UniformSpace.Completion.completionSeparationQuotientEquiv (α : Type u) [UniformSpace α] : Completion (SeparationQuotient α) ≃ Completion α

The isomorphism between the completion of a uniform space and the completion of its separation quotient.

theorem UniformSpace.Completion.uniformContinuous_completionSeparationQuotientEquiv {α : Type u_1} [UniformSpace α] : UniformContinuous ⇑(completionSeparationQuotientEquiv α)

theorem UniformSpace.Completion.uniformContinuous_completionSeparationQuotientEquiv_symm {α : Type u_1} [UniformSpace α] : UniformContinuous ⇑(completionSeparationQuotientEquiv α).symm

noncomputable def UniformSpace.Completion.extension₂ {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {γ : Type u_3} [UniformSpace γ] (f : α → β → γ) : Completion α → Completion β → γ

Extend a two variable map to the Hausdorff completions.

theorem UniformSpace.Completion.extension₂_coe_coe {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {γ : Type u_3} [UniformSpace γ] {f : α → β → γ} [T0Space γ] (hf : UniformContinuous₂ f) (a : α) (b : β) : Completion.extension₂ f ↑a ↑b = f a b

theorem UniformSpace.Completion.uniformContinuous_extension₂ {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {γ : Type u_3} [UniformSpace γ] (f : α → β → γ) [CompleteSpace γ] : UniformContinuous₂ (Completion.extension₂ f)

noncomputable def UniformSpace.Completion.map₂ {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {γ : Type u_3} [UniformSpace γ] (f : α → β → γ) : Completion α → Completion β → Completion γ

Lift a two variable map to the Hausdorff completions.

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

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

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