Documentation

Mathlib.Topology.UniformSpace.Completion

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 (ie. 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:

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.uniform_space.basic.

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

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.

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

    The pairs of Cauchy filters generated by a set.

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      theorem CauchyFilter.monotone_gen {α : Type u} [UniformSpace α] :
      Monotone CauchyFilter.gen
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      • One or more equations did not get rendered due to their size.
      theorem CauchyFilter.mem_uniformity' {α : Type u} [UniformSpace α] {s : Set (CauchyFilter α × CauchyFilter α)} :
      s uniformity (CauchyFilter α) t, t uniformity α ∀ (f g : CauchyFilter α), t f ×ˢ g(f, g) s
      def CauchyFilter.pureCauchy {α : Type u} [UniformSpace α] (a : α) :

      Embedding of α into its completion CauchyFilter α

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        theorem CauchyFilter.denseRange_pureCauchy {α : Type u} [UniformSpace α] :
        DenseRange CauchyFilter.pureCauchy
        theorem CauchyFilter.denseInducing_pureCauchy {α : Type u} [UniformSpace α] :
        DenseInducing CauchyFilter.pureCauchy
        theorem CauchyFilter.denseEmbedding_pureCauchy {α : Type u} [UniformSpace α] :
        DenseEmbedding CauchyFilter.pureCauchy
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        def CauchyFilter.extend {α : Type u} [UniformSpace α] {β : Type v} [UniformSpace β] (f : αβ) :
        CauchyFilter αβ

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

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          theorem CauchyFilter.extend_pureCauchy {α : Type u} [UniformSpace α] {β : Type v} [UniformSpace β] [SeparatedSpace β] {f : αβ} (hf : UniformContinuous f) (a : α) :

          Hausdorff completion of α

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            The map from a uniform space to its completion.

            porting note: this was added to create a target for the @[coe] attribute.

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            • α = Quotient.mk' CauchyFilter.pureCauchy
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              Automatic coercion from α to its completion. Not always injective.

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

              The Haudorff completion as an abstract completion.

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              • One or more equations did not get rendered due to their size.
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                The uniform bijection between a complete space and its uniform completion.

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                  theorem UniformSpace.Completion.denseRange_coe₂ {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] :
                  DenseRange fun x => (α x.fst, β x.snd)
                  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 x.fst, β x.snd.fst, γ x.snd.snd)
                  theorem UniformSpace.Completion.induction_on {α : Type u_1} [UniformSpace α] {p : UniformSpace.Completion αProp} (a : UniformSpace.Completion α) (hp : IsClosed {a | p a}) (ih : (a : α) → p (α a)) :
                  p a
                  theorem UniformSpace.Completion.induction_on₂ {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {p : UniformSpace.Completion αUniformSpace.Completion βProp} (a : UniformSpace.Completion α) (b : UniformSpace.Completion β) (hp : IsClosed {x | p x.fst x.snd}) (ih : (a : α) → (b : β) → p (α a) (β 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 : UniformSpace.Completion αUniformSpace.Completion βUniformSpace.Completion γProp} (a : UniformSpace.Completion α) (b : UniformSpace.Completion β) (c : UniformSpace.Completion γ) (hp : IsClosed {x | p x.fst x.snd.fst x.snd.snd}) (ih : (a : α) → (b : β) → (c : γ) → p (α a) (β b) (γ c)) :
                  p a b c
                  theorem UniformSpace.Completion.ext {α : Type u_1} [UniformSpace α] {Y : Type u_4} [TopologicalSpace Y] [T2Space Y] {f : UniformSpace.Completion αY} {g : UniformSpace.Completion αY} (hf : Continuous f) (hg : Continuous g) (h : ∀ (a : α), f (α a) = g (α a)) :
                  f = g
                  theorem UniformSpace.Completion.ext' {α : Type u_1} [UniformSpace α] {Y : Type u_4} [TopologicalSpace Y] [T2Space Y] {f : UniformSpace.Completion αY} {g : UniformSpace.Completion αY} (hf : Continuous f) (hg : Continuous g) (h : ∀ (a : α), f (α a) = g (α a)) (a : UniformSpace.Completion α) :
                  f a = g a
                  def UniformSpace.Completion.extension {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] (f : αβ) :

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

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

                    Completion functor acting on morphisms

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

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

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                      • One or more equations did not get rendered due to their size.
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                        def UniformSpace.Completion.extension₂ {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {γ : Type u_3} [UniformSpace γ] (f : αβγ) :

                        Extend a two variable map to the Hausdorff completions.

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                          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 : αβγ} [SeparatedSpace γ] (hf : UniformContinuous₂ f) (a : α) (b : β) :
                          UniformSpace.Completion.extension₂ f (α a) (β b) = f a b
                          def UniformSpace.Completion.map₂ {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {γ : Type u_3} [UniformSpace γ] (f : αβγ) :

                          Lift a two variable map to the Hausdorff completions.

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                            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 : αβγ} {a : δUniformSpace.Completion α} {b : δUniformSpace.Completion β} (ha : Continuous a) (hb : Continuous b) :
                            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 : αβγ) (hf : UniformContinuous₂ f) :
                            UniformSpace.Completion.map₂ f (α a) (β b) = γ (f a b)