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Mathlib.Topology.Category.CompHaus.Basic

The category of Compact Hausdorff Spaces #

We construct the category of compact Hausdorff spaces. The type of compact Hausdorff spaces is denoted CompHaus, and it is endowed with a category instance making it a full subcategory of TopCat. The fully faithful functor CompHausTopCat is denoted compHausToTop.

Note: The file Topology/Category/Compactum.lean provides the equivalence between Compactum, which is defined as the category of algebras for the ultrafilter monad, and CompHaus. CompactumToCompHaus is the functor from Compactum to CompHaus which is proven to be an equivalence of categories in CompactumToCompHaus.isEquivalence. See topology/category/Compactum.lean for a more detailed discussion where these definitions are introduced.

structure CompHaus :
Type (u_1 + 1)
  • toTop : TopCat

    The underlying topological space of an object of CompHaus.

  • is_compact : CompactSpace s.toTop

    The underlying topological space is compact.

  • is_hausdorff : T2Space s.toTop

    The underlying topological space is T2.

The type of Compact Hausdorff topological spaces.

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    A constructor for objects of the category CompHaus, taking a type, and bundling the compact Hausdorff topology found by typeclass inference.

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      @[simp]
      theorem CompHaus.coe_of (X : Type u_1) [TopologicalSpace X] [CompactSpace X] [T2Space X] :
      (CompHaus.of X).toTop = X
      theorem CompHaus.isClosedMap {X : CompHaus} {Y : CompHaus} (f : X Y) :

      Any continuous function on compact Hausdorff spaces is a closed map.

      Any continuous bijection of compact Hausdorff spaces is an isomorphism.

      noncomputable def CompHaus.isoOfBijective {X : CompHaus} {Y : CompHaus} (f : X Y) (bij : Function.Bijective f) :
      X Y

      Any continuous bijection of compact Hausdorff spaces induces an isomorphism.

      Instances For
        @[simp]
        theorem CompHaus.isoOfHomeo_inv {X : CompHaus} {Y : CompHaus} (f : X.toTop ≃ₜ Y.toTop) :
        @[simp]
        theorem CompHaus.isoOfHomeo_hom {X : CompHaus} {Y : CompHaus} (f : X.toTop ≃ₜ Y.toTop) :
        def CompHaus.isoOfHomeo {X : CompHaus} {Y : CompHaus} (f : X.toTop ≃ₜ Y.toTop) :
        X Y

        Construct an isomorphism from a homeomorphism.

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          @[simp]
          theorem CompHaus.homeoOfIso_apply {X : CompHaus} {Y : CompHaus} (f : X Y) (a : (CategoryTheory.forget CompHaus).obj X) :
          ↑(CompHaus.homeoOfIso f) a = f.hom a
          @[simp]
          def CompHaus.homeoOfIso {X : CompHaus} {Y : CompHaus} (f : X Y) :
          X.toTop ≃ₜ Y.toTop

          Construct a homeomorphism from an isomorphism.

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            @[simp]
            theorem CompHaus.isoEquivHomeo_symm_apply {X : CompHaus} {Y : CompHaus} (f : X.toTop ≃ₜ Y.toTop) :
            CompHaus.isoEquivHomeo.symm f = CompHaus.isoOfHomeo f
            @[simp]
            theorem CompHaus.isoEquivHomeo_apply {X : CompHaus} {Y : CompHaus} (f : X Y) :
            CompHaus.isoEquivHomeo f = CompHaus.homeoOfIso f
            def CompHaus.isoEquivHomeo {X : CompHaus} {Y : CompHaus} :
            (X Y) (X.toTop ≃ₜ Y.toTop)

            The equivalence between isomorphisms in CompHaus and homeomorphisms of topological spaces.

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              @[simp]
              theorem compHausToTop_obj (self : CompHaus) :
              compHausToTop.obj self = self.toTop

              The fully faithful embedding of CompHaus in TopCat.

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                @[simp]
                theorem stoneCechObj_toTop_α (X : TopCat) :
                (stoneCechObj X).toTop = StoneCech X

                (Implementation) The object part of the compactification functor from topological spaces to compact Hausdorff spaces.

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                  noncomputable def stoneCechEquivalence (X : TopCat) (Y : CompHaus) :

                  (Implementation) The bijection of homsets to establish the reflective adjunction of compact Hausdorff spaces in topological spaces.

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                    The Stone-Cech compactification functor from topological spaces to compact Hausdorff spaces, left adjoint to the inclusion functor.

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                      theorem topToCompHaus_obj (X : TopCat) :
                      (topToCompHaus.obj X).toTop = StoneCech X

                      The category of compact Hausdorff spaces is reflective in the category of topological spaces.

                      An explicit limit cone for a functor F : J ⥤ CompHaus, defined in terms of TopCat.limitCone.

                      Instances For