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Mathlib.Topology.Category.TopCat.Basic

Category instance for topological spaces #

We introduce the bundled category TopCat of topological spaces together with the functors TopCat.discrete and TopCat.trivial from the category of types to TopCat which equip a type with the corresponding discrete, resp. trivial, topology. For a proof that these functors are left, resp. right adjoint to the forgetful functor, see Mathlib.Topology.Category.TopCat.Adjunctions.

def TopCat :
Type (u + 1)

The category of topological spaces and continuous maps.

Equations
Instances For
    Equations
    • One or more equations did not get rendered due to their size.
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    Equations
    • X.topologicalSpaceUnbundled = X.str
    instance TopCat.instFunLike (X : TopCat) (Y : TopCat) :
    FunLike (X Y) X Y
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    instance TopCat.instContinuousMapClass (X : TopCat) (Y : TopCat) :
    ContinuousMapClass (X Y) X Y
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    • =
    @[simp]
    theorem TopCat.id_app (X : TopCat) (x : X) :
    @[simp]
    theorem TopCat.comp_app {X : TopCat} {Y : TopCat} {Z : TopCat} (f : X Y) (g : Y Z) (x : X) :
    @[simp]
    theorem TopCat.coe_comp {X : TopCat} {Y : TopCat} {Z : TopCat} (f : X Y) (g : Y Z) :
    @[simp]
    theorem TopCat.hom_inv_id_apply {X : TopCat} {Y : TopCat} (f : X Y) (x : X) :
    f.inv (f.hom x) = x
    @[simp]
    theorem TopCat.inv_hom_id_apply {X : TopCat} {Y : TopCat} (f : X Y) (y : Y) :
    f.hom (f.inv y) = y

    Construct a bundled Top from the underlying type and the typeclass.

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    Instances For
      Equations
      • X.topologicalSpace_coe = X.str
      @[reducible, inline]
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      • X.topologicalSpace_forget = X.str
      @[simp]
      theorem TopCat.coe_of (X : Type u) [TopologicalSpace X] :
      (TopCat.of X) = X
      @[simp]
      theorem TopCat.coe_of_of {X : Type u} {Y : Type u} [TopologicalSpace X] [TopologicalSpace Y] {f : C(X, Y)} {x : (CategoryTheory.forget TopCat).obj (TopCat.of X)} :
      f x = f x

      Replace a function coercion for a morphism TopCat.of X ⟶ TopCat.of Y with the definitionally equal function coercion for a continuous map C(X, Y).

      theorem TopCat.hom_apply {X : TopCat} {Y : TopCat} (f : X Y) (x : X) :
      f x = f.toFun x

      The discrete topology on any type.

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

        The trivial topology on any type.

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        • One or more equations did not get rendered due to their size.
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          @[simp]
          theorem TopCat.isoOfHomeo_inv {X : TopCat} {Y : TopCat} (f : X ≃ₜ Y) :
          (TopCat.isoOfHomeo f).inv = f.symm
          @[simp]
          theorem TopCat.isoOfHomeo_hom {X : TopCat} {Y : TopCat} (f : X ≃ₜ Y) :
          (TopCat.isoOfHomeo f).hom = f
          def TopCat.isoOfHomeo {X : TopCat} {Y : TopCat} (f : X ≃ₜ Y) :
          X Y

          Any homeomorphisms induces an isomorphism in Top.

          Equations
          • TopCat.isoOfHomeo f = { hom := f, inv := f.symm, hom_inv_id := , inv_hom_id := }
          Instances For
            @[simp]
            theorem TopCat.homeoOfIso_symm_apply {X : TopCat} {Y : TopCat} (f : X Y) (a : Y) :
            (TopCat.homeoOfIso f).symm a = f.inv a
            @[simp]
            theorem TopCat.homeoOfIso_apply {X : TopCat} {Y : TopCat} (f : X Y) (a : X) :
            (TopCat.homeoOfIso f) a = f.hom a
            def TopCat.homeoOfIso {X : TopCat} {Y : TopCat} (f : X Y) :
            X ≃ₜ Y

            Any isomorphism in Top induces a homeomorphism.

            Equations
            • TopCat.homeoOfIso f = { toFun := f.hom, invFun := f.inv, left_inv := , right_inv := , continuous_toFun := , continuous_invFun := }
            Instances For
              @[simp]
              theorem TopCat.of_isoOfHomeo {X : TopCat} {Y : TopCat} (f : X ≃ₜ Y) :
              @[simp]
              @[deprecated TopCat.isOpenEmbedding_iff_comp_isIso]

              Alias of TopCat.isOpenEmbedding_iff_comp_isIso.

              @[deprecated TopCat.isOpenEmbedding_iff_isIso_comp]

              Alias of TopCat.isOpenEmbedding_iff_isIso_comp.