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Mathlib.Topology.Defs.Induced

Induced and coinduced topologies #

In this file we define the induced and coinduced topologies, as well as topology inducing maps, topological embeddings, and quotient maps.

Main definitions #

def TopologicalSpace.induced {X : Type u_1} {Y : Type u_2} (f : XY) (t : TopologicalSpace Y) :

Given f : X → Y and a topology on Y, the induced topology on X is the collection of sets that are preimages of some open set in Y. This is the coarsest topology that makes f continuous.

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    def TopologicalSpace.coinduced {X : Type u_1} {Y : Type u_2} (f : XY) (t : TopologicalSpace X) :

    Given f : X → Y and a topology on X, the coinduced topology on Y is defined such that s : Set Y is open if the preimage of s is open. This is the finest topology that makes f continuous.

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      structure RestrictGenTopology {X : Type u_1} [tX : TopologicalSpace X] (S : Set (Set X)) :

      We say that restrictions of the topology on X to sets from a family S generates the original topology, if either of the following equivalent conditions hold:

      • a set which is relatively open in each s ∈ S is open;
      • a set which is relatively closed in each s ∈ S is closed;
      • for any topological space Y, a function f : X → Y is continuous provided that it is continuous on each s ∈ S.
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        theorem RestrictGenTopology.isOpen_of_forall_induced {X : Type u_1} [tX : TopologicalSpace X] {S : Set (Set X)} (self : RestrictGenTopology S) (u : Set X) :
        (sS, IsOpen (Subtype.val ⁻¹' u))IsOpen u
        theorem inducing_iff {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] (f : XY) :
        structure Inducing {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] (f : XY) :

        A function f : X → Y between topological spaces is inducing if the topology on X is induced by the topology on Y through f, meaning that a set s : Set X is open iff it is the preimage under f of some open set t : Set Y.

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          theorem Inducing.induced {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] {f : XY} (self : Inducing f) :

          The topology on the domain is equal to the induced topology.

          structure Embedding {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : XY) extends Inducing :

          A function between topological spaces is an embedding if it is injective, and for all s : Set X, s is open iff it is the preimage of an open set.

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            theorem Embedding.inj {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : XY} (self : Embedding f) :

            A topological embedding is injective.

            theorem openEmbedding_iff {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] (f : XY) :
            structure OpenEmbedding {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] (f : XY) extends Embedding :

            An open embedding is an embedding with open range.

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              theorem OpenEmbedding.isOpen_range {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] {f : XY} (self : OpenEmbedding f) :

              The range of an open embedding is an open set.

              theorem closedEmbedding_iff {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] (f : XY) :
              structure ClosedEmbedding {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] (f : XY) extends Embedding :

              A closed embedding is an embedding with closed image.

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                theorem ClosedEmbedding.isClosed_range {X : Type u_1} {Y : Type u_2} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] {f : XY} (self : ClosedEmbedding f) :

                The range of a closed embedding is a closed set.

                def QuotientMap {X : Type u_3} {Y : Type u_4} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] (f : XY) :

                A function between topological spaces is a quotient map if it is surjective, and for all s : Set Y, s is open iff its preimage is an open set.

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