identification topology


Let f be a function from a topological spaceMathworldPlanetmath X to a set Y. The identification topology on Y with respect to f is defined to be the finest topology on Y such that the function f is continuousMathworldPlanetmathPlanetmath.

Theorem 1.

Let f:X→Y be defined as above. The following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    𝒯 is the identification topology on Y.

  2. 2.

    U⊆Y is open under 𝒯 iff f-1⁢(U) is open in X.

Proof.

(1.⇒2.) If U is open under 𝒯, then f-1⁢(U) is open in X as f is continuous under 𝒯. Now, suppose U is not open under 𝒯 and f-1⁢(U) is open in X. Let ℬ be a subbase of 𝒯. Define ℬ′:=ℬ∪{U}. Then the topology 𝒯′ generated by ℬ′ is a strictly finer topology than 𝒯 making f continuous, a contradictionMathworldPlanetmathPlanetmath.

(2.⇒1.) Let 𝒯 be the topology defined by 2. Then f is continuous. Suppose 𝒯′ is another topology on Y making f continuous. Let U be 𝒯′-open. Then f-1⁢(U) is open in X, which implies U is 𝒯-open. Thus 𝒯′⊆𝒯 and 𝒯 is finer than 𝒯′. ∎

Remarks.

  • •

    𝒮={f⁢(V)∣V⁢ is open in ⁢X} is a subbasis for f⁢(X), using the subspace topology on f⁢(X) of the identification topology on Y.

  • •

    More generally, let Xi be a family of topological spaces and fi:Xi→Y be a family of functions from Xi into Y. The identification topology on Y with respect to the family fi is the finest topology on Y making each fi a continuous function. In literature, this topology is also called the final topology.

  • •

    The dual concept of this is the initial topology.

  • •

    Let f:X→Y be defined as above. Define binary relationMathworldPlanetmath ∼ on X so that x∼y iff f⁢(x)=f⁢(y). Clearly ∼ is an equivalence relation. Let X* be the quotient X/∼. Then f induces an injective map f*:X*→Y given by f*⁢([x])=f⁢(x). Let Y be given the identification topology and X* the quotient topology (induced by ∼), then f* is continuous. Indeed, for if V⊆Y is open, then f-1⁢(V) is open in X. But then f-1⁢(V)=⋃f*-1⁢(V), which implies f*-1⁢(V) is open in X*. Furthermore, the argumentMathworldPlanetmath is reversible, so that if U is open in X*, then so is f*⁢(U) open in Y. Finally, if f is surjectivePlanetmathPlanetmath, so is f*, so that f* is a homeomorphism.

Title identification topology
Canonical name IdentificationTopology
Date of creation 2013-03-22 14:41:26
Last modified on 2013-03-22 14:41:26
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 11
Author rspuzio (6075)
Entry type Definition
Classification msc 54A99
Synonym final topology
Related topic InitialTopology