characterization of compactly generated space

We give equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath conditions when a Hausdorff topological space is compactly generated. First we need a definition.

Definition. Let X be a T1 space and Y a topological spaceMathworldPlanetmath. We define Ck(X,Y) to be the set of functions f:XY such that for every compactPlanetmathPlanetmath, closed setPlanetmathPlanetmath KX the restrictionPlanetmathPlanetmathPlanetmath f|K is continuousPlanetmathPlanetmath.

Clearly, C(X,Y)Ck(X,Y) for such spaces X,Y. With this we have the following theorem.

Theorem. Let X be a Hausdorff space. Then the following conditions are equivalent.

i) X is compactly generated

ii) X carries the final topology generated by the family of inclusion mappings (iK:KX)KXcompact.

iii) For every topological space Y we have Ck(X,Y)=C(X,Y).

iv) X is an image of a locally compact space under a quotient mapping.

Remark. It follows easily from this that if there is a quotient mapping f:XY which maps a compactly generated space X onto a Hausdorff space Y then Y is compactly generated.

Title characterization of compactly generated space
Canonical name CharacterizationOfCompactlyGeneratedSpace
Date of creation 2013-03-22 19:10:04
Last modified on 2013-03-22 19:10:04
Owner karstenb (16623)
Last modified by karstenb (16623)
Numerical id 4
Author karstenb (16623)
Entry type Theorem
Classification msc 54E99
Defines C_k(X
Defines Y)