characterization of compactly generated space

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

Definition. Let $X$ be a $T_{1}$ space and $Y$ a topological space. We define $C_{k}(X,Y)$ to be the set of functions $f\colon X\to Y$ such that for every compact, closed set $K\subset X$ the restriction $f_{|K}$ is continuous.

Clearly, $C(X,Y)\subset C_{k}(X,Y)$ 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 $(i_{K}\colon K\to X)_{K\subset X\ \text{compact}}$ .

iii) For every topological space $Y$ we have $C_{k}(X,Y)=C(X,Y)$ .

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 $f\colon X\to Y$ 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)