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 $f:X\to Y$ 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 ${(i_K:K\to X)}_{K\subset X\text{compact}}$.

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 $f:X\to Y$ which maps a compactly generated space $X$ onto a Hausdorff space $Y$ then $Y$ is compactly generated.