Compactly generated space

Remarks

If $X$ is Hausdorff, the condition is equivalent to the classical one. A subset $A \subseteq X$ is closed whenever $A \cap C$ is closed in $C$ for every compact subset $C \subseteq X$.

Example: every metric space is compactly generated. If $A$ is not closed, there is a sequence $a_n \in A$ converging to some $x \notin A$. The set $K = \set{x} \cup \set{a_n : n \ge 1}$ is compact, and $A \cap K$ is not closed in $K$.

Non-example: an uncountable set $X$ with the cocountable topology. Its compact subsets are finite, so every test map $u$ has finite image. Finite subsets of $X$ are closed and discrete, so every $u^{-1}(A)$ is closed. But not every subset of $X$ is closed.