Compactly Generated

Subjects: Topology
Links: Compactness-Type Properties, Compactness

Def: A toplogical space is said to be compactly generated if it has the following property: if $A$ is any subset of $X$ whose intersection with each compact subset $K\subseteq X$ is closed in $K$, then $A$ is closed in $X$.

Lemma: First countable spaces and locally compact spaces are compactly generated.