I am using the following definitions (from Wikipedia):
- A space $X$ is locally compact if every $x \in X$ has a compact neighborhood;
- A space $X$ is compactly generated if a subset $A \subseteq X$ is closed if and only if $A \cap K \subseteq K$ is closed in every compact subset $K \subseteq X.$
According to Wikipedia's article on compactly generated spaces,
Every locally compact space is compactly generated.
I proved this claim for locally compact Hausdorff spaces, where compact subsets are also closed. My proof doesn't work in a non-Hausdorff case. I would like help with this case.