Compactly detected closed-set theorem in a locally compact Hausdorff space
= Compactly detected closed-set theorem in a locally compact Hausdorff space
If $X$ is locally compact Hausdorff and $A\cap K$ is closed in every compact subset $K$, then $A$ is closed. Around each point outside $A$, use a compact neighbourhood and the relative openness of its complement of $A\cap K$.