TITLE: A subset of a metric space is closed iff its intersection with every compact subset is closed
QUESTION [7 upvotes]: I want to show that a subset of a metric space $X$ is closed iff its intersection with every compact subset of $X$  is closed

REPLY [3 votes]: If $F\subset{X}$ is closed and $C\subset{X}$ is compact, then $C$ is also closed (compact in Hausdorff is closed), and hence the intersection $F\cap{C}$ is closed as well.