TITLE: If every compact set is closed, then is the space Hausdorff?
QUESTION [17 upvotes]: I know that in a Hausdorff space, every compact set is closed.
However, is it true that if every compact set is closed, then the space is necessarily Hausdorff?

REPLY [26 votes]: I’m answering the question in the title. Let $X$ be an uncountable set, and let $\tau$ be the co-countable topology on $X$. The compact sets in $\langle X,\tau\rangle$ are precisely the finite sets, which are all closed, but $X$ is not Hausdorff.