TITLE: Every compact subset must be closed?
QUESTION [5 upvotes]: This is an exercise from a topological book.
In $T_1$ space, every compact subset must be closed? For any two compact subset, their intersection must be compact?
Thanks for any help:)

REPLY [4 votes]: I would like to advertise the following result,   and some  consequences, just in case it is  not so well reported: 
Theorem: Let $B,C$ be compact subsets of $X,Y$ respectively and let $\mathcal W$ be a cover of $B \times C$ by sets open in $X \times Y$. Then $B,C$ have open neighbourhoods $U,V$ respectively such that $U \times V$ is covered by a finite number of sets of $\mathcal W$. 
This has a number of consequences from "the product of two compact spaces is compact" to "a compact subset of a Hausdorff space is closed". (Two others are listed in my book "Topology and groupoids", p. 86, but I forget where I first learned of the theorem, back in the 1960s!)