Let p : Y -> X be an n-sheeted covering map, where X and Y are topological spaces. If X is compact, prove that Y is compact.

I realize that this seems like a very simple problem, but I want to stress the lack of assumptions on X and Y. For example, this is very easy to prove if we can assume that X and Y are metrizable, for sequential compactness is then equivalent to compactness and it is easy to lift sequential compactness from X to Y.

I asked three people in person this question and all of them immediately made the assumption that X and Y are metrizable, so I feel like I should put in this warning here that they are not.

Hausdorff? If so, then I can't see what goes wrong with the natural approach: take an open cover of Y, push it down to an open cover of $X$ (because $p$ is surjective it will be open) take a finite subcover downstairs and lift it up with multiplicity $n$ to a finite subcover upstairs. What have I missed? – Yemon Choi Jul 16 '10 at 4:31