I'm trying to prove:
A surjective local homeomorphism $p : E \to B$ is a finite covering map if $E$ is compact Hausdorff and $B$ is Hausdorff.
I assumed $p$ is not a finite covering map, so that there is some point $b_0$ in $B$ such that $p^{-1}(b_0)$ is infinite. then I'm going to construct an open cover for $E$, somehow induced by $p^{-1}(b_0)$, which has no finite subcover.
But I have no more idea to construct such open cover. and I'm not sure this is the right way. also, I wonder why the Hausdorff condition is needed. How can I proceed here?