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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?

Juno Seo
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    Don't directly construct a cover. Use that any infinite subset of a compact space has an accumulation point. If infinite, $p^{-1}(b_0)$ has an accumulation point, $c$, but then $p$ can't be locally injective near $c$, so this is a contradiction. – jgon Sep 17 '19 at 17:53

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