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Let $f: X \rightarrow Y$ be a local homeomorphism with X, Y connected, locally path connected, Hausdorff and with X also compact. Then f is also a covering with finite fibers.

I know how to show that the fibers are finite. Given that f is a surjection, I know how to show that f is a covering map.

How do I show that f is surjective?

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up vote 5 down vote accepted

I think I have it. The image of X under f is open (use "local homeomorphism") and closed (use "compact" and "Hausdorff") in Y and since Y is connected this shows that the image is the whole of Y.

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