When is a local homeomorphism a covering map?

Question

if X and Y are Hausdorff spaces, $f:X \to Y$ is a local homeomorphism, X is compact, and Y is connected, is f a covering map?

It seems to be, and I almost have a proof, but I'm stuck at the very end of it:

I've already proved that f is surjective (using the connectedness), and that for each $y \in Y$, $f^{-1}(y)$ is finite. Because X is compact, there exists a finite open cover of X by $ \lbrace U_i \rbrace $ such that $f(U_i)$ is open and $ \left. f \right|_{U_i}:U_i \to f(U_i) $ is a homeomorphism. For each $y \in Y$, we choose the subset $ \lbrace U_{i_j} \rbrace $ such that $y \in U_{i_j}$, and then define $V = \bigcap_{j=1}^k f(U_{i_j})$, and $U'_j = U_{i_j} \bigcap f^{-1}(V)$.

... and this is were I got stuck. I really want to write that $f^{-1}(V) = \bigcup_{j=1}^k U'_j$ (more or less proving it's a covering map), but I can't justify that, and I actually think that it's not true. I think I might need an extra step, and to take an even smaller neighborhood of y, in order to make sure that extra sets from $ \lbrace U_i \rbrace $ didn't sneak into $f^{-1}(V)$.

Any help would be greatly appreciated as I've already spent several hours working on this problem.

Answer 1

For $y \in Y$, let $\{x_1, \dots, x_n\}= f^{-1}(y)$ (the $x_i$ all being different points). Choose pairwise disjoint neighborhoods $U_1, \dots, U_n$ of $x_1, \dots, x_n$, respectively (using the Hausdorff property).

By shrinking the $U_i$ further, we may assume that each one is mapped homeomorphically onto some neighborhood $V_i$ of $y$.

Now let $C = X \setminus (U_1 \cup \dots \cup U_n)$ and set $$V = (V_1 \cap \dots \cap V_n)\setminus f(C)$$

If I'm not mistaken this $V$ should be an evenly covered nbh of $y$.

Answer 2

cp. Fulton, Algebraic Topology, Proposition 19.3, p.266. He uses the compactness of X. But a problem in the John Lee's book Introduction to Topological Manifolds is this (Problem 11-9): Show that a proper local homeomorphism between connected, locally path-connected, compactly generated Hausdorff spaces is a covering map.