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 $ \{ U_i \}$ such that $f(U_i)$ is open and $f |_{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

Here is a complete solution, said slightly differently than, but in the same spirit as, Sam's solution.

1. Show that $f$ is surjective. We use the fact that $Y$ is connected and Hausdorff. Local homeomorphisms are open, so $U=f(X)$ is an open subset of $Y$. Since $X$ is compact, $f(X)$ is compact, and $Y$ Hausdorff implies that compact subsets are closed. So, $V=Y\setminus f(X)$ is also open. If $f$ were not surjective, then $V\neq \emptyset$, and $U,V$ would be separating sets for $Y$, contradicting connectedness of $Y$. We conclude that $f$ is surjective.

2. For each $y\in Y$, $f^{-1}(y)$ is finite. Again using $Y$ Hausdorff, $\{y\}$ is closed, so $f^{-1}(y)$ is a closed subset of the compact space $X$, hence compact. For each $x\in f^{-1}(y)$, let $U_x$ be a neighborhood of $x$ where $f$ restricts to a homeomorphism. Such neighborhoods exist by the assumption that $f$ is a local homeomorphism. Then $\{U_x : x\in f^{-1}(y)\}$ is an open cover of $f^{-1}(y)$, hence has a finite subcover which we label $\{U_i\}_{i=1}^n$. The map $f$ is injective on each $U_i$, thus only contains one pre-image of $y$. Hence $y$ has finitely many pre-images in $X$.

3. Get an evenly covered neighborhood of $y$. Keeping the cover $\{U_i\}$ from the previous step, $V = \bigcap_{i=1}^n{f(U_i)}$ is an open neighborhood of $y$. Then $\{f^{-1}(V)\cap U_i\}$ is a disjoint collection of open neighborhoods, each homeomorphic to $V$ under $f$ since the restriction of a homeomorphism to a subspace is a homeomorphism. Thus, $V$ is an evenly covered neighborhood of $y$.

Therefore, $f$ is a covering map.

Answer 3

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.