Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 16 down vote accepted

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 $y$.

share|cite|improve this answer
Seems right to me. f is a closed map (compactness+hausdoff), C is closed so f(C) is closed, so V is open, and $f^{-1}(y)\bigcap C = \emptyset$ so $y \in V$, and so it's indeed an open nbh of y. $f^{-1}(V) \subset U_1 \cup \ldots \cup U_n$ by definition, and so $f^{-1}(V) = (U_1 \cap f^{-1}(V)) \cup \ldots \cup (U_1 \cap f^{-1}(V))$ when $U_i \cap f^{-1}(V) \cong V$. – Or Sharir Jun 17 '11 at 23:23
Right, looks like you finished the proof. :) – Sam Jun 17 '11 at 23:31
@Sam why $f^{-1}(y)$ is finite? where is the compactness is nedded to argue that? – Un Chien Andalou Feb 28 '13 at 6:05
@CityOfGod: Since $f$ is a local homeomorphism, the set $f^{-1}(y)$ consists of isolated points. Since $X$ is compact, this can only be the case, if $f^{-1}(y)$ is finite. – Sam Feb 28 '13 at 6:13
thank you very much sam! – Un Chien Andalou Feb 28 '13 at 6:37

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.

share|cite|improve this answer

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.

share|cite|improve this answer
I think the second part of your answer would better fit as a separate question. If you ask it, just link here and say that it is related but not exactly the same. – t.b. Mar 31 '12 at 14:03

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.