Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am in need of solution or tip for this question. I thank you.

Let $ p: \widetilde X \to X $ be a covering space with $ p^{-1}(x) $ finite and nonempty for all $ x \in X$. Show that $ \widetilde X$ is compact Hausdorff iff $ X$ is compact Hausdorff.

share|improve this question
2  
I changed your title to be more descriptive. Using a descriptive title will attract more relevant users to your question, as opposed to a generic "I need a solution". –  Ted Dec 16 '12 at 7:49
add comment

1 Answer

up vote 4 down vote accepted

(Giving just hints for now; but if they’re not clear after you’ve thought about them a bit, I’m happy to expand in more detail if you’d like.)

For Hausdorffness: given two distinct points $x, y \in \tilde{X}$, consider their images $p(x)$, $p(y)$. Are these also distinct? If so, then you can separate them with open sets in $X$; use these to construct open sets separating $x$ and $y$ in $\tilde{X}$. Otherwise, if $p(x) = p(y)$, then you can use the covering space condition to find open neighbourhoods separating $x$ and $y$.

For compactness: suppose given an open covering $\mathcal{U}$ of $\tilde{X}$. Call an open set $W \subseteq X$ “$\mathcal{U}$-good” if $p^{-1}(W)$ is a disjoint union of homeomorphic copies of $W$ (as in the definition of covering space), and moreover, each of these disjoint copies is a subset of some $U \in \mathcal{U}$. Now, show that the $\mathcal{U}$-good sets form an open cover of $X$. Thus some finite set of $\mathcal{U}$-good sets covers $X$; from these, build a finite subcover of $U$.


I found both of these using the principle of “think how you could make use of the assumptions”. For the second, for instance, we start with an open cover of $\tilde{X}$, and want a subcover. We know we’ll need somehow to use compactness of $X$, so we can expect the proof may well go something like: from $U$, construct some open cover of $X$; from a finite subcover of that, build a finite subcover of $\tilde{X}$. With the first cover of $X$ I tried (the images of the sets in $\mathcal{U}$), I couldn’t get the second step to work, so I thought: how can I refine the construction so that from the finite cover of $X$, I can get back to a finite subcover of $\mathcal{U}$?

share|improve this answer
add comment

Your Answer

 
discard

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.