Timeline for Compactness and Covering Spaces
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Jul 16, 2010 at 15:48 | comment | added | BMann | I think a fix of the proof above goes as follows: Let $U_\gamma$ be a cover of $Y$. Choose a refinement $V_\alpha$ of this cover so that each $V_\alpha$ is small enough to be homeomorphic to its image under the covering map. It suffices to show that $V_\alpha$ has a finite subcover. Since the sets $p(V_\alpha)$ form an open cover of $X$, they have a finite subcover, $p(V_\alpha)_\beta$, each of which has $n$ lifts. The lifts of this subcover provide a finite subcover of $Y$. | |
| Jul 16, 2010 at 11:26 | answer | added | Georges Elencwajg | timeline score: 1 | |
| Jul 16, 2010 at 5:13 | answer | added | Greg Kuperberg | timeline score: 12 | |
| Jul 16, 2010 at 5:12 | comment | added | Yemon Choi | @Tyler: good point, I was being over hasty. Seeing as my topology is rusty: by an n-sheeted covering, do we mean that (a) p is a quotient map of topological spaces; (b) each point $x\in X$ has an open neighbourhood $U$ suchthat $p^{-1}U)$ is the disjoint union of $n$ open sets, each of which is mapped homeomorphically onto $U$? | |
| Jul 16, 2010 at 4:59 | answer | added | Dick Palais | timeline score: 4 | |
| Jul 16, 2010 at 4:57 | comment | added | Dylan Moreland | If I recall correctly, you don't need Hausdorff. | |
| Jul 16, 2010 at 4:50 | comment | added | Tyler Lawson | @Yemon: What does "lift it up with multiplicity n" mean? How do you choose sets from the original cover to cover your new cover? (And I apologize for that sentence.) | |
| Jul 16, 2010 at 4:31 | comment | added | Yemon Choi | Are you assuming X and Y are also Hausdorff? If so, then I can't see what goes wrong with the natural approach: take an open cover of Y, push it down to an open cover of $X$ (because $p$ is surjective it will be open) take a finite subcover downstairs and lift it up with multiplicity $n$ to a finite subcover upstairs. What have I missed? | |
| Jul 16, 2010 at 4:27 | history | asked | Eric Haengel | CC BY-SA 2.5 |