Locally connected and compact Hausdorff space invariant of continuous mappings

Asked Jun 08 '13 at 07:09

Active Jun 14 '13 at 22:14

Viewed 248 times

I am looking for a reference (not proof) to the following theorem:

If $X$ is a compact and locally connected topological space, Y is a Hausdorff topological space, $f:X\to Y$ is continuous and $f(X)=Y$, then $Y$ is locally connected.

I found reference in Kuratowski, Topology II for the case where $X$ is metrizable. Any one knows about any reference to this case?

edited Jun 14 '13 at 22:14

Dan Rust

30,108

asked Jun 08 '13 at 07:09

Michal

$X$ compact and $Y$ hausdorf means the map is a quotient map, right? is local connectivity preserved under quotients? – citedcorpse Jun 08 '13 at 07:21
Yes, local connectivity is preserved under quotients. I have citation on that (after Engelking) Whyburn, On quasi-compact mappings, Duke Math J.19 (1952), 445-446. – Michal Jun 08 '13 at 07:35
oh, are you looking for like first recorded references? – citedcorpse Jun 08 '13 at 07:37
no, I need to give any reliable referential support for this fact, not proving it myself. I think your solution is good enough as it's really elementary to notice that $f$ is quotient. – Michal Jun 08 '13 at 07:48
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I added the reference request tag so that it's more clear you're looking for a published proof, and not just a proof. – Dan Rust Jun 14 '13 at 22:15
I once stumbled upon a proof for this fact in Willard, General Topology. – Stefan Hamcke Jun 14 '13 at 23:13
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What's wrong with just sketching a short proof (if there is one)? – tomasz Jun 15 '13 at 00:43
@StefanH. Indeed, Theorem 27.12. – Martin Jun 15 '13 at 07:12

Locally connected and compact Hausdorff space invariant of continuous mappings

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