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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?

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  • $\begingroup$ $X$ compact and $Y$ hausdorf means the map is a quotient map, right? is local connectivity preserved under quotients? $\endgroup$ – citedcorpse Jun 8 '13 at 7:21
  • $\begingroup$ 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. $\endgroup$ – Michal Jun 8 '13 at 7:35
  • $\begingroup$ oh, are you looking for like first recorded references? $\endgroup$ – citedcorpse Jun 8 '13 at 7:37
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    $\begingroup$ I added the reference request tag so that it's more clear you're looking for a published proof, and not just a proof. $\endgroup$ – Dan Rust Jun 14 '13 at 22:15
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    $\begingroup$ What's wrong with just sketching a short proof (if there is one)? $\endgroup$ – tomasz Jun 15 '13 at 0:43

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