# Locally connected and compact Hausdorff space invariant of continuous mappings

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?

• $X$ compact and $Y$ hausdorf means the map is a quotient map, right? is local connectivity preserved under quotients? – citedcorpse Jun 8 '13 at 7:21
• 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 8 '13 at 7:35
• oh, are you looking for like first recorded references? – citedcorpse Jun 8 '13 at 7:37
• 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
• What's wrong with just sketching a short proof (if there is one)? – tomasz Jun 15 '13 at 0:43