TITLE: Locally compact Hausdorff space that is not normal
QUESTION [21 upvotes]: What is a good example of a locally compact Hausdorff space that is not normal? It seems to be well-known that not all locally compact Hausdorff spaces are normal (and only a weaker version of Urysohn's Lemma holds in general in the locally compact Hausdorff case). However, I can't seem to think of any examples that demonstrate this, and I have tried all of the "standard" topological counterexamples such as the long line, etc.

REPLY [5 votes]: Let me give a construction that gives many locally compact spaces that are not normal. If $X$ is a completely regular space and $C$ is a compactification of $X$, then $X$ is paracompact if and only if $X\times C$ is normal. Therefore if $X$ is locally compact but not paracompact, then $X\times C$ is locally compact but not normal. Furthermore, the paracompact locally compact spaces are precisely the spaces that can be partitioned into a collection of $\sigma$-compact clopen sets (recall that a space is $\sigma$-compact if and only if it can be written as a countable union of compact sets). In particular, if $X$ is a connected locally compact space that is not $\sigma$-compact and $C$ is a compactification of $X$, then $X\times C$ is not normal.