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.
[Math] Locally compact Hausdorff space that is not normal
examplesgn.general-topology
Best Answer
I think the Tychonoff plank serves as an example. It is obtained by taking the product of the two ordinals $\omega_1+1$ and $\omega+1$, each with the order topology, and removing the corner point $(\omega_1,\omega)$. The product is a compact Hausdorff space, so the plank, as an open subspace, is locally compact. But it is not normal, because the "edges" $\omega_1\times\{\omega\}$ and $\{\omega_1\}\times\omega$ don't have disjoint neighborhoods.