# Locally compact $T_2$ but not $T_4$ spaces

Does there exist an example of locally compact Hausdorff (each two points can be separated by disjoint neighborhoods) but not normal Hausdorff (each two closed subsets can be separated by disjoint neighborhoods) topological space?

Even if there is, does it carry the Tietz extension property (any continuous function to $\mathbb{R}$ on closed subset can be extended to the whole space i.e. the sheaf of continuous $\mathbb{R}$-functions is soft)?

There are quite a few of them: an easy way to find them: use $\pi$-base which incorporates the spaces from "Counterexamples in Topology" by Steen and Seebach.

The deleted Tychonoff plank $(\omega_1 + 1) \times (\omega + 1) \setminus \{\omega_1, \omega)\}$ is the classical example.

$\omega_1 \times [0,1]^{[0,1]}$ is another relatively easy example, if you know the relevant theory: It's locally compact and $T_2$ as a product of two such spaces. It's not normal, because it contains $P= \omega_1 \times (\omega_1+1)$ as a closed subspace (by Tikhonoff's embedding theorem, as the weight of $\omega_1 + 1$ is at most $\mathfrak{c}$, it can be embedded into $[0,1]^{[0,1]}$ and compact subsets of $[0,1]^{[0,1]}$ are closed) and $P$ is not normal (classical; $X \times \beta(X)$ is normal iff $X$ is paracompact and $\beta\omega_1 = \omega_1 +1$, or use a pushing down lemma argument).

But a Mrówka $\Psi$-space based on a MAD family is also an example. See this blog for proofs and definitions. This is not normal as it's pseudocompact and $T_3$ but not countably compact. Or use Jones' lemma.

I also like the rational sequence topology, it's not normal by a Jones' lemma argument. And it's the simplest to describe (no ordinals or MAD families).

The Tietze extension property for $X$ is equivalent to $X$ being normal. So these spaces don't obey it.

• You could also quote the result that a space is hereditarily normal iff every open subspace is normal. So a locally compact normal space X that is not hereditarily normal has a non-normal open subspace Y. And Y is locally compact (because it is open in X, and X is both normal and locally compact). – DanielWainfleet May 14 '17 at 10:09
• @DanielWainfleet I don't follow. You mean that any compact $T_2$ but not $T_5$ space contains an example as asked (which is true)? – Henno Brandsma May 14 '17 at 10:13
• If $X$ is normal and locally compact but not $T_5$ then an open non-normal subspace $Y$ is an example of a locally compact non-normal Tychonoff ( and a fortiori) Hausdorff space. – DanielWainfleet May 14 '17 at 10:27
• @DanielWainfleet and all examples are of this form. It doesn't give us "new" or "concrete" examples. – Henno Brandsma May 14 '17 at 10:35