# Locally Compact Hausdorff Space That is Not Normal

Someone told me that locally compact Hausdorff spaces (unlike compact ones) need not be normal. Can one give me please such an example? Thank you.

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Spacebook, a database-driven version of Steen and Seebach's Counterexamples in Topology, lists the following locally compact Hausdorff (i.e. T2) spaces that are not normal.

Deleted Tychonoff Plank

Open Uncountable Ordinal Crossed with Uncountable Cartesian Product of Unit Interval

Rational Sequence Topology

Thomas’s Plank

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The space $X$ described in this answer is a fairly simple separable, locally compact Tikhonov space that is not normal, provided that the set $\mathscr{D}$ is chosen to have cardinality $2^\omega=\mathfrak c$, as described at the end of the answer. The non-normality follows from Jones’s lemma, which implies that a separable, normal space cannot contain a closed discrete subset of of cardinality $2^\omega$.