[Math] Example of a separable, locally-compact metric space which is not $\sigma$-compact

Question

I am looking for an example of a separable, locally-compact metric space which is not $\sigma$-compact.

At first I thought I could show that if a metric space is separable and locally-compact, then it must be $\sigma$-compact. But I could not show it and I haven't found any theorem that implies that. So I figured there must be an example of a space which is metrizable, separable, locally-compact but not $\sigma$-compact. Clearly such a space cannot be compact, so I am looking for a locally-compact non-compact metric space.

Can someone give me such an example?

Answer 1

Such a space does not exist.

Note that every second-countable locally compact Hausdorff space $X$ is σ-compact.

• Proof. The family $\mathcal B$ of all open $U \subseteq X$ with compact closure forms a base for $X$ by local compactness. By second-countability some countable subfamily $\mathcal B_0$ of $\mathcal B$ is itself a base for $X$. But then $X = \bigcup \{ \overline U : U \in \mathcal B_0 \}$, a countable union of compact sets, so $X$ is σ-compact.

Now recall that separability is equivalent to second-countability in metric spaces.