Locally compact Hausdorff space with every open set NOT $\sigma$-compact.

Question

My major example of locally compact hausdorff spaces come from $\mathbb{R}^k$. Clearly $\mathbb{R}^k$ is sigma compact, but what are some examples of spaces that aren't $\sigma$-compact?

Answer 1

The disjoint union of uncountably many copies of $\mathbb{R}^k$ is a locally compact Hausdorff space which is not $\sigma$-compact.

I don't think you can find a locally compact Hausdorff space for which every (nonempty) open set fails to be $\sigma$-compact. This is because you will always be able to construct compact sets $K_i$ and open $U_i$ such that $$ K_1 \subseteq U_1 \subseteq K_2 \subseteq U_2 \subseteq K_3 \subseteq \ldots$$ and then $\bigcup U_i = \bigcup K_i$ is a $\sigma$ compact open set.

Answer 2

Another classic to the rescue: $\omega_1$ (the first uncountable ordinal in the order topology) as well as the long line, which are locally compact and not $\sigma$-compact. The latter is even locally homeomorphic to $\mathbb{R}$ and also connected.

A more trivial one: an uncountable discrete space will do, as the only compact sets are the finite ones. Generalising this: any disjoint sum $\oplus_{i \in I} X_i$ where all $X_i$ are (locally) compact and $I$ is uncountable.

Mike F already pointed out that in any locally compact space we will have open $\sigma$-compact subsets.