[Math] What can we say about a locally compact Hausdorff space whose every open subset is sigma compact

Question

What can we say about a locally compact Hausdorff space whose every open subset is sigma compact? Can we say that it is metrizable or second countable?

thanks

Answer 1

We can say neither. A classical counterexample (due to Alexandroff and Urysohn) is the so-called Double Arrow space (description here; I believe that in Counterexamples in Topology it is called the Weak Parallel Line topology), which is compact Hausdorff (it's ordered even) and perfectly normal space. The latter implies all open subsets are $F_\sigma$ and thus $\sigma$-compact in particular. But it is not second countable (and thus not metrisable) as e.g. the upper half contains (a homeomorphic copy of) the Sorgenfrey line.