TITLE: What can we say about a locally compact Hausdorff space whose every open subset is sigma compact?
QUESTION [8 upvotes]: 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

REPLY [2 votes]: I investigated this question as follows. Suppose that $X$ is a locally compact Hausdorff space whose every open subset is $\sigma$-compact. Then Alexandroff compactification $\alpha X$ should be a compact Hausdorff space whose every open subset is $\sigma$-compact. A space whose every open subset is $F_\sigma$ is called perfect. It is known that a compact Hausdorff space with a perfect square (even with a $G_\delta$-diagonal) is metrizable (see, for instance,  [Gru, 2.13]). I tried to prove or to find a theorem claimed that each Hausdorff compact perfect  space is metrizable and at last I found a paper [John], which may contain an example different from Double Arrow space.
And even we cannot say that $X$ is metrizable, we can say some things about it knowing that $X$ is a perfect compact space without a point. 
Suppose that $X$ has a $G_\delta$-diagonal. It seems the following.
There is a countable family $\mathcal F$ of closed subsets of $\alpha X$ such that
$\bigcup\mathcal F\cap (X\times X)=\Delta\backslash\{(\alpha,\alpha)\}$,
where $\Delta$ is a diagonal of the space $\alpha X$. Also there is a countable
family $\mathcal F_X$ of closed subsets of the
space $\alpha X$ such that $\bigcup\mathcal F_X=\alpha X\backslash\{\alpha\}$.
Put $\mathcal F'=\{F\cap F_1\times F_2:F\in\mathcal F, F_1,F_2\in\mathcal F_X\}\cup \{F\times\{\alpha\}\cup \{\alpha\}\times F:F\in\mathcal F_X\}$. Then
$\bigcup\mathcal F'=\Delta\backslash \{(\alpha,\alpha)\}$, and therefore the space $\alpha X$ is metrizable.
References 
[Gru] Gary Gruenhage, Generalized Metric Spaces, in K. Kunen and J. E. Vaughan, Handbook of set theoretic topology, Elsevier, 1984.
[John] Roy A. Johnson. A Compact Non-Metrizable Space Such That Every Closed Subset is a G-Delta. The American Mathematical Monthly, Vol. 77, No. 2, pp. 172–176.