I'm reading Guedj and Zeriahi's Degenerate Complex Monge-Ampère Equations Chapter 4 which talks about capacities. Specifically Corollary 4.13 claims that when $X$ is a locally compact Hausdorff $\sigma$-compact (this conditions is not stated in the book but definitely necessary I think) topological space every open subset $A \subseteq X$ is $\mathcal{K}$-analytic, see below for the definitions.
The book claims that any open subset will be $\sigma$-compact but this doesn't seem to be true. There can still be hope that this works though, since $\mathcal{K}$-analytic spaces don't need to be $\sigma$-compact in general. Is the book's claim true?
Definitions:
- A subset $A$ of a topological space $X$ is called $\mathcal{F}_{\sigma\delta}$ if it is a countable intersection of countable unions of closed sets.
- A topological space $X$ is a $\mathcal{K}_{\sigma\delta}$ space if it is homeomorphic to a $\mathcal{F}_{\sigma\delta}$-subset of a compact space $Y$.
- A topological space is $\mathcal{K}$-analytic if there is a $\mathcal{K}_{\sigma\delta}$-space that continuously surjects onto it.
Remark: If you add the extra assumption of $X$ being 2nd-countable (which is a property inherited to open subsets and implies $\sigma$-compactness for LCH spaces) then indeed every open subset is $\sigma$-compact.
