Every open set is a countable union of compact sets. [In which kind of topological spaces this is true] Let $X$ be a Hausdorff topological space. 
Under which hypotheses on $X$ every open subset can be written as a union of countably many compact sets. I was wondering if $\sigma$-locally compact is a sufficient condition and, more generally, if there exists a characterization of this kind of topological spaces.
 A: That $X$ is $\sigma$-locally compact (by definition: both locally compact and $\sigma$-compact) is not enough, as shown by an example like $\beta \mathbb{N}$, the Čech-Stone compactification of the integers, which is moreover separable. I think $I^I$ is an example too, BTW. 
If every open set is $\sigma$-compact (hence Lindelöf) it means that $X$ must be itself $\sigma$-compact, hereditarily Lindelöf and also perfectly normal. And having those properties plus local compactness implies that every open set is $\sigma$-compact. But without that we have spaces like $\mathbb{Q}$ which is obeys what you want but is not locally compact. 
So there is a characterisation within the class of locally compact Hausdorff spaces (namely equivalent to being hereditarily Lindelöf), but in general I'm not so sure. There are hereditarily Lindelöf Hausdorff spaces that don't obey your property, like the space of irrationals $\mathbb{P}$ (as a subspace of $\mathbb{R}$ of course, homeomorphic to $\omega^\omega$). We need enough compact subsets, but a countable Hausdorff space like $\mathbb{Q}$ automatically satisfies it, without having that many compact subsets.
