Is there a name for a topological space $X$ which satisfies the following condition:
Every closed set in $X$ is contained in a countable union of compact sets
Does Baire space satisfy this condition?
This property is equivalent to $\sigma$-compactness, which says that the space itself is a countable union of compact subsets. If your property holds for a space $X$, then since $X$ is a closed subspace of itself, it is contained in a countable union of compact subsets. Conversely, if $X$ is $\sigma$-compact, then your property holds because every subset is contained in a countable union of compact subsets.