As was recommended for me in here I would like to share the following question with you: Is there a name for a topological space $X$ which satisfies the following condition:
Every closed subset $A\subsetneq X$ is contained in a countable union of compact sets
Does Baire space satisfy this condition?
Thank you!
A space $\mathbf{X}$ which is a countable union of compact sets is called a $K_\sigma$-space, and as the intersection of a closed set and a compact set is compact, any closed subspace of a $K_\sigma$-space is $K_\sigma$ again. Baire space $\mathbb{N}^\mathbb{N}$ is not $K_\sigma$.
I am a bit puzzled though by the requirement $A \subsetneq \mathbf{X}$ though (rather than $A \subseteq \mathbf{X})$. In case of Baire space this makes no difference, as e.g. $\mathbb{N}^\mathbb{N} \times \mathbb{N}^\mathbb{N} \cong \mathbb{N}^\mathbb{N}$.