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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!

Community
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topsi
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    Note that if the space itself can be written as a union of two closed sets (not even disjoint, just both nonempty) then you get $\sigma$-compactness again. – Asaf Karagila Mar 13 '14 at 11:20
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    Note that any closed $F \subseteq \omega^\omega$ with nonempty interior cannot be covered by countably many compact sets, by the same argument I used earlier. – user642796 Mar 13 '14 at 11:28

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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. $\mathbb{N}^\mathbb{N} \times \mathbb{N}^\mathbb{N} \cong \mathbb{N}^\mathbb{N}$.

Arno
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