# Is there a name for a topological space $X$ in which very closed set is contained in a countable union of compact sets?

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?

Thank you!

• The Baire space clearly doesn't satisfy that. By your previous question. Remember that the Baire space is closed in itself. – Asaf Karagila Mar 12 '14 at 19:49
• yes, ok Thank you! – topsi Mar 13 '14 at 10:21

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.
• I see Thank you. As a matter of fact, I meant "Every closed set $A$ in $X$, where $A \neq X$ is contained in a countable union of compact sets".. Should I open another post for it? – topsi Mar 13 '14 at 10:23