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!
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2The 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
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yes, ok Thank you! – topsi Mar 13 '14 at 10:21
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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.
Jack Lee
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1I 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
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1You probably should. Off the top of my head, I don't know the answer to that one. – Jack Lee Mar 13 '14 at 11:06