Hello,
I'm wondering if there exists general results insuring exhaustion by compact sets of a given topological space ?
nicolas
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David White's result may be generalized to necessary and sufficient conditions for when a locally compact space is $\sigma$-compact. A locally compact space is $\sigma$-compact if and only if it is Lindelof. In particular, since every second countable space is Lindelof, every locally compact and second countable space is $\sigma$-compact. Also, the paracompact locally compact spaces are precisely the free unions of $\sigma$-compact locally compact spaces. In particular, every connected locally compact paracompact space is $\sigma$-compact. More generally, a locally compact space is $\sigma$-compact if and only if it is paracompact and cannot be partitioned into uncountably many clopen sets. See the topology book by Dugundji for proofs of these facts. |
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On page 289 of Munkres, Exercise 10 proves that if $X$ is locally compact and second countable then $X$ is $\sigma$-compact. Hopefully this is good enough for whatever application you have in mind. Incidentally, the wikipedia page on $\sigma$-compactness is pretty decent and there's a link to it from the page on "Exhaustion by compact sets" |
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