Hello,
I'm wondering if there exists general results insuring exhaustion by compact sets of a given topological space ?
nicolas
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1$\begingroup$ The question is not clear at all. Every topological space is the union of compact sets, namely its finite subsets. $\endgroup$– Stefan GeschkeCommented Feb 18, 2013 at 13:26
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$\begingroup$ Hello Nicolas and welcome to MO. Your question does not seem like a research level question for which this forum is intended to, please see the FAQ. And as Stefan said, also your question is not very clear, please see the how to ask -section. However you might be looking for information about the concept of sigma-compactness, of which information can be found for example from Wikipedia. $\endgroup$– Rami LuistoCommented Feb 18, 2013 at 13:59
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$\begingroup$ Sorry for the fuzzy formulation of the question; i thought the terminology "exhaustion by compact set" was clear. Thanks for the hint about sigma-compactness, it is precisely what i meant. Therefore my question is "What general criteria are known to insure sigma-compactness ?" $\endgroup$– Loic RosnayCommented Feb 18, 2013 at 14:21
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$\begingroup$ Cf. this related question on Math.SE: math.stackexchange.com/questions/1360900/… $\endgroup$– Chill2MachtCommented Feb 11, 2022 at 23:13
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2 Answers
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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"
answered Feb 18, 2013 at 15:35