Hello,
I'm wondering if there exists general results insuring exhaustion by compact sets of a given topological space ?
nicolas
Hello, I'm wondering if there exists general results insuring exhaustion by compact sets of a given topological space ? nicolas 


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. 


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" 

