Hello, 


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


nicolas



General criteria for exhaustion by compact sets
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.


