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A topological space is called $\sigma$-compact if it is a countable union of compact sets.

Us12
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  • Please give the definition of paracompactness and state the question explicitly. I presume that you are wanting a proof of the statement in the title? – user558840 Oct 02 '20 at 22:37
  • @user558840: Paracompactness is a standard, basic topological property; there is no need to define it in the question. – Brian M. Scott Oct 02 '20 at 22:38
  • locally compact is overkill: regular is enough. Without regularity there are plenty of counterexamples, see $\pi$-base e.g. – Henno Brandsma Oct 02 '20 at 22:53
  • @Brian M. Scott My question on paracompactness was partly for pedagogic reasons and partly to encourage a new contributor by showing an interest in the question. Also I seldom use paracompactness and I struggle to remember the definition (and may be typical of many users in this respect). – user558840 Oct 03 '20 at 07:29
  • Please write the question in the question body itself. – Toby Mak Oct 04 '20 at 03:22

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HINT: A $\sigma$-compact space is easily shown to be Lindelöf, and a locally compact Hausdorff space is regular. Finally, regular Lindelöf spaces are paracompact.

Brian M. Scott
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