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A search through this site yielded plenty of examples where a space is locally compact but not sigma compact, but not the other way around. The fact that there is a term of sigma locally compact suggests that there are counter-examples, but I am having trouble finding one. I would appreciate any such counter-examples.

Also I was wondering about the validity if the following condition would imply local compactness: Let $X$ be a topological space such that $X=\cup_{n=1}^\infty K_n$ where $K_n$ is compact. If $\{K_n \}_{n=1}^\infty$ is locally finite, then $X$ is locally compact. This argument seems to work even if we have a general locally finite compact cover (every set is compact) and not just countable. Under such conditions can we conclude local compactness?

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    $\begingroup$ Just as a hint for the future: There is a search engine allowing to search for topological spaces via their properties. See for instance topology.jdabbs.com/… $\endgroup$ – PhoemueX Jan 20 '20 at 12:45
  • $\begingroup$ I did not know about that site, thank you! $\endgroup$ – Keen-ameteur Jan 20 '20 at 12:56
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For the first part: $\mathbb Q$ with the usual topology is sigma compact but not locally compact.

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