# Example of a sigma compact space which is not locally compact

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?

• 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/… – PhoemueX Jan 20 '20 at 12:45
• I did not know about that site, thank you! – Keen-ameteur Jan 20 '20 at 12:56

For the first part: $$\mathbb Q$$ with the usual topology is sigma compact but not locally compact.