Definition:Sigma-Locally Compact Space

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$T$ is $\sigma$-locally compact if and only if:

$T$ is $\sigma$-compact
$T$ is locally compact

That is, $T$ is $\sigma$-locally compact if and only if:

$T$ is the union of countably many compact subspaces
every point of $S$ has a local basis $\BB$ such that all elements of $\BB$ are compact.

Also defined as

Some sources define a $\sigma$-locally compact space as $\mathsf{Pr} \infty \mathsf{fWiki}$ defines a weakly $\sigma$-locally compact space:

a weakly $\sigma$-locally compact space is one which is:
weakly locally compact.

There appears to be no appreciation anywhere on Internet-accessible sources that there are two such differing definitions, or that they define different concepts.

The difference arises from the frequent confusion between our definitions of a weakly locally compact space and a locally compact space, the difference between which is again frequently omitted in the literature.

It is the aim of $\mathsf{Pr} \infty \mathsf{fWiki}$ to ensure that these subtle differences are documented, and the terms used consistently.

Hence $\mathsf{Pr} \infty \mathsf{fWiki}$ has coined the term weakly $\sigma$-locally compact space for the latter concept.

Also see

  • Results about $\sigma$-locally compact spaces can be found here.