Definition:Sigma-Locally Compact Space

Definition
$T$ is $\sigma$-locally compact :
 * $T$ is $\sigma$-compact
 * $T$ is locally compact

That is, $T$ is $\sigma$-locally compact :
 * $T$ is the union of countably many compact subspaces
 * every point of $S$ has a local basis $\mathcal B$ such that all elements of $\mathcal B$ are compact.

Note that by Sigma-Locally Compact Space is Weakly Sigma-Locally Compact, the one implies the other.

Also defined as
Some sources define a $\sigma$-locally compact space as defines a weakly $\sigma$-locally compact space:
 * a weakly $\sigma$-locally compact space is one which is:
 * $\sigma$-compact
 * 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 are again frequently omitted in the literature.

It is the aim of to ensure that these subtle differences are documented, and the terms used consistently.

Hence has coined the term weakly $\sigma$-locally compact space for the latter concept.

Also see

 * Definition:Weakly $\sigma$-Locally Compact Space


 * Locally Compact Space is Weakly Locally Compact


 * Weakly Sigma-Locally Compact Hausdorff Space is Sigma-Locally Compact


 * Sigma-Local Compactness in Hausdorff Space