# Definition:Sigma-Locally Compact Space

## Definition

$T$ 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:

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.