# Definition:Weakly Sigma-Locally Compact Space

## Definition

Let $T = \struct {S, \tau}$ be a topological space.

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

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

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

- it is the union of countably many compact subspaces
- every point of $S$ is contained in a compact neighborhood.

## Also known as

Most sources, when defining this concept, refer to it as a **$\sigma$-locally compact space**.

However, it is more usual to find a **$\sigma$-locally compact space** defined as:

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 $\mathsf{Pr} \infty \mathsf{fWiki}$ 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**, reserving the term **$\sigma$-locally compact space** for the object based on the locally compact space.

## Also see

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

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Localized Compactness Properties