Definition:Weakly Sigma-Locally Compact Space

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

Then $T$ is weakly $\sigma$-locally compact :
 * $T$ is $\sigma$-compact
 * $T$ is weakly locally compact.

That is, $T$ is weakly $\sigma$-locally compact :
 * 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:
 * $\sigma$-compact
 * 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, reserving the term $\sigma$-locally compact space for the object based on the locally compact space.

Also see

 * Definition:Sigma-Locally Compact Space


 * Locally Compact Space is Weakly Locally Compact,


 * $\sigma$-Locally Compact Space is Weakly $\sigma$-Locally Compact


 * Weakly $\sigma$-Locally Compact Hausdorff Space is $\sigma$-Locally Compact