Definition:Weakly Sigma-Locally Compact Space

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Definition

Let $T = \left({S, \tau}\right)$ 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:

$\sigma$-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 $\mathsf{Pr} \infty \mathsf{fWiki}$ 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