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