Definition:Weakly Sigma-Locally Compact Space

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

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

That is, $T$ is $\sigma$-locally compact :
 * it is the union of countably many compact sets.
 * every point of $S$ is contained in a compact neighborhood.

Alternative Definition
There is an alternative definition, based on local compactness which is often seen:

$T$ is $\sigma$-locally compact :
 * $T$ is $\sigma$-compact
 * $T$ is locally compact

That is, $T$ is $\sigma$-locally compact :
 * it is the union of countably many compact sets.
 * every point of $S$ has a local basis $\mathcal B$ such that all elements of $\mathcal B$ are compact.

However, this definition is equivalent to the main definition only when $T$ is a $T_2$ (Hausdorff) space.

Note that by Locally Compact implies Weakly Locally Compact, the one implies the other.

Also see

 * Sigma-Local Compactness in Hausdorff Space