Definition:Weakly Sigma-Locally Compact Space

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

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

That is, $T$ is $\sigma$-locally compact iff:
 * 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.

Alternative Definition
There is an alternative definition, based on the alternative definition of locally compact which is often seen:

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

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

Note that if there exists a local basis of compact sets, there is a compact neighborhood (any of the sets that form the local basis).

Thus the main definition trivially implies the alternative one.

Also see

 * Sigma-Local Compactness in Hausdorff Space