\begin{definition}[Definition:Sigma-Locally Compact Space]
$T$ is '''$\sigma$-locally compact''' {{iff}}:
:$T$ is $\sigma$-compact
:$T$ is locally compact
That is, $T$ is '''$\sigma$-locally compact''' {{iff}}:
:$T$ is the union of countably many compact subspaces
:every point of $S$ has a local basis $\BB$ such that all elements of $\BB$ are  compact.
\end{definition}