Weakly Sigma-Locally Compact iff Weakly Locally Compact and Lindelöf

Theorem
Let $T = \left({X, \vartheta}\right)$ be a topological space.

Then $T$ is locally compact and Lindelöf iff $T$ is $\sigma$-locally compact.

Proof
Let $T = \left({X, \vartheta}\right)$ be $\sigma$-locally compact.

Then by definition:


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

From Sigma-Compact Space is Lindelöf it follows directly that:
 * $T$ is Lindelöf
 * $T$ is locally compact.

Now let $T = \left({X, \vartheta}\right)$ be locally compact and Lindelöf.

By definition:
 * $T$ is locally compact iff every point of $X$ is contained in a compact neighborhood.
 * $T$ is Lindelöf iff every open cover of $X$ has a countable subcover.

Thus the interiors of the compact neighborhoods are an open cover of $X$.

As $T$ is Lindelöf, this cover has a countable subcover.

Thus $T$ is the union of countably many compact sets.

That is, $T$ is $\sigma$-compact, and so $\sigma$-locally compact.