Weakly Sigma-Locally Compact iff Weakly Locally Compact and Lindelöf

Theorem
Let $T = \struct {S, \tau}$ be a topological space.


 * $(1): \quad T$ is weakly $\sigma$-locally compact
 * $(2): \quad T$ is weakly locally compact and Lindelöf

1 implies 2
Let $T = \struct {S, \tau}$ be weakly $\sigma$-locally compact.

Then by definition:


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

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

2 implies 1
Now let $T = \struct {S, \tau}$ be weakly locally compact and Lindelöf.

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

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

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

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

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

Also see

 * Sequence of Implications of Local Compactness Properties