Weakly Sigma-Locally Compact iff Weakly Locally Compact and Lindelöf

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Theorem

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


The following are equivalent:

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


Proof

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.

$\Box$


2 implies 1

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

By definition:

$T$ is weakly locally compact if and only if every point of $S$ is contained in a compact neighborhood.
$T$ is Lindelöf if and only if 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.

$\blacksquare$


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