# 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.

$\blacksquare$

## Also see

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Localized Compactness Properties