Compact Space is Weakly Sigma-Locally Compact

Theorem

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

Then $T$ is a weakly $\sigma$-locally compact space.

Proof

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

We have that:

Compact Space is Sigma-Compact

Compact Space is Weakly Locally Compact

Hence by definition $T$ is weakly $\sigma$-locally compact space.

$\blacksquare$

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