Sequence of Implications of Local Compactness Properties

Theorem

Let $P_1$ and $P_2$ be compactness properties and let:

$P_1 \implies P_2$

mean:

If a topological space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$.

Then the following sequence of implications holds:

Compact		$\implies$		Strongly Locally Compact
$\Big\Downarrow$				$\Big\Downarrow$
Weakly $\sigma$-Locally Compact		$\implies$		Weakly Locally Compact		$\Longleftarrow$		Locally Compact
$\Big\Downarrow$
$\sigma$-Compact
$\Big\Downarrow$
Lindelöf Space

Proof

The relevant justifications are listed as follows:

• Compact Space is Strongly Locally Compact.

• Strongly Locally Compact Space is Weakly Locally Compact.

• Locally Compact Space is Weakly Locally Compact.

• Compact Space is Weakly $\sigma$-Locally Compact.

• A weakly $\sigma$-locally compact is both weakly locally compact and $\sigma$-compact by definition.

• $\sigma$-Compact Space is Lindelöf.

$\blacksquare$

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