Sequence of Implications of Metric Space 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:

Sequentially Compact		$\implies$		$\sigma$-Locally Compact		$\implies$		Locally Compact
$\Big\Updownarrow$				$\Big\Downarrow$				$\Big\Updownarrow$
Countably Compact				$\sigma$-Compact				Strongly Locally Compact
$\Big\Updownarrow$				$\Big\Downarrow$
Compact				Separable
$\Big\Updownarrow$				$\Big\Updownarrow$
Weakly Countably Compact				Lindelöf
				$\Big\Updownarrow$
				Second-Countable

Proof

The relevant justifications are listed as follows:

• Countably Compact Metric Space is Compact.
• Countably Compact Metric Space is Sequentially Compact.
• Sequentially Compact Metric Space is Compact.
• Weakly Countably Compact Metric Space is Countably Compact.

• Compact Space is Sigma-Locally Compact.

• $\sigma$-Locally Compact is both Locally Compact and $\sigma$-Compact by definition.

• $\sigma$-Compact Space is Lindelöf.
• Lindelöf Metric Space is Second-Countable.
• Separable Metric Space is Second-Countable.

• Locally Compact Metric Space is Strongly Locally Compact.

$\blacksquare$

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 5$