Sequence of Implications of Global 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:

Proof

The relevant justifications are listed as follows:

• Sequentially Compact Space is Countably Compact.

• Compact Space is Countably Compact.

• Compact Space is $\sigma$-Compact.

• $\sigma$-Compact Space is Lindelöf.

• Countably Compact Space is Weakly Countably Compact.

• Countably Compact Space is Pseudocompact.

$\blacksquare$

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: Global Compactness Properties