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.