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 metric space $M$ satsifies property $P_1$, then $M$ also satisfies property $P_2$.

Then the following sequence of implications holds:

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.


 * By definition, a $\sigma$-locally compact space is both locally compact and $\sigma$-compact.


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


 * Locally Compact Metric Space is Strongly Locally Compact.