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:

Proof
The relevant justifications are listed as follows:


 * Compact Space is Strongly Locally Compact.


 * Strongly Locally Compact Space is Locally Compact.


 * Compact Space is $\sigma$-Locally Compact.


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


 * $\sigma$-Compact Space is Lindelöf.