Sequence of Implications of Local Compactness Properties

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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:

Compact $\implies$ Strongly Locally Compact
$\Big\Downarrow$ $\Big\Downarrow$
Weakly $\sigma$-Locally Compact $\implies$ Weakly Locally Compact $\Longleftarrow$ Locally Compact
$\Big\Downarrow$
$\sigma$-Compact
$\Big\Downarrow$
Lindelöf Space


Proof

The relevant justifications are listed as follows:

$\blacksquare$


Sources