Compactness Properties in T3 Spaces

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Theorem

Let $P_1$ and $P_2$ be compactness properties and let:

$P_1 \implies P_2$

mean:

If a $T_3$ space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$.


Then the following sequence of implications holds:


Second-Countable $\implies$ Lindelöf
$\Big\Downarrow$ $\Big\Downarrow$
$\Downarrow$ Fully $T_4$ $\iff$ Paracompact
$\Big\Downarrow$ $\Big\Downarrow$
$T_5$ $\implies$ $T_4$


Proof

The justifications are listed as follows:

Second-Countable $T_3$ Space is $T_5$
Second-Countable Space is Lindelöf
Lindelöf $T_3$ Space is Paracompact
$T_3$ Space is Fully $T_4$ iff Paracompact
Fully $T_4$ Space is $T_4$ Space
$T_5$ Space is $T_4$ Space



Sources