# Compactness Properties in T3 Spaces

## 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