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:

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