Compactness Properties in Hausdorff Spaces

Theorem
Let $P_1$ and $P_2$ be compactness properties and let:
 * $P_1 \implies P_2$

mean:
 * If a $T_2$ (Hausdorff) space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$.

Then the following sequence of implications holds:

Proof
From $T_2$ Space is $T_1$ Space and $T_1$ Space is $T_0$ Space, we note that in a $T_2$ (Hausdorff) space:


 * A fully $T_4$ space is a fully normal space


 * A $T_4$ space is a normal space


 * A $T_{3 \frac 1 2}$ space is a Tychonoff space


 * A $T_3$ space is a regular space

all by definition.

From Hausdorff Space is Fully $T_4$ iff Paracompact we have that:
 * Fully $T_4$ $\iff$ Paracompact.

From that same result and Fully $T_4$ Space is $T_4$ Space:
 * Paracompact $\implies$ $T_4$

From Normal Space is Tychonoff Space it follows that:
 * $T_4$ $\implies$ $T_{3 \frac 1 2}$

From First-Countable Space is Sequentially Compact iff Countably Compact and Sequentially Compact Hausdorff Space is Regular it follows that:


 * First-Countable and Countably Compact $\implies$ $T_3$

The further justifications are listed as follows:


 * $\sigma$-Locally Compact Hausdorff Space is $T_4$


 * Weakly $\sigma$-Locally Compact implies Weakly Locally Compact by definition.


 * $T_{3 \frac 1 2}$ Space is $T_3$ Space