Compactness Properties in Hausdorff 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 $T_2$ (Hausdorff) space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$.


Then the following sequence of implications holds:


Fully $T_4$ $\iff$ Paracompact
$\Big\Downarrow$
Weakly $\sigma$-Locally Compact $\implies$ $T_4$
$\Big\Downarrow$ $\Big\Downarrow$
Weakly Locally Compact $\implies$ $T_{3 \frac 1 2}$
$\Big\Downarrow$
First-Countable and Countably Compact $\implies$ $T_3$


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



Sources