# 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 $T_4$ space is a normal 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:

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

$\blacksquare$

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Compactness Properties and the $T_i$ Axioms