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.
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$\blacksquare$
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