Compactness Properties in T3 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 $T_3$ space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$.
Then the following sequence of implications holds:
Second-Countable | $\implies$ | Lindelöf | |||
$\Big\Downarrow$ | $\Big\Downarrow$ | ||||
$\Downarrow$ | Fully $T_4$ $\iff$ Paracompact | ||||
$\Big\Downarrow$ | $\Big\Downarrow$ | ||||
$T_5$ | $\implies$ | $T_4$ |
Proof
The justifications are listed as follows:
$\blacksquare$
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