Sequence of Implications of Paracompactness Properties
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Theorem
Let $P_1$ and $P_2$ be paracompactness properties and let:
- $P_1 \implies P_2$
mean:
- If a topological space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$.
Then the following sequence of implications holds:
Compact | $\implies$ | Countably Compact | |||||||
$\Big\Downarrow$ | $\Big\Downarrow$ | ||||||||
Fully Normal | $\implies$ | Paracompact | $\implies$ | Countably Paracompact | |||||
$\Big\Downarrow$ | $\Big\Downarrow$ | $\Big\Downarrow$ | |||||||
Fully $T_4$ | Metacompact | $\implies$ | Countably Metacompact | ||||||
$\Big\Downarrow$ | |||||||||
$T_4$ |
Proof
The relevant justifications are listed as follows:
- Fully Normal Space is Fully $T_4$ 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: Paracompactness