Sequence of Implications of Metric Space Compactness Properties
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Theorem
Let $P_1$ and $P_2$ be compactness properties and let:
- $P_1 \implies P_2$
mean:
- If a metric space $M$ satsifies property $P_1$, then $M$ also satisfies property $P_2$.
Then the following sequence of implications holds:
Sequentially Compact | $\implies$ | Weakly $\sigma$-Locally Compact | $\implies$ | Weakly Locally Compact | |||||
$\Big\Updownarrow$ | $\Big\Downarrow$ | $\Big\Updownarrow$ | |||||||
Countably Compact | $\sigma$-Compact | Strongly Locally Compact | |||||||
$\Big\Updownarrow$ | $\Big\Downarrow$ | ||||||||
Compact | Separable | ||||||||
$\Big\Updownarrow$ | $\Big\Updownarrow$ | ||||||||
Weakly Countably Compact | Lindelöf | ||||||||
$\Big\Updownarrow$ | |||||||||
Second-Countable |
Proof
The relevant justifications are listed as follows:
- Metric Space is Compact iff Countably Compact.
- Metric Space is Countably Compact iff Sequentially Compact.
- Metric Space is Weakly Countably Compact iff Countably Compact.
- By definition, a weakly $\sigma$-locally compact space is both weakly locally compact and $\sigma$-compact.
- $\sigma$-Compact Space is Lindelöf.
- Metric Space is Lindelöf iff Second-Countable.
- Metric Space is Separable iff Second-Countable.
$\blacksquare$
Axiom of Countable Choice
This theorem depends on the Axiom of Countable Choice.
Although not as strong as the Axiom of Choice, the Axiom of Countable Choice is similarly independent of the Zermelo-Fraenkel axioms.
As such, mathematicians are generally convinced of its truth and believe that it should be generally accepted.
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Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces