Heine-Borel Theorem/Metric Space
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< Heine-Borel Theorem(Redirected from Heine-Borel Theorem (General Case))
Contents |
Theorem
A metric space is compact iff it is both complete and totally bounded.
Proof
Necessary Condition
This follows directly from:
$\Box$
Sufficient Condition
This follows directly from:
- Complete and Totally Bounded Metric Space is Sequentially Compact
- Sequentially Compact Metric Space is Compact
$\blacksquare$
Axiom of Countable Choice
This theorem depends on the Axiom of Countable Choice, by way of Complete and Totally Bounded Metric Space is Sequentially Compact.
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.
Source of Name
This entry was named for Heinrich Eduard Heine and Émile Borel.
