Heine-Borel Theorem/Metric Space

Theorem

A metric space is compact if and only if it is both complete and totally bounded.

Proof

Necessary Condition

This follows directly from:

• Compact Metric Space is Complete
• Compact Metric Space is Totally Bounded

$\Box$

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.

Also see

• Heine-Borel Theorem
• Heine-Borel Theorem: Real Line

Sources

• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): compact