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A compact metric space is complete: every Cauchy sequence converges.
Compactness implies completeness: Let be a metric space and let be compact (with the induced metric). Then is a complete metric space: every Cauchy sequence in is a convergent sequence with limit in .
Together with compactness implies total boundedness, this yields the metric equivalence compact iff complete and totally bounded.
A metric space is a pair where is a set and is a metric on .
Every metric space determines a topological space via the metric-induced topology, whose basic neighborhoods are open balls.
A compact set is a subset of a topological space such that every open cover of contains a finite subcover: whenever is a family of open sets with , there exist indices such that
A complete metric space is a metric space in which every Cauchy sequence converges (as a convergent sequence) to a point of .
A Cauchy sequence in a metric space is a sequence such that for every there exists with
A convergent sequence in a topological space converges to a point if for every neighborhood of , there exists such that for all .
In a metric space , this is equivalent to . In a Hausdorff space, limits of convergent sequences are unique (see uniqueness of limits).
Compactness implies total boundedness: Let be a metric space and let be compact. Then is totally bounded: for every there exists a finite set such that
where denotes the open ball.
Compact iff complete and totally bounded: Let be a metric space. Then the following are equivalent: