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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. Then , with the induced metric, is complete: every Cauchy sequence in converges to a point of .
Together with compactness implies total boundedness, this yields the metric equivalence compact iff complete and totally bounded.
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
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: