Compact metric space

A compact metric space is a metric space that is compact. Equivalently, it is complete and totally bounded.

A metric space is compact exactly when it is both complete and totally bounded. For the nontrivial direction, successively choose a point from a nested sequence of finite $2^{-n}$-nets; a diagonal construction gives a Cauchy subsequence of every sequence, and completeness gives its limit.

A metric space is totally bounded when, for every $\epsilon >0$, it is covered by finitely many open balls of radius $\epsilon$.

A metric space is totally bounded when, for every $\epsilon >0$, finitely many open balls of radius $\epsilon$ cover it. Equivalently it has a finite $\epsilon$-net at every positive scale.