# Totally Bounded Space - Definition For A Metric Space

Definition For A Metric Space

A metric space is totally bounded if and only if for every real number, there exists a finite collection of open balls in of radius whose union contains . Equivalently, the metric space is totally bounded if and only if for every, there exists a finite cover such that the radius of each element of the cover is at most . This is equivalent to the existence of a finite ε-net.

Each totally bounded space is bounded (as the union of finitely many bounded sets is bounded), but the converse is not true in general. For example, an infinite set equipped with the discrete metric is bounded but not totally bounded.

If M is Euclidean space and d is the Euclidean distance, then a subset (with the subspace topology) is totally bounded if and only if it is bounded.

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