In mathematics, particularly in the field of functional analysis and metric spaces, a subset \( S \) of a metric space \( (X, d) \) is said to be **totally bounded** if, for every \( \epsilon > 0 \), there exists a finite cover of \( S \) by open balls of radius \( \epsilon \).
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A metric space is totally bounded when, for every , it is covered by finitely many open balls of radius .
A metric space is totally bounded when, for every , finitely many open balls of radius cover it. Equivalently it has a finite -net at every positive scale.