Totally bounded space
= Totally bounded space
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= Total boundedness
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= Totally bounded
{synonym}
A <metric space> is totally bounded when, for every $\varepsilon>0$, it is covered by finitely many open balls of radius $\varepsilon$.
A <metric space> is totally bounded when, for every $\varepsilon>0$, finitely many open balls of radius $\varepsilon$ cover it. Equivalently it has a finite $\varepsilon$-net at every positive scale.