# Totally bounded

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Demote once content added - MOVE TO OWN PAGE, as there are probably other kinds of totally bounded

## Definition

A metric space [ilmath](X,d)[/ilmath] is *totally bounded* if^{[1]}:

- [ilmath]\forall\epsilon>0\exists n\in\mathbb{N}\exists\{B_i\}_{i=1}^n\text{ of}[/ilmath] open balls[ilmath]\text{ of radius }\epsilon[X\subseteq\cup_{i=1}^n B_i][/ilmath], that is:
- [ilmath]\forall\epsilon>0[/ilmath] there exists a finite collection of open balls, each of radius [ilmath]\epsilon[/ilmath], such that the family of balls cover [ilmath]X[/ilmath]