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 $(X,d)$ is totally bounded if:

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