Definition:Totally Bounded Metric Space/Definition 1

Definition
Let $M = \left({A, d}\right)$ be a metric space.

$M$ is totally bounded iff:
 * for every $\epsilon \in \R_{>0}$ there exists a finite $\epsilon$-net for $M$.

That is, $M$ is totally bounded iff:
 * for every $\epsilon \in \R_{>0}$ there exists a finite set of points $x_1, \ldots, x_n \in A$ such that:
 * $\displaystyle A = \bigcup_{i \mathop = 1}^n B_\epsilon \left({x_i}\right)$
 * where $B_\epsilon \left({x_i}\right)$ denotes the open $\epsilon$-ball of $x_i$.

That is: $M$ is totally bounded iff, given any $\epsilon \in \R_{>0}$, one can find a finite number of open $\epsilon$-balls which cover $A$.

Also known as
A totally bounded space is also referred to as a precompact space.

Also see

 * Equivalence of Definitions of Total Boundedness