Definition:Totally Bounded Metric Space/Definition 2

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

$M$ totally bounded iff:
 * for every $\epsilon \in \R_{>0}$ there exist finitely many points $x_0, \dots, x_n \in A$ such that:
 * $\displaystyle \inf_{0 \mathop \le i \mathop \le n} d \left({x_i, x}\right) \le \epsilon$
 * for all $x \in A$.

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

Also see

 * Equivalence of Definitions of Total Boundedness