Definition:Totally Bounded Metric Space/Definition 2

Definition
A metric space $M = \struct {A, d}$ is totally bounded :
 * for every $\epsilon \in \R_{>0}$ there exist finitely many points $x_0, \dots, x_n \in A$ such that:
 * $\displaystyle \inf_{0 \mathop \le i \mathop \le n} \map d {x_i, x} \le \epsilon$
 * for all $x \in A$.

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

Also see

 * Equivalence of Definitions of Totally Bounded Metric Space