# Definition:Totally Bounded Metric Space/Definition 2

## Definition

A metric space $M = \struct {A, d}$ is totally bounded if and only if:

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

• Results about totally bounded metric spaces can be found here.