# Definition:Totally Bounded Metric Space/Definition 2

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## 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.