Definition:Totally Bounded Metric Space/Definition 1

Definition
A metric space $M = \struct {A, d}$ is totally bounded :
 * for every $\epsilon \in \R_{>0}$ there exists a finite $\epsilon$-net for $M$.

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

That is: $M$ is totally bounded, 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 metric space is also referred to as a precompact space.

Also see

 * Equivalence of Definitions of Totally Bounded Metric Space