Definition:Totally Bounded Metric Space

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

We say that $X$ is totally bounded iff for every $\epsilon > 0$ it has a finite $\epsilon$-net.

That is, iff for every $\epsilon > 0$ there exists a finite set of points $x_1, \ldots, x_n \in X$ such that:
 * $\displaystyle X = \bigcup_{i=1}^n N_\epsilon \left({x_i}\right)$

where $N_\epsilon \left({x_i}\right)$ represents the open $\epsilon$-ball neighborhood of $x_i$.

That is: $X$ is totally bounded iff, given any $\epsilon > 0$, one can find a finite number of $\epsilon$-balls which cover $X$.

An alternative term for totally bounded is precompact.

Alternative Definition
A metric space $\left({S, d}\right)$ is called totally bounded iff for every $\epsilon > 0$ there exist finitely many points $x_0, \dots, x_n \in S$ such that
 * $\displaystyle \inf_{0 \le i \le n} d \left({x_i, x}\right) \le \epsilon$

for all $x \in S$.

Equivalence of Definitions
These definitions are shown to be equivalent by Equivalence of Definitions of Total Boundedness.

Also see

 * Any totally bounded metric space is also bounded, but the converse is not true. (The simplest example is a countable set with the discrete metric.)


 * A metric space is compact if and only if it is complete and totally bounded.