Definition:Totally Bounded Metric Space

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

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

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

where $$B \left({x_i, \epsilon}\right)$$ represents the ball of center $$x_i$$ and radius $$\epsilon$$.

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

An alternative term for totally bounded is precompact.

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

for all $$x \in S$$.

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