Definition:Net (Metric Space)

Definition
Let $M$ be a metric space.

Let $\epsilon > 0$.

An $\epsilon$-net for $M$ is a subset $S \subseteq M$ such that:
 * $\displaystyle M \subseteq \bigcup_{x \mathop \in S} B_\epsilon \left({x}\right)$

where $B_\epsilon \left({x}\right)$ is the open $\epsilon$-ball of $x$.

That is, it is a subset of $M$ such that the set of all open $\epsilon$-balls of each element of that set forms a cover for $M$.