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:
 * $\displaystyle M \subseteq \bigcup_{x \in S} N_\epsilon \left({x}\right)$

where $N_\epsilon \left({x}\right)$ is the $\epsilon$-neighborhood of $x$.

Finite Net
A finite $\epsilon$-net for $M$ is an $\epsilon$-net for $M$ which is finite.