Epsilon-net

An epsilon-net for a subset $A\subseteq X$ in a metric space $(X,d)$ is a subset $F\subseteq X$ such that for every $a\in A$ there exists $f\in F$ with

Equivalently, $A\subseteq \bigcup_{f\in F} B(f,\varepsilon)$, where $B(f,\varepsilon)$ is an open ball .

Finite epsilon-nets are the basic finiteness objects behind total boundedness .

Examples:

• For $A=[0,1]\subseteq\mathbb{R}$ and $\varepsilon>0$, a finite set of equally spaced points $\{0,\varepsilon,2\varepsilon,\dots,N\varepsilon\}$ with $N\varepsilon\ge 1$ is an epsilon-net for $A$.
• On an infinite set with the discrete metric, any epsilon-net with $\varepsilon<1$ must contain every point of $A$, so $A$ has no finite epsilon-net unless it is finite.