Totally bounded set

A totally bounded set is a subset $A\subseteq X$ of a metric space $(X,d)$ such that for every $\varepsilon>0$ there exist points $x_1,\dots,x_n\in X$ with

where $B(x_k,\varepsilon)$ is the open ball of radius $\varepsilon$ around $x_k$.

Total boundedness is equivalent to the existence of finite epsilon-nets at every scale, and (together with completeness ) it characterizes compactness in metric spaces.

Examples:

• In $\mathbb{R}$ with the usual metric, the interval $[0,1]$ is totally bounded.
• The set of integers $\mathbb{Z}\subseteq\mathbb{R}$ (usual metric) is not totally bounded.