Totally bounded set

Let $(X,d)$ be a metric space and let $E\subseteq X$. The set $E$ is totally bounded if for every $\varepsilon>0$ there exist points $x_1,\dots,x_N\in X$ such that

(see open ball ).

Equivalently: for every $\varepsilon>0$, $E$ has a finite $\varepsilon$-net, i.e. a finite subset $F\subseteq X$ such that every point of $E$ is within distance $<\varepsilon$ of some $f\in F$.

Total boundedness is a “precompactness” condition: in a complete metric space , $E$ is relatively compact (its closure is compact ) iff it is totally bounded.

Examples:

• In $\mathbb{R}^k$, every bounded set is totally bounded.
• The set $\mathbb{Z}\subset\mathbb{R}$ is not totally bounded: for $\varepsilon<1/2$, infinitely many $\varepsilon$-balls are needed.
• Any compact set is totally bounded.