Total boundedness characterization via ε-nets

Let $(X,d)$ be a metric space and let $E\subseteq X$.

An $\varepsilon$-net for $E$ is a finite set $\{x_1,\dots,x_N\}\subseteq X$ such that $E\subseteq \bigcup_{j=1}^N B (x_j,\varepsilon).$

Proposition: The following are equivalent:

• $E$ is totally bounded , meaning: for every $\varepsilon>0$ there exist $x_1,\dots,x_N\in X$ such that $E\subseteq \bigcup_{j=1}^N B(x_j,\varepsilon)$.
• For every $\varepsilon>0$, $E$ admits a finite $\varepsilon$-net.

(These are the same statement in slightly different language; “$\varepsilon$-net” is the packaging.)

Total boundedness is stronger than boundedness and is one of the two metric ingredients (the other is completeness ) that together characterize compactness in metric spaces.

Examples:

• The interval $[0,1]\subset\mathbb{R}$ is totally bounded: cover it by finitely many intervals of length $\varepsilon$.
• The set $\mathbb{Z}\subset\mathbb{R}$ is not totally bounded: for $\varepsilon<1/2$ the balls $B(n,\varepsilon)$ are disjoint, so no finite subcover exists.