A subset $E$ of the metric space $(X,\rho)$ is totally bounded if for all $\varepsilon > 0$, there exists a finite set ${x_i}_{i=1}^n$ of points in $X$ such that \(E \subseteq \bigcup_{i=1}^n B(x_i, \varepsilon),\) where $B(x_i, \varepsilon) = {x \in X \mid \rho(x_i,x)< \varepsilon}.$ That is, for every $\varepsilon > 0$, $E$ can be covered by a finite set of balls of radius $\varepsilon$.
Wikidata ID: Q1362228