Let $(M,d)$ be a [metric space](/page/Metric%20Space), and let $A \subset M$. Say that $A$ is [totally bounded](/page/Totally%20Bounded) in $M$ if for every $\rho > 0$ there exist an integer $N \geq 0$ and points $x_1,\dots,x_N \in M$ such that
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\begin{align*}
A \subset \bigcup_{i=1}^{N} B(x_i,\rho)
\end{align*}