Let $\mathcal{C}$ be a class of compact metric spaces, regarded up to isometry and equipped with the Gromov-Hausdorff metric. Then $\mathcal{C}$ is precompact in the Gromov-Hausdorff topology if and only if the following two conditions hold:
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1. There exists $D < \infty$ such that $\operatorname{diam}(X) \leq D$ for every $X \in \mathcal{C}$.
2. For every $\varepsilon > 0$, there exists $N(\varepsilon) \in \mathbb{N}$ such that every $X \in \mathcal{C}$ contains an $\varepsilon$-net with at most $N(\varepsilon)$ points.
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Here an $\varepsilon$-net in a [metric space](/page/Metric%20Space) $(X,d_X)$ is a subset $A \subset X$ such that for every $x \in X$ there exists $a \in A$ with $d_X(x,a) < \varepsilon$.