Gromov's compactness theorem

Gromov's compactness theorem states that a family of compact metric spaces with uniformly bounded diameters and uniformly controlled covering numbers is precompact in the Gromov–Hausdorff topology.

Gromov–Hausdorff distance: A metric measuring how closely two compact metric spaces approximate one another after isometric embedding into a common metric space. It defines the topology in which the theorem guarantees convergent subsequences.
Compact metric space: A metric space in which every open cover has a finite subcover. The theorem is commonly stated for families of compact metric spaces.
Cheeger–Gromov convergence: A form of smooth convergence for Riemannian manifolds, using diffeomorphisms on suitable regions. It retains differentiable structure that Gromov–Hausdorff convergence alone may discard.
Gromov–Hausdorff topology: The topology on compact metric spaces induced by Gromov–Hausdorff distance. Precompactness is relative to this topology, not to a topology on coordinates or embeddings.
Covering number: The least number of balls of a specified radius needed to cover a metric space. Uniform bounds on these numbers prevent spaces from developing arbitrarily fine complexity.
Diameter: The supremum of distances between pairs of points in a metric space. A uniform diameter bound prevents the spaces from spreading without limit.
Ricci curvature: A tensor that describes how volumes and geodesics in a Riemannian manifold deviate from Euclidean behavior. Ricci curvature bounds often combine with diameter bounds to control geometric limits.
Hausdorff distance: A distance between subsets of a metric space determined by their greatest point-to-set distance. It compares subsets in one ambient space, unlike Gromov–Hausdorff distance between spaces.
Metric entropy: A measure of the number of small-scale balls required to cover a metric space or set. Its scale-dependent bounds express the theorem's uniform covering hypothesis.
Precompactness: A property meaning that the closure of a set in a specified topology is compact. This is the precise compactness conclusion asserted for the family.

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