Gromov–Hausdorff convergence

Gromov–Hausdorff convergence describes when compact metric spaces become arbitrarily close up to isometry, measured by embedding them in a common metric space and comparing their images.

Gromov–Hausdorff distance: A metric on isometry classes of compact metric spaces, defined by minimizing Hausdorff distance between isometric copies in a common space. Convergence is exactly the claim that this distance to the proposed limit tends to zero.
Metric space: A set equipped with a distance function satisfying nonnegativity, identity, symmetry, and the triangle inequality. Gromov–Hausdorff convergence compares spaces through their distance functions.
Gromov compactness theorem: A theorem giving subsequential compactness for families of compact metric spaces under uniform diameter and covering bounds. It supplies conditions under which a sequence has a Gromov–Hausdorff convergent subsequence.
Measured Gromov–Hausdorff convergence: A convergence notion for metric measure spaces that tracks both distances and measures. It strengthens metric convergence when the distribution of mass must survive in the limit.
Hausdorff distance: A distance between subsets that measures how far each set lies from the nearest point of the other. The Gromov–Hausdorff distance applies this set distance to embedded copies of spaces.
Isometry: A distance-preserving bijection between metric spaces. The convergence notion treats isometric spaces as identical, regardless of their point labels or presentation.
Cheeger–Gromov convergence: A smooth convergence notion for Riemannian manifolds using diffeomorphisms on increasingly large regions. It retains differentiable structure that Gromov–Hausdorff convergence can discard.
Intrinsic flat convergence: A convergence notion for oriented metric spaces based on flat distance between their isometric embeddings. It can disregard regions of small volume that remain visible in Gromov–Hausdorff distance.
Metric correspondence: A relation between two sets that pairs every point of each set with at least one point of the other. Correspondences encode approximate pointwise matches and yield an equivalent way to measure Gromov–Hausdorff distance.
Compact space: A topological space in which every open cover has a finite subcover. Compactness ensures the standard Gromov–Hausdorff distance is finite and supports subsequential limit theorems.

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