Метрикалық кеңістіктердің Gromov-Hausdorff жинақтылығы

Notion for convergence of metric spaces
In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence.

The Gromov–Hausdorff distance was introduced by David Edwards in 1975, and it was later rediscovered and generalized by Mikhail Gromov in 1981. This distance measures how far two compact metric spaces are from being isometric. If X and Y are two compact metric spaces, then dGH (X, Y) is defined to be the infimum of all numbers dH(f(X), g(Y)) for all (compact) metric spaces M and all isometric embeddings f : X → M and g : Y → M. Here dH denotes Hausdorff distance between subsets in M and the isometric embedding is understood in the global sense, i. e. it must preserve all distances, not only infinitesimally small ones; for example no compact Riemannian manifold admits such an embedding into Euclidean space of the same dimension. The Gromov–Hausdorff distance turns the set of all isometry classes of compact metric spaces into a metric space, called Gromov–Hausdorff space, and it therefore defines a notion of convergence for sequences of compact metric spaces, called Gromov–Hausdorff convergence. A metric space to which such a sequence converges is called the Gromov–Hausdorff limit of the sequence.

The Gromov–Hausdorff space is path connected, complete, and separable. It is also geodesic, i. e., any two of its points are the endpoints of a minimizing geodesic. In the global sense, the Gromov–Hausdorff space is totally heterogeneous, i. e., its isometry group is trivial, but locally there are many nontrivial isometries.

The pointed Gromov–Hausdorff convergence is an analog of Gromov–Hausdorff convergence appropriate for non compact spaces. A pointed metric space is a pair (X,p) consisting of a metric space X and point p in X. A sequence (Xn, pn) of pointed metric spaces converges to a pointed metric space (Y, p) if, for each R > 0, the sequence of closed R balls around pn in Xn converges to the closed R ball around p in Y in the usual Gromov–Hausdorff sense.