Metric space
Hausdorff and Gromov–Hausdorff distance

The idea of spaces of mathematical objects can also be applied to subsets of a metric space, as well as metric spaces themselves. Hausdorff and Gromov–Hausdorff distance define metrics on the set of compact subsets of a metric space and the set of compact metric spaces, respectively.

Suppose (M, d) is a metric space, and let S be a subset of M. The distance from S to a point x of M is, informally, the distance from x to the closest point of S. However, since there may not be a single closest point, it is defined via an infimum: d (x, S) = inf { d (x, s): s ∈ S }. {\displaystyle d(x,S)=\inf\{d(x,s):s\in S\}.} In particular, d (x, S) = 0 {\displaystyle d(x,S)=0} if and only if x belongs to the closure of S. Furthermore, distances between points and sets satisfy a version of the triangle inequality: d (x, S) ≤ d (x, y) + d (y, S), {\displaystyle d(x,S)\leq d(x,y)+d(y,S),} and therefore the map d S: M → R {\displaystyle d_{S}:M\to \mathbb {R} } defined by d S (x) = d (x, S) {\displaystyle d_{S}(x)=d(x,S)} is continuous. Incidentally, this shows that metric spaces are completely regular.

Given two subsets S and T of M, their Hausdorff distance is d H (S, T) = max { sup { d (s, T): s ∈ S }, sup { d (t, S): t ∈ T } }. {\displaystyle d_{H}(S,T)=\max\{\sup\{d(s,T):s\in S\},\sup\{d(t,S):t\in T\}\}.} Informally, two sets S and T are close to each other in the Hausdorff distance if no element of S is too far from T and vice versa. For example, if S is an open set in Euclidean space T is an ε-net inside S, then d H (S, T) < ε {\displaystyle d_{H}(S,T)<\varepsilon }. In general, the Hausdorff distance d H (S, T) {\displaystyle d_{H}(S,T)} can be infinite or zero. However, the Hausdorff distance between two distinct compact sets is always positive and finite. Thus the Hausdorff distance defines a metric on the set of compact subsets of M.

The Gromov–Hausdorff metric defines a distance between (isometry classes of) compact metric spaces. The Gromov–Hausdorff distance between compact spaces X and Y is the infimum of the Hausdorff distance over all metric spaces Z that contain X and Y as subspaces. While the exact value of the Gromov–Hausdorff distance is rarely useful to know, the resulting topology has found many applications.
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