Metric Space - Distance Between Points and Sets; Hausdorff Distance and Gromov Metric

Distance Between Points and Sets; Hausdorff Distance and Gromov Metric

A simple way to construct a function separating a point from a closed set (as required for a completely regular space) is to consider the distance between the point and the set. If (M,d) is a metric space, S is a subset of M and x is a point of M, we define the distance from x to S as

where represents the infimum.

Then d(x, S) = 0 if and only if x belongs to the closure of S. Furthermore, we have the following generalization of the triangle inequality:

which in particular shows that the map is continuous.

Given two subsets S and T of M, we define their Hausdorff distance to be

where represents the supremum.

In general, the Hausdorff distance d_H(S,T) can be infinite. Two sets are close to each other in the Hausdorff distance if every element of either set is close to some element of the other set.

The Hausdorff distance d_H turns the set K(M) of all non-empty compact subsets of M into a metric space. One can show that K(M) is complete if M is complete. (A different notion of convergence of compact subsets is given by the Kuratowski convergence.)

One can then define the Gromov–Hausdorff distance between any two metric spaces by considering the minimal Hausdorff distance of isometrically embedded versions of the two spaces. Using this distance, the set of all (isometry classes of) compact metric spaces becomes a metric space in its own right.