Editing Metric space (section)

=== 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 distance|Hausdorff]] and [[Gromov–Hausdorff convergence|Gromov–Hausdorff distance]] define metrics on the set of compact subsets of a metric space and the set of compact metric spaces, respectively.

{{anchor|Distance to a set}}
Suppose {{math|(''M'', ''d'')}} is a metric space, and let {{mvar|S}} be a subset of {{mvar|M}}.  The ''distance from {{mvar|S}} to a point {{mvar|x}} of {{mvar|M}}'' is, informally, the distance from {{mvar|x}} to the closest point of {{mvar|S}}.  However, since there may not be a single closest point, it is defined via an [[infimum]]:
<math display="block">d(x,S) = \inf\{d(x,s) : s \in S \}.</math>
In particular, <math>d(x, S)=0</math> if and only if {{mvar|x}} belongs to the [[closure (topology)|closure]] of {{mvar|S}}. Furthermore, distances between points and sets satisfy a version of the triangle inequality:
<math display="block">d(x,S) \leq d(x,y) + d(y,S),</math>
and therefore the map <math>d_S:M \to \R</math> defined by <math>d_S(x)=d(x,S)</math> is continuous.  Incidentally, this shows that metric spaces are [[completely regular]].

Given two subsets {{mvar|S}} and {{mvar|T}} of {{mvar|M}}, their ''Hausdorff distance'' is
<math display="block">d_H(S,T) = \max \{ \sup\{d(s,T) : s \in S \} , \sup\{ d(t,S) : t \in T \} \}.</math>
Informally, two sets {{mvar|S}} and {{mvar|T}} are close to each other in the Hausdorff distance if no element of {{mvar|S}} is too far from {{mvar|T}} and vice versa.  For example, if {{mvar|S}} is an open set in Euclidean space {{mvar|T}} is an [[Delone set|ε-net]] inside {{mvar|S}}, then <math>d_H(S,T)<\varepsilon</math>.  In general, the Hausdorff distance <math>d_H(S,T)</math> 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 {{mvar|M}}.

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