Editing Metric space (section)

===Metrics on multisets===
The notion of a metric can be generalized from a distance between two elements to a number assigned to a multiset of elements. A [[multiset]] is a generalization of the notion of a [[set (mathematics)|set]] in which an element can occur more than once.  Define the multiset union <math>U=XY</math> as follows: if an element {{mvar|x}} occurs {{mvar|m}} times in {{mvar|X}} and {{mvar|n}} times in {{mvar|Y}} then it occurs {{math|''m'' + ''n''}} times in {{mvar|U}}.  A function {{mvar|d}} on the set of nonempty finite multisets of elements of a set {{mvar|M}} is a metric{{sfn|Vitányi|2011}} if
# <math>d(X)=0</math> if all elements of {{mvar|X}} are equal and <math>d(X) > 0</math> otherwise ([[positive definiteness]])
# <math>d(X)</math> depends only on the (unordered) multiset {{mvar|X}} ([[symmetry]])
# <math>d(XY) \leq d(XZ)+d(ZY)</math> ([[triangle inequality]])
By considering the cases of axioms 1 and 2 in which the multiset {{mvar|X}} has two elements and the case of axiom 3 in which the multisets {{mvar|X}}, {{mvar|Y}}, and {{mvar|Z}} have one element each, one recovers the usual axioms for a metric.  That is, every multiset metric yields an ordinary metric when restricted to sets of two elements.

A simple example is the set of all nonempty finite multisets <math>X</math>  of integers with <math>d(X)=\max (X)- \min (X)</math>. More complex examples are [[information distance]] in multisets;{{sfn|Vitányi|2011}} and [[normalized compression distance]] (NCD) in multisets.{{sfn|Cohen|Vitányi|2012}}