Editing Disjoint sets (section)

==Generalizations==
[[File:Disjoint sets.svg|thumb|A disjoint collection of sets]]

This definition of disjoint sets can be extended to [[family of sets|families of sets]] and to [[indexed family|indexed families]] of sets. 
By definition, a collection of sets is called a ''[[family of sets]]'' (such as the [[power set]], for example). In some sources this is a set of sets, while other sources allow it to be a [[multiset]] of sets, with some sets repeated.
An {{em|[[indexed family]] of sets}} <math>\left(A_i\right)_{i \in I},</math> is by definition a set-valued [[Function (mathematics)|function]] (that is, it is a function that assigns a set <math>A_i</math> to every element <math>i \in I</math> in its domain) whose domain <math>I</math> is called its {{em|[[index set]]}} (and elements of its domain are called {{em|indices}}).

There are two subtly different definitions for when a family of sets <math>\mathcal{F}</math> is called '''pairwise disjoint'''. According to one such definition, the family is disjoint if each two sets in the family are either identical or disjoint. This definition would allow pairwise disjoint families of sets to have repeated copies of the same set. According to an alternative definition, each two sets in the family must be disjoint; repeated copies are not allowed. The same two definitions can be applied to an indexed family of sets: according to the first definition, every two distinct indices in the family must name sets that are disjoint or identical, while according to the second, every two distinct indices must name disjoint sets.<ref name="douglas">{{citation|title=A Transition to Advanced Mathematics|first1=Douglas|last1=Smith|first2=Maurice|last2=Eggen|first3=Richard|last3=St. Andre|publisher=Cengage Learning|year=2010|isbn=978-0-495-56202-3|page=95|url=https://books.google.com/books?id=jJUs0ZDOOHoC&pg=PA95}}.</ref> For example, the family of sets {{nowrap|1={ {0, 1, 2}, {3, 4, 5}, {6, 7, 8}, ... } }} is disjoint according to both definitions, as is the family {{nowrap|1={ {..., −2, 0, 2, 4, ...}, {..., −3, −1, 1, 3, 5} } }} of the two parity classes of integers. However, the family <math>(\{n + 2k \mid k\in\mathbb{Z}\})_{n \in \{0, 1, \ldots, 9\}}</math> with 10 members has five repetitions each of two disjoint sets, so it is pairwise disjoint under the first definition but not under the second.

Two sets are said to be [[almost disjoint sets]] if their intersection is small in some sense. For instance, two [[infinite set]]s whose intersection is a [[finite set]] may be said to be almost disjoint.<ref>{{citation|title=Combinatorial Set Theory: With a Gentle Introduction to Forcing|series=Springer monographs in mathematics|first=Lorenz J.|last=Halbeisen|publisher=Springer|year=2011|isbn=9781447121732|page=184|url=https://books.google.com/books?id=NZVb54INnywC&pg=PA184}}.</ref>

In [[topology]], there are various notions of [[separated sets]] with more strict conditions than disjointness. For instance, two sets may be considered to be separated when they have disjoint [[closure (topology)|closure]]s or disjoint [[neighbourhood (mathematics)|neighborhoods]]. Similarly, in a [[metric space]], [[positively separated sets]] are sets separated by a nonzero [[metric space|distance]].<ref>{{citation|title=Metric Spaces|volume=57|series=Cambridge Tracts in Mathematics|first=Edward Thomas|last=Copson|publisher=Cambridge University Press|year=1988|isbn=9780521357326|page=62|url=https://books.google.com/books?id=egc5AAAAIAAJ&pg=PA62}}.</ref>