Editing Almost disjoint sets (section)

==Definition==
The most common choice is to take "small" to mean [[finite set|finite]]. In this case, two sets are almost disjoint if their intersection is finite, i.e. if

:<math>\left|A \cap B\right| < \infty.</math>

(Here, '|''X''{{hairsp}}|' denotes the [[cardinality]] of ''X'', and '< ∞' means 'finite'.) For example, the [[closed interval]]s [0, 1] and [1, 2] are almost disjoint, because their intersection is the finite set {1}. However, the [[unit interval]] [0, 1] and the set of [[rational number]]s '''Q''' are not almost disjoint, because their intersection is infinite.

This definition extends to any collection of sets. A collection of sets is '''pairwise almost disjoint''' or '''mutually almost disjoint''' if any two ''distinct'' sets in the collection are almost disjoint. Often the prefix 'pairwise' is dropped, and a pairwise almost disjoint collection is simply called "almost disjoint".

Formally, let ''I'' be an [[index set]], and for each ''i'' in ''I'', let ''A''<sub>''i''</sub> be a set. Then the collection of sets {''A''<sub>''i''</sub> : ''i'' in ''I''{{hairsp}}} is almost disjoint if for any ''i'' and ''j'' in ''I'',

:<math>A_i \ne A_j \quad \implies \quad \left|A_i \cap A_j\right| < \infty.</math>

For example, the collection of all lines through the origin in [[Euclidean plane|'''R'''<sup>2</sup>]] is almost disjoint, because any two of them only meet at the origin. If {''A''<sub>''i''</sub>{{hairsp}}} is an almost disjoint collection consisting of more than one set, then clearly its intersection is finite:

:<math>\bigcap_{i \in I} A_i < \infty.</math>

However, the [[converse (logic)|converse]] is not true—the intersection of the collection 
:<math>\{\{1, 2, 3,\ldots\}, \{2, 3, 4,\ldots\}, \{3, 4, 5,\ldots\},\ldots\}</math> 
is [[empty set|empty]], but the collection is ''not'' almost disjoint; in fact, the intersection of ''any'' two distinct sets in this collection is infinite.

The possible cardinalities of a maximal almost disjoint family (commonly referred to as a MAD family) on the set <math>\omega</math> of the [[natural number]]s has been the object of intense study.<ref>[[Eric van Douwen]]. The Integers and Topology. In K. Kunen and J.E. Vaughan (eds) ''Handbook of Set-Theoretic Topology.''  North-Holland, Amsterdam, 1984.</ref><ref name = "jech" /> The minimum infinite such [[cardinal number|cardinal]] is one of the classical [[cardinal characteristics of the continuum]].<ref>{{cite book | last = Vaughan | first = Jerry E. | chapter = Chapter 11: Small uncountable cardinals and topology | editor1-last = van Mill | editor1-first = Jan | editor2-last = Reed | editor2-first = George M. | title = Open Problems in Topology | pages = [https://archive.org/details/openproblemsinto0000unse/page/196 196–218] | year = 1990 | publisher = [[North-Holland Publishing Company]] | location = Amsterdam | isbn = 0-444-88768-7 | url = https://archive.org/details/openproblemsinto0000unse/page/196 | format = PDF }}</ref><ref>{{cite book | last = Blass | first = Andreas | authorlink = Andreas Blass | chapter = Chapter 6 : Combinatorial Cardinal Characteristics of the Continuum | editor1-last = Foreman | editor1-first = Matthew | editor1-link = Matthew Foreman | editor2-last = Kanamori | editor2-first = Akihiro | editor2-link = Akihiro Kanamori | title = Handbook of Set Theory | volume = 1 | pages = 395–490 | date = January 12, 2010 | publisher = [[Springer Science+Business Media|Springer]] | isbn = 1-4020-4843-2 | url = http://www.math.lsa.umich.edu/~ablass/hbk.pdf | format = PDF}}</ref>