Editing Singleton (mathematics) (section)

==Properties==
Within the framework of [[Zermelo–Fraenkel set theory]], the [[axiom of regularity]] guarantees that no set is an element of itself. This implies that a singleton is necessarily distinct from the element it contains,<ref name="Stoll"/> thus 1 and <math>\{1\}</math> are not the same thing, and the [[empty set]] is distinct from the set containing only the empty set. A set such as <math>\{\{1, 2, 3\}\}</math> is a singleton as it contains a single element (which itself is a set, but not a singleton).

A set is a singleton [[if and only if]] its [[cardinality]] is {{num|1}}. In [[Set-theoretic definition of natural numbers|von Neumann's set-theoretic construction of the natural numbers]], the number 1 is ''defined'' as the singleton <math>\{0\}.</math>

In [[axiomatic set theory]], the existence of singletons is a consequence of the [[axiom of pairing]]: for any set ''A'', the axiom applied to ''A'' and ''A'' asserts the existence of <math>\{A, A\},</math> which is the same as the singleton <math>\{A\}</math> (since it contains ''A'', and no other set, as an element).

If ''A'' is any set and ''S'' is any singleton, then there exists precisely one [[Function (mathematics)|function]] from ''A'' to ''S'', the function sending every element of ''A'' to the single element of ''S''. Thus every singleton is a [[terminal object]] in the [[category of sets]].

A singleton has the property that every function from it to any arbitrary set is injective. The only non-singleton set with this property is the [[empty set]].

Every singleton set is an [[ultra prefilter]]. If <math>X</math> is a set and <math>x \in X</math> then the upward of <math>\{x\}</math> in <math>X,</math> which is the set <math>\{S \subseteq X : x \in S\},</math> is a [[Ultrafilter (set theory)#principal|principal]] [[Ultrafilter (set theory)|ultrafilter]] on <math>X</math>. Moreover, every principal ultrafilter on <math>X</math> is necessarily of this form.<ref>{{cite book
 | last1 = Dolecki | first1 = Szymon
 | last2 = Mynard | first2 = Frédéric
 | doi = 10.1142/9012
 | isbn = 978-981-4571-52-4
 | location = Hackensack, New Jersey
 | mr = 3497013
 | pages = 27–54
 | publisher = World Scientific Publishing
 | title = Convergence Foundations of Topology
 | year = 2016}}</ref> The [[ultrafilter lemma]] implies that non-[[Ultrafilter (set theory)#principal|principal]] ultrafilters exist on every [[infinite set]] (these are called {{em|[[Free ultrafilter (set theory)|free ultrafilters]]}}). 
Every [[Net (mathematics)|net]] valued in a singleton subset <math>X</math> of is an [[Ultranet (math)|ultranet]] in <math>X.</math>

The [[Bell number]] integer sequence counts the number of [[partitions of a set]] ({{OEIS2C|A000110}}), if singletons are excluded then the numbers are smaller ({{OEIS2C|A000296}}).