View source for Singleton (mathematics)

{{Short description|Set with exactly one element}}
{{Probability fundamentals}}

In [[Mathematics|mathematics]], a '''singleton''', also known as a '''unit set'''<ref name="Stoll">{{Cite book  | last = Stoll  | first = Robert  | title = Sets, Logic and Axiomatic Theories  | publisher = W. H. Freeman and Company  | year = 1961  | pages = 5–6  }}</ref> or '''one-point set''', is a [[Set (mathematics)|set]] with [[Uniqueness quantification|exactly one]] [[Element (mathematics)|element]]. For example, the set <math>\{0\}</math> is a singleton whose single element is <math>0</math>.

==Properties==
Within the framework of [[Zermelo–Fraenkel set theory]], the [[Axiom of regularity|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 {1} are not the same thing, and the [[Empty set|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, however, not a singleton).

A set is a singleton [[If and only if|if and only if]] its [[Cardinality|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|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|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|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>{{sfn|Dolecki|Mynard|2016|pp=27–54}} Moreover, every principal ultrafilter on <math>X</math> is necessarily of this form.{{sfn|Dolecki|Mynard|2016|pp=27–54}} The ultrafilter lemma implies that non-[[Ultrafilter (set theory)#principal|principal]] ultrafilters exist on every [[Infinite set|infinite set]] (these are called {{em|free ultrafilters}}). 
Every [[Net (mathematics)|net]] valued in a singleton subset <math>X</math> of is an 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}}).

==In category theory==

Structures built on singletons often serve as terminal objects or zero objects of various categories:
* The statement above shows that the singleton sets are precisely the terminal objects in the category '''[[Category of sets|Set]]''' of [[Set (mathematics)|set]]s. No other sets are terminal.
* Any singleton admits a unique [[Topological space|topological space]] structure (both subsets are open). These singleton topological spaces are terminal objects in the category of topological spaces and [[Continuous function|continuous function]]s. No other spaces are terminal in that category.
* Any singleton admits a unique [[Group (mathematics)|group]] structure (the unique element serving as [[Identity element|identity element]]). These singleton groups are zero objects in the category of groups and [[Group homomorphism|group homomorphism]]s. No other groups are terminal in that category.

==Definition by indicator functions==
Let {{mvar|S}} be a [[Class (set theory)|class]] defined by an [[Indicator function|indicator function]] 
<math display=block>b : X \to \{0, 1\}.</math>
Then {{mvar|S}} is called a ''singleton'' if and only if there is some <math>y \in X</math> such that for all <math>x \in X,</math>
<math display=block>b(x) = (x = y).</math>

==Definition in ''Principia Mathematica''==
The following definition was introduced by [[Biography:Alfred North Whitehead|Whitehead]] and [[Biography:Bertrand Russell|Russell]]<ref>{{cite book | first=Alfred North | last=Whitehead |author2=Bertrand Russell  | year=1910 | title=[[Principia Mathematica]]|volume=I | page=37 }}</ref> 
:<math>\iota</math>'''‘'''<math>x = \hat{y}(y = x)</math> '''Df.'''
The symbol <math>\iota</math>'''‘'''<math>x</math> denotes the singleton <math>\{x\}</math> and <math>\hat{y}(y = x)</math> denotes the class of objects identical with <math>x</math> aka <math>\{y : y=x\}</math>.  
This occurs as a definition in the introduction, which, in places, simplifies the argument in the main text, where it occurs as proposition 51.01 (p.357 ibid.).
The proposition is subsequently used to define the [[Cardinal number|cardinal number]] 1 as
:<math>1=\hat{\alpha}((\exists x)\alpha=\iota</math>'''‘'''<math>x)</math> '''Df.'''  
That is, 1 is the class of singletons. This is definition 52.01 (p.363 ibid.)

==See also==

* {{annotated link|Class (set theory)}}
* {{annotated link|Isolated point}}
* {{annotated link|Uniqueness quantification}}

==References==

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{{reflist}}
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* {{Dolecki Mynard Convergence Foundations Of Topology}} <!-- {{sfn|Dolecki|Mynard|2016|p=}} -->


[[Category:Basic concepts in set theory]]
[[Category:1 (number)]]

{{Sourceattribution|Singleton (mathematics)}}