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| In [[mathematics]], a '''singleton''', also known as a '''unit set''',<ref name="Stoll">{{Cite book | last = Stoll | first = Robert | authorlink = | coauthors = | title = Sets, Logic and Axiomatic Theories | publisher = W. H. Freeman and Company | series = | volume = | edition = | year = 1961 | location = | pages = 5–6 | language = | url = | doi = | id = | isbn = | mr = | zbl = | jfm = }}</ref> is a [[Set (mathematics)|set]] with [[unique|exactly one]] element. For example, the set {0} is a singleton.
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| The term is also used for a 1-[[tuple]] (a [[sequence]] with one element).
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| ==Properties==
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| 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 {1} are not the same thing, and the [[empty set]] is distinct from the set containing only the empty set. A set such as <nowiki>{{1, 2, 3}}</nowiki> is a singleton as it contains a single element (which itself is a set, however, not a singleton).
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| A set is a singleton [[if and only if]] its [[cardinality]] is {{num|1}}. In the [[Set-theoretic definition of natural numbers|standard set-theoretic construction of the natural numbers]], the number 1 is ''defined'' as the singleton {0}.
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| 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 {''A'', ''A''}, which is the same as the singleton {''A''} (since it contains ''A'', and no other set, as an element).
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| 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]].
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| == Applications ==
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| In [[topology]], a space is a [[T1 space]] if and only if every singleton is [[closed set|closed]].
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| Structures built on singletons often serve as [[terminal object]]s or [[zero object]]s of various [[category (category theory)|categories]]:
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| * 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.
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| * Any singleton can be turned into a [[topological space]] in just one way (all subsets are open). These singleton topological spaces are terminal objects in the category of topological spaces and continuous functions. No other spaces are terminal in that category.
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| * Any singleton can be turned into a [[group (mathematics)|group]] in just one way (the unique element serving as [[identity element]]). These singleton groups are [[Initial object|zero object]]s in the category of groups and [[group homomorphism]]s. No other groups are terminal in that category.
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| == Definition by indicator functions ==
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| Let <math>S</math> be a [[Class (set theory)|class]] defined by an [[indicator function]]
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| :<math>b: X \to \{0, 1\}</math>. | |
| Then <math>S</math> is called a '''''singleton''''' if and only if there is some {{math|''y'' ∈ ''X''}} such that for all {{math|''x'' ∈ ''X''}},
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| :<math>b(x) = (x = y) \,</math>. | |
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| Traditionally, this definition was introduced by [[Alfred North Whitehead|Whitehead]] and [[Bertrand Russell|Russell]]<ref>{{cite book | first=Alfred North | last=Whitehead | coauthors=Bertrand Russell | year=1910 | title=[[Principia Mathematica]] | page=37 }}</ref> along with the definition of the [[1 (number)|natural number 1]], as
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| :<math>1 \ \overset{\underset{\mathrm{def}}{}}{=} \ \hat{\alpha}\{(\exists x) . \alpha = \iota \jmath x\}</math>, where <math>\iota \jmath x \ \overset{\underset{\mathrm{def}}{}}{=} \ \hat{y}(y = x)</math>. | |
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| ==See also==
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| *[[Class (set theory)]]
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| ==References==
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| <div class="references"><references/></div>
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| [[Category:Basic concepts in set theory]]
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| [[Category:One]]
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