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Singleton (mathematics)

From Wikipedia, the free encyclopedia
Set with exactly one element
For a sequence with one member, see1-tuple.

Inmathematics, asingleton (also known as aunit set[1] orone-point set) is aset withexactly oneelement. For example, the set{0}{\displaystyle \{0\}} is a singleton whose single element is0{\displaystyle 0}.

Properties

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Within the framework ofZermelo–Fraenkel set theory, theaxiom of regularity guarantees that no set is an element of itself. This implies that a singleton is necessarily distinct from the element it contains,[1] thus 1 and{1}{\displaystyle \{1\}} are not the same thing, and theempty set is distinct from the set containing only the empty set. A set such as{{1,2,3}}{\displaystyle \{\{1,2,3\}\}} is a singleton as it contains a single element (which itself is a set, but not a singleton).

A set is a singletonif and only if itscardinality is1. Invon Neumann's set-theoretic construction of the natural numbers, the number 1 isdefined as the singleton{0}.{\displaystyle \{0\}.}

Inaxiomatic set theory, the existence of singletons is a consequence of theaxiom of pairing: for any setA, the axiom applied toA andA asserts the existence of{A,A},{\displaystyle \{A,A\},} which is the same as the singleton{A}{\displaystyle \{A\}} (since it containsA, and no other set, as an element).

IfA is any set andS is any singleton, then there exists precisely onefunction fromA toS, the function sending every element ofA to the single element ofS. Thus every singleton is aterminal object in thecategory 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 theempty set.

Every singleton set is anultra prefilter. IfX{\displaystyle X} is a set andx∈X{\displaystyle x\in X} then the upward of{x}{\displaystyle \{x\}} inX,{\displaystyle X,} which is the set{S⊆X:x∈S},{\displaystyle \{S\subseteq X:x\in S\},} is aprincipalultrafilter onX{\displaystyle X}. Moreover, every principal ultrafilter onX{\displaystyle X} is necessarily of this form.[2] Theultrafilter lemma implies that non-principal ultrafilters exist on everyinfinite set (these are calledfree ultrafilters). Everynet valued in a singleton subsetX{\displaystyle X} of is anultranet inX.{\displaystyle X.}

TheBell number integer sequence counts the number ofpartitions of a set (OEIS: A000110), if singletons are excluded then the numbers are smaller (OEIS: A000296).

In category theory

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Structures built on singletons often serve asterminal objects orzero objects of variouscategories:

  • The statement above shows that the singleton sets are precisely the terminal objects in the categorySet ofsets. No other sets are terminal.
  • Any singleton admits a uniquetopological space structure (both subsets are open). These singleton topological spaces are terminal objects in the category of topological spaces andcontinuous functions. No other spaces are terminal in that category.
  • Any singleton admits a uniquegroup structure (the unique element serving asidentity element). These singleton groups arezero objects in the category of groups andgroup homomorphisms. No other groups are terminal in that category.

Definition by indicator functions

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LetS be aclass defined by anindicator functionb:X→{0,1}.{\displaystyle b:X\to \{0,1\}.}ThenS is called asingleton if and only if there is somey∈X{\displaystyle y\in X} such that for allx∈X,{\displaystyle x\in X,}b(x)=(x=y).{\displaystyle b(x)=(x=y).}

Definition inPrincipia Mathematica

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The following definition was introduced inPrincipia Mathematica byWhitehead andRussell[3]

ι{\displaystyle \iota }‘x=y^(y=x){\displaystyle x={\hat {y}}(y=x)}Df.

The symbolι{\displaystyle \iota }‘x{\displaystyle x} denotes the singleton{x}{\displaystyle \{x\}} andy^(y=x){\displaystyle {\hat {y}}(y=x)} denotes the class of objects identical withx{\displaystyle x} aka{y:y=x}{\displaystyle \{y:y=x\}}. 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 thecardinal number 1 as

1=α^((∃x)α=ι{\displaystyle 1={\hat {\alpha }}((\exists x)\alpha =\iota }‘x){\displaystyle x)}Df.

That is, 1 is the class of singletons. This is definition 52.01 (p. 363 ibid.)

See also

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References

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  1. ^abStoll, Robert (1961).Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. pp. 5–6.
  2. ^Dolecki, Szymon; Mynard, Frédéric (2016).Convergence Foundations of Topology. Hackensack, New Jersey: World Scientific Publishing. pp. 27–54.doi:10.1142/9012.ISBN 978-981-4571-52-4.MR 3497013.
  3. ^Whitehead, Alfred North; Bertrand Russell (1910).Principia Mathematica. Vol. I. p. 37.


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