Singleton (mathematics)

In mathematics, a singleton is a set with exactly one element. For example, the set {0} is a singleton. Note that a set such as Template:1,2,3 is also a singleton: the only element is a set (which itself is however not a singleton).

A set is a singleton if and only if its cardinality is 1. In the set-theoretic construction of the natural numbers, the number 1 is defined as the singleton {0}.

In axiomatic set theory, the existence of singletons is a consequence of the axiom of empty set and the axiom of pairing: the former yields the empty set {}, and the latter, applied to the pairing of {} and {}, yields the singleton {{}}.

If A is any set and S is any singleton, then there exists precisely one function from A to S, the function sending every element of A to the one element of S.

Structures built on singletons often serve as terminal objects or zero objects of various categories:

• The statement above shows that every singleton S is a terminal object in the category of sets and functions.
• 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.
• Any singleton can be turned into a group in just one way (the unique element serving as identity element). These singleton groups are zero objects in the category of groups and group homomorphisms.