Singleton (mathematics)

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 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.

The Bell number integer sequence counts the number of partitions of a set (), if singletons are excluded then the numbers are smaller ().

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

Then S is called a singleton if and only if there is some yX such that for all xX,

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