Singleton (mathematics)

Properties

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, thus 1 and ${\displaystyle \{1\}}$ are not the same thing, and the empty set is distinct from the set containing only the empty set. A set such as ${\displaystyle \{\{1,2,3\}\}}$ is a singleton as it contains a single element (which itself is a set, but not a singleton).

A set is a singleton if and only if its cardinality is 1. In von Neumann's set-theoretic construction of the natural numbers, the number 1 is defined as the singleton ${\displaystyle \{0\}.}$

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 ${\displaystyle \{A,A\},}$ which is the same as the singleton ${\displaystyle \{A\}}$ (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 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.

Every singleton set is an ultra prefilter. If ${\displaystyle X}$ is a set and ${\displaystyle x\in X}$ then the upward of ${\displaystyle \{x\}}$ in ${\displaystyle X,}$ which is the set ${\displaystyle \{S\subseteq X:x\in S\},}$ is a principal ultrafilter on ${\displaystyle X}$. Moreover, every principal ultrafilter on ${\displaystyle X}$ is necessarily of this form. The ultrafilter lemma implies that non-principal ultrafilters exist on every infinite set (these are called free ultrafilters). Every net valued in a singleton subset ${\displaystyle X}$ of is an ultranet in ${\displaystyle X.}$

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