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 {{1,2,3}} is also a singleton: the only element is a set (which itself is however not a singleton). Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In predicate logic and technical fields that depend on it, uniqueness quantification, or unique existential quantification, is an attempt to formalise the notion of something being true for exactly one thing, or exactly one thing of a certain type. ...

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 mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... Look up one in Wiktionary, the free dictionary. ... In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...

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 {{}}. Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of empty set is one of the axioms of Zermelo_Fraenkel set theory. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo-Fraenkel set theory. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...

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. Partial plot of a function f. ...

In topology, a space is a T1 space if and only if every singleton is closed. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... The title given to this article is incorrect due to technical limitations. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ...

Structures built on singletons often serve as terminal objects or zero objects of various categories: In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ...

• The statement above shows that the singleton sets are precisely the terminal objects in the category Set of sets. No other sets are terminal.
• 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.
• 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. No other groups are terminal in that category.