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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 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. No other sets are terminal in that category.
- 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 groupIn mathematics, a group is a set, together with a binary operation satisfying certain axioms, detailed below. The branch of mathematics which studies groups is called group theory. The historical origin of group theory goes back to the works of Evariste G in just one way (the unique element serving as identity elementIn mathematics, an identity element (or neutral element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them. The term identity element is often shortened to ident). These singleton groups are zero objects in the category of groups and group homomorphismGiven two groups G ) and H ·), a group homomorphism from G ) to H ·) is a function h : G H such that for all u and v in G it holds that : h ''u v h ''u · h ''v From this property, one can deduce that h maps the identity element e of G to the identity elems. No other groups are terminal in that category.
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