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Concept
Singleton (mathematics)
Summary
In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set is a singleton whose single element is .
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 {1} are not the same thing, and the empty set is distinct from the set containing only the empty set. A set such as is a singleton as it contains a single element (which itself is a set, however, not a singleton).
A set is a singleton if and only if its cardinality is . In von Neumann's set-theoretic construction of the natural numbers, the number 1 is defined as the singleton
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 which is the same as the singleton (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 .
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 is a set and then the upward of in which is the set is a principal ultrafilter on Moreover, every principal ultrafilter on is necessarily of this form. The ultrafilter lemma implies that non-principal ultrafilters exist on every infinite set (these are called ).
Every net valued in a singleton subset of is an ultranet in
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 :
The statement above shows that the singleton sets are precisely the terminal objects in the category of sets.
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