Singleton (mathematics)

Singleton (mathematics)

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a singleton, also known as a unit set, is a set with exactly one element. For example, the set {0} is a singleton.

The term is also used for a 1-tuple
Tuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...

 (a sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

 with one element).

Properties


Note that a set such as is also a singleton: the only element is a set (which itself is however not a singleton). A singleton is distinct from the element it contains, thus 1 and {1} are not the same thing.

A set is a singleton if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

 its cardinality is . In the set-theoretic construction of the natural numbers
Set-theoretic definition of natural numbers
Several ways have been proposed to define the natural numbers using set theory.- The contemporary standard :In standard, Zermelo-Fraenkel set theory 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 pairing
Axiom of pairing
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.- Formal statement :...

: for any set A that axiom applied to A and A asserts the existence of {A,A}, which is the same as the singleton {A} (since it contains A, and no other set, as element).

If A is any set and S is any singleton, then there exists precisely one function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 from A to S, the function sending every element of A to the one element of S.

Applications


In topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

, a space is a T1 space
T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...

 if and only if every singleton is closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

.

Structures built on singletons often serve as terminal objects or zero objects of various categories:
  • The statement above shows that the singleton sets are precisely the terminal objects in the category Set
    Category of sets
    In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...

    of sets. No other sets are terminal.
  • Any singleton can be turned into a topological space
    Topological space
    Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

     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
    Group (mathematics)
    In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

     in just one way (the unique element serving as identity element
    Identity element
    In mathematics, an identity 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...

    ). These singleton groups are zero object
    Initial object
    In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...

    s in the category of groups and group homomorphism
    Group homomorphism
    In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...

    s. No other groups are terminal in that category.

Definition by indicator functions


Let be a class
Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...

 defined by an indicator function.
Then is called a singleton if and only if, for all ,
for some .

Traditionally, this definition was introduced by Whitehead
Alfred North Whitehead
Alfred North Whitehead, OM FRS was an English mathematician who became a philosopher. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education...

 and Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...

 along with the definition of the natural number 1, as, where .