Singleton (mathematics)

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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 $0$.

# Properties

Within the framework of
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
, 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 are not the same thing, and the
empty set In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
its
cardinality In mathematics, the cardinality of a set (mathematics), set is a measure of the number of Element (mathematics), elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19 ...
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
category of sets In the mathematical field of category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundatio ...
. 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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
. Every singleton set is an ultra prefilter. If $X$ is a set and $x \in X$ then the upward of $\$ in $X,$ which is the set $\,$ is a principal ultrafilter on $X.$ Moreover, every principal ultrafilter on $X$ is necessarily of this form. The ultrafilter lemma implies that non- principal ultrafilters exist on every
infinite set In set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a bran ...
(these are called ). Every net valued in a singleton subset $X$ of is an ultranet in $X.$ The Bell number integer sequence counts the number of partitions of a set (), if singletons are excluded then the numbers are smaller ().

# In category theory

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 of sets. No other sets are terminal. * Any singleton admits a unique topological space structure (both subsets are open). These singleton topological spaces are terminal objects in the category of topological spaces and
continuous function In mathematics, a continuous function is a function (mathematics), function such that a continuous variation (that is a change without jump) of the argument of a function, argument induces a continuous variation of the Value (mathematics), value ...
s. No other spaces are terminal in that category. * Any singleton admits a unique group structure (the unique element serving as
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a Set (mathematics), set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in alge ...
). These singleton groups are zero objects in the category of groups and
group homomorphism In mathematics, given two group (mathematics), groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function (mathematics), function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' ...
s. No other groups are terminal in that category.

# Definition by indicator functions

Let be a class defined by an indicator function $b : X \to \.$ Then is called a ''singleton'' if and only if there is some $y \in X$ such that for all $x \in X,$ $b(x) = (x = y).$

# Definition in ''Principia Mathematica''

The following definition was introduced by Whitehead and Russell :$\iota$$x = \hat\left(y = x\right)$ Df. The symbol $\iota$$x$ denotes the singleton $\$ and $\hat\left(y = x\right)$ denotes the class of objects identical with $x$ aka $\$. This occurs as a definition in the introduction, which, in places, simplifies the argument in the main text, where it occurs as proposition 51.01 (p.357 ibid.). The proposition is subsequently used to define the
cardinal number In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
1 as :$1=\hat\left(\left(\exists x\right)\alpha=\iota$$x\right)$ Df. That is, 1 is the class of singletons. This is definition 52.01 (p.363 ibid.)