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
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
with
exactly one element. For example, the set
is a singleton whose single element is
.
Properties
Within the framework of
Zermelo–Fraenkel set theory, the
axiom of regularity
In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set ''A'' contains an element that is disjoint from ''A''. In first-order logic, the a ...
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 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 bic ...
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
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. It was introduced by as a special case of his axiom of elementary set ...
: 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.
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 ().
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
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
s. No other sets are terminal.
* Any singleton admits a unique
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
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 such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s. No other spaces are terminal in that category.
* Any singleton admits a unique
group structure (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.
Definition by indicator functions
Let be a
class
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differently ...
defined by an
indicator function
Then is called a ''singleton'' if and only if there is some
such that for all
Definition in ''Principia Mathematica''
The following definition was introduced by
Whitehead and
Russell
:
‘
Df.
The symbol
‘
denotes the singleton
and
denotes the class of objects identical with
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, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
1 as
:
‘
Df.
That is, 1 is the class of singletons. This is definition 52.01 (p.363 ibid.)
See also
*
*
References
*
{{Set theory
Basic concepts in set theory
1 (number)