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Singleton (mathematics)
Singleton (mathematics)
Singleton (mathematics)
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In mathematics, a singleton (also known as a unit set[1] or one-point set) is a set with exactly one element. For example, the set is a singleton whose single element is .

Properties

[edit]

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,[1] 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, but not a singleton).

A set is a singleton if and only if its cardinality is 1. 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.

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.[2] The ultrafilter lemma implies that non-principal ultrafilters exist on every infinite set (these are called free ultrafilters). 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 (OEIS: A000110), if singletons are excluded then the numbers are smaller (OEIS: A000296).

In category theory

[edit]

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 functions. 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

[edit]

Let S be a class defined by an indicator function Then S is called a singleton if and only if there is some such that for all

Definition in Principia Mathematica

[edit]

The following definition was introduced in Principia Mathematica by Whitehead and Russell[3]

‘ 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 1 as

‘ Df.

That is, 1 is the class of singletons. This is definition 52.01 (p. 363 ibid.)

See also

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Singleton (mathematics)

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from Grokipedia
In set theory, a singleton, also known as a unit set, is a set consisting of exactly one element, denoted as {x} where xx is that element. This distinguishes it from the empty set, which has no elements, and from sets with multiple elements. A key property of singletons is that the set {x} is not identical to the element xx itself; for instance, the singleton containing the integer 5, {5}, is a distinct mathematical object from the number 5. Singletons have cardinality 1, meaning they contain precisely one member, and they form the basic building blocks for constructing larger sets through operations like unions and power sets. In foundational mathematics, singletons are essential for defining more complex structures, such as ordered pairs via the Kuratowski definition ⟨a,b⟩={{a},{a,b}}\langle a, b \rangle = \{\{a\}, \{a, b\}\}, which relies on singletons to encode relations and functions. Beyond , singletons play a significant role in other areas of . In , a space is T1 (or Fréchet) if and only if every singleton set is closed, ensuring points can be separated by open sets. In the discrete topology on a set XX, the collection of all singletons {{x}∣x∈X}\{\{x\} \mid x \in X\} forms a basis for the . These properties highlight the singleton's utility in distinguishing topological spaces and analyzing their separation axioms.

Fundamentals

Definition

In , a singleton, also known as a unit set, is a set containing exactly one element. This distinguishes it as the simplest nonempty set, where the single element can be any object within the of , such as , another set, or an abstract entity. Formally, for any element aa, the singleton denoted {a}\{a\} is the set uniquely characterized by the property that aa is its only member: that is, a∈{a}a \in \{a\} and for every y≠ay \neq a, y∉{a}y \notin \{a\}. Equivalently, a set SS is a singleton if there exists exactly one xx such that x∈Sx \in S, and no other elements belong to SS. The notation {a}\{a\} employs standard set-builder notation, relying on the membership relation ∈\in to specify containment. Examples illustrate this concept clearly. The set {0}\{0\} is a singleton whose sole element is the integer zero, while {∅}\{\emptyset\} is a singleton containing the empty set ∅\emptyset as its only member. It is crucial to emphasize that the singleton {a}\{a\} is not identical to the element aa itself; rather, {a}\{a\} is a set that has aa as a member. Singletons thus provide a foundational mechanism for uniquely identifying and isolating individual elements within broader set structures.

Construction in Set Theory

In Zermelo–Fraenkel set theory with the (ZFC), the existence of singleton sets is guaranteed by the , which states that for any sets xx and yy, there exists a set uu such that ∀z(z∈u↔(z=x∨z=y))\forall z (z \in u \leftrightarrow (z = x \lor z = y)). Applying this axiom with x=yx = y yields the set {x,x}\{x, x\}, which, by the , is identical to the singleton {x}\{x\} containing xx as its sole element. The singleton axiom schema, which asserts the existence of {x}\{x\} for every set xx via comprehension {y∣y=x}\{y \mid y = x\}, is not a primitive axiom in ZFC but can be derived using the together with the . Specifically, first form the pair {x,x}\{x, x\} using ; then apply separation to this pair with the property ϕ(y)≡y=x\phi(y) \equiv y = x, yielding {y∈{x,x}∣y=x}={x}\{y \in \{x, x\} \mid y = x\} = \{x\}. Uniqueness follows from , as any two sets with the same unique element must coincide: if A={x}A = \{x\} and B={x}B = \{x\}, then ∀z(z∈A↔z∈B)\forall z (z \in A \leftrightarrow z \in B). The is not required for this derivation, though it facilitates further constructions involving singletons. Although some weaker set theories replace pairing with a dedicated singleton axiom, in ZFC the pairing-based approach ensures singletons without introducing additional primitives. Historically, Ernst Zermelo's original 1908 axiomatization explicitly included an of elementary sets to guarantee the existence of singletons: for any object aa, the set {a}\{a\} with aa as its sole member exists. This , part of Zermelo's effort to resolve paradoxes like Russell's while enabling Cantor's transfinite constructions, directly provided singletons alongside the , differing from the derived approach in modern ZFC.

Properties

Basic Properties

In Zermelo-Fraenkel set theory with the axiom of choice (ZFC), a singleton set {x}\{x\} is distinct from its sole element xx, as the axiom of regularity (also known as the axiom of foundation) prohibits any set from being an element of itself. This axiom ensures that every non-empty set AA has an element y∈Ay \in A such that no element of yy belongs to AA, thereby preventing membership cycles like x∈{x}x \in \{x\}, which would imply {x}=x\{x\} = x. The axiom of regularity thus guarantees the well-foundedness of the membership relation, distinguishing sets from their elements and enabling the hierarchical structure of the cumulative hierarchy of sets. Singletons are constructed using the axiom of pairing, which asserts the existence of {x,x}={x}\{x, x\} = \{x\} for any set xx. A key relational property of singletons is their subset behavior: for any sets {x}\{x\} and AA, {x}⊆A\{x\} \subseteq A if and only if x∈Ax \in A. This equivalence follows directly from the definition of subset inclusion, where every element of the singleton—namely xx—must belong to AA. Consequently, every singleton is a minimal non-empty subset of any set containing its element, as no proper non-empty subset of {x}\{x\} exists. This property underscores the atomic role of singletons in the subset lattice of the power set P(A)\mathcal{P}(A). Functions defined on singletons exhibit straightforward injectivity: any function f:{x}→Bf: \{x\} \to B is injective, since the domain has only one element, precluding the possibility of distinct elements mapping to the same output. The image of such a function is precisely {f(x)}\{f(x)\}, a singleton in BB, and this mapping is unique up to the choice of f(x)f(x). This injectivity highlights how singletons as domains simplify function theory, ensuring one-to-one correspondences without additional constraints. In the context of filters on the power set P(X)\mathcal{P}(X), singletons generate principal ultrafilters: for x∈Xx \in X, the principal ultrafilter at xx consists of all subsets of XX that contain xx, forming a maximal filter closed under finite intersections and supersets. This ultrafilter is principal precisely because it is generated by the singleton {x}\{x\}, distinguishing it from non-principal ultrafilters that contain no finite sets. Such structures are fundamental in extending notions of "largeness" in set-theoretic and .

Cardinality and Structure

The cardinality of a singleton set {x}\{x\} is 1, as it contains exactly one element and admits a with any other one-element set, such as {∅}\{\emptyset\}. This finite distinguishes singletons from empty sets ( 0) and larger finite or infinite sets. In , provides a measure of set size independent of order or , and for singletons, it serves as the foundational unit for building larger cardinals. In the Von Neumann construction of ordinals, the cardinal number 1 is represented as the singleton {0}\{0\}, where 0 is defined as the empty set ∅\emptyset. This approach defines each ordinal as the set of all preceding ordinals: starting with 0=∅0 = \emptyset, the successor ordinal 1 is {0}={∅}\{0\} = \{\emptyset\}, establishing a transitive, well-ordered structure that embeds the natural numbers within the ordinals. This construction, introduced by John von Neumann in the 1920s, ensures that finite ordinals coincide with the natural numbers while extending to transfinite cases. The of a singleton {x}\{x\} consists of all , namely {∅,{x}}\{\emptyset, \{x\}\}, which has exactly two elements and illustrates the cardinal 21=22^1 = 2. This operation highlights the structural growth from a singleton: it includes the as the unique improper and the singleton itself as the only proper . In general, the 2∣S∣2^{|S|} exceeds ∣S∣|S| for any set SS, with the singleton case providing the simplest demonstration of . In cardinal arithmetic, a singleton contributes additively as 1, so the cardinality of the disjoint union {x}∪A\{x\} \cup A equals 1+∣A∣1 + |A|, where ∣A∣|A| is the of set AA. For finite ∣A∣=n|A| = n, this yields n+1n + 1; for infinite κ\kappa, 1+κ=κ1 + \kappa = \kappa, as the singleton is absorbed. Multiplicatively, the product 1⋅κ=κ1 \cdot \kappa = \kappa reflects the bijection between {x}×A\{x\} \times A and AA. These operations underscore the singleton's role in scaling set sizes without altering infinite cardinalities.

Formalizations

In Principia Mathematica

In Principia Mathematica, and denote the singleton containing an element xx of a given type using the iota operator as ι′x\iota ' x, which forms the unit class consisting solely of xx. This notation, introduced in section *51·11, serves as a shorthand for the class abstraction z^(z=x)\hat{z}(z = x), where z^\hat{z} is the class-forming operator applied to the propositional function z=xz = x, meaning the class of all zz identical to xx. The propositional function here is of higher type than xx, ensuring that the singleton ι′x\iota ' x is a construct of the appropriate ramified type. This type-theoretic framework, central to 's ramified theory of types, prevents paradoxes like Russell's by stratifying expressions: an element xx of type α\alpha belongs to a singleton of type α+1\alpha + 1, treating singletons as higher-order entities rather than unrestricted sets. Russell and Whitehead emphasize this in their discussion of classes, where propositional functions define extensions without self-reference, as in the derivation: ι′x=z^⋅z=x:⊃:ϕz≡ϕx\iota ' x = \hat{z} \cdot z = x : \supset : \phi z \equiv \phi x, linking membership in the singleton to the identity with xx. The cardinal number 1 is explicitly defined as the class of all singletons, i.e., the collection of all unit classes, which corresponds to the type comprising one-element classes across the type hierarchy. This definition, given in *52·01, underscores the role of singletons in cardinal arithmetic, where 1 denotes equivalence under similarity to any singleton.

Using Indicator Functions

In , a singleton set can be characterized using , also known as characteristic functions, which provide a functional representation of subsets. The χA:X→{0,1}\chi_A: X \to \{0,1\} of a subset A⊆XA \subseteq X is defined by χA(x)=1\chi_A(x) = 1 if x∈Ax \in A and χA(x)=0\chi_A(x) = 0 otherwise. For the singleton {y}⊆X\{y\} \subseteq X, the corresponding χ{y}\chi_{\{y\}} satisfies χ{y}(x)=1\chi_{\{y\}}(x) = 1 if x=yx = y and 00 otherwise, making {y}\{y\} precisely the support of χ{y}\chi_{\{y\}}—the set of points where the function is nonzero. This characterization can be expressed formally as {y}={x∈X∣χ{y}(x)=1},\{y\} = \{ x \in X \mid \chi_{\{y\}}(x) = 1 \}, where χ{y}\chi_{\{y\}} denotes the associated with {y}\{y\}. This equivalence highlights how singletons emerge naturally as the loci where the attains its value of 1. More generally, within any universe XX, the singleton {y}\{y\} arises as the inverse image of the singleton {1}\{1\} under the χ{y}:X→{0,1}\chi_{\{y\}}: X \to \{0,1\}, i.e., {y}=χ{y}−1({1}).\{y\} = \chi_{\{y\}}^{-1}(\{1\}). This perspective underscores the duality between subsets of XX and functions from XX to {0,1}\{0,1\}, with singletons corresponding to functions that are 1 at exactly one point. As an example, consider X=RX = \mathbb{R}. The singleton {0}\{0\} corresponds to the indicator function χ{0}:R→{0,1}\chi_{\{0\}}: \mathbb{R} \to \{0,1\} defined by χ{0}(x)=1\chi_{\{0\}}(x) = 1 if x=0x = 0 and 00 otherwise, whose support is exactly {0}\{0\}.

In Category Theory

Terminal Objects in Set

In the category Set of sets and functions, a terminal object is an object TT such that for every set AA, there exists exactly one morphism (function) from AA to TT. Any singleton set {∗}\{*\}, where ∗* is an arbitrary element, serves as such a terminal object. The unique morphism !:A→{∗}! : A \to \{*\} is the constant function that maps every element of AA to the single element ∗* in the singleton, regardless of the choice of ∗*. This ensures existence, as one can always define such a constant map for any nonempty or empty set AA (in the empty case, it is the empty function). Uniqueness follows directly from the singleton having only one possible codomain element: any function to {∗}\{*\} must send all inputs to that sole element, leaving no other options. This property can be expressed hom-set theoretically: for any set AA, ∣Hom⁡Set(A,{∗})∣=1.|\operatorname{Hom}_{\mathbf{Set}}(A, \{*\})| = 1. The proof relies on the fact that the only function possible is the constant one, as the image must be contained in the singleton. The terminality of singletons in Set connects to broader set-theoretic properties, such as the injectivity of functions originating from singletons. Specifically, any morphism f:{∗}→Af : \{*\} \to A is injective, as it embeds the single element into AA uniquely, reflecting the universal mapping property in reverse.

General Category Perspective

In , the singleton set from the generalizes to the notion of a terminal object in an arbitrary category C\mathcal{C}, defined as an object TT such that for every object XX in C\mathcal{C}, there exists exactly one X→TX \to T. This property abstracts the "one-point" nature of singletons, where the uniqueness of the encodes the idea of a unique "point" to map to, without reference to underlying set structure. Examples illustrate this generalization across different categories. In the category Top\mathbf{Top} of topological spaces and continuous functions, a singleton space (a single point with the indiscrete topology) serves as a terminal object, as every topological space admits a unique continuous map to it via the constant function. Similarly, in the category Ab\mathbf{Ab} of abelian groups and group homomorphisms, the trivial group {0}\{0\} functions as a zero object, which is both initial and terminal, since every abelian group has a unique homomorphism to {0}\{0\} (the zero map). However, terminal objects do not always correspond directly to singleton sets in their respective categories. For instance, in poset categories—where objects are elements of a and morphisms are order relations—a terminal object is an upper bound, specifically a greatest element, which may exist without the category's structure resembling a simple one-element set. The universal property of a terminal object, namely the existence and uniqueness of morphisms into it from all other objects, provides an abstract characterization of singleton-like structures, emphasizing their in unifying diverse mathematical contexts through categorical duality and limits. If a terminal object exists, it is unique up to unique , ensuring a canonical "one-point" representative in the category.

Applications in Other Areas

Topology

In topological spaces, singletons play a key role in separation axioms, particularly those ensuring that points can be distinguished via open sets. A topological space is defined as a T_1 space (also known as a Fréchet space) if and only if every singleton set {x} is closed. This property arises because, for any distinct points x and y in the space, there exists an open set containing y but not x, allowing the complement of {x} to be expressed as a union of such open sets, making it open and thus {x} closed. Hausdorff spaces, or T_2 spaces, provide a stronger separation condition where, for any distinct points x and y, there exist disjoint open neighborhoods U containing x and V containing y. Since Hausdorff spaces satisfy the T_1 axiom, singletons {x} are closed in such spaces, with the complement X \setminus {x} being open as the union of all open neighborhoods of points other than x that avoid x. This separation ensures that points behave as isolated in terms of closure properties, facilitating continuity and arguments in analysis. In the discrete topology on a set X, every , including each singleton {x}, is both open and closed (clopen). The collection of all singletons forms a basis for this , generating all subsets as unions thereof, which underscores its coarsest structure where no nontrivial convergence occurs. For a concrete example, consider the real line \mathbb{R} equipped with the standard generated by open intervals. Here, the singleton {0} is closed because its complement \mathbb{R} \setminus {0} = (-\infty, 0) \cup (0, \infty) is a union of open intervals, hence open; however, {0} is not open, as no open interval around 0 is contained solely within it. This illustrates how singletons in metric-induced topologies like the standard one on \mathbb{R} are closed but generally not open, reflecting the space's connectedness.

Measure Theory

In measure theory, the singleton set {x}\{x\} in a measurable space (X,Σ)(X, \Sigma) is closely tied to the Dirac measure δx\delta_x, defined by δx(A)=1\delta_x(A) = 1 if x∈Ax \in A and δx(A)=0\delta_x(A) = 0 otherwise, for any A∈ΣA \in \Sigma. This measure concentrates its entire mass of 1 at the point xx, making it a fundamental example of an atomic measure supported solely on the singleton. The facilitates integration over test functions, where for a f:X→Rf: X \to \mathbb{R}, the satisfies ∫Xf dδx=f(x).\int_X f \, d\delta_x = f(x). This property arises because the measure assigns positive weight only to sets containing xx, effectively evaluating ff at that point. The can be expressed using the of {x}\{x\}, though this representation is secondary to its direct definition via set inclusion. In probability theory, the measure of singleton events varies by the underlying space: under the Lebesgue measure on R\mathbb{R}, every singleton {x}\{x\} has measure zero, reflecting the continuous nature where points carry no "length." Conversely, in discrete settings equipped with the counting measure on a finite set, the singleton {x}\{x\} has measure 1, as the counting measure assigns 1 to each individual element. This distinction underscores how singletons behave differently across measure spaces, with zero measure in diffuse continua and unit measure in atomic discrete structures.

Convexity

In a vector space over the real numbers R\mathbb{R} or the complex numbers C\mathbb{C}, a singleton set {x}\{x\} is convex. This holds trivially because the definition of a convex set requires that for any two points in the set, the line segment joining them lies entirely within the set; with only one point, there are no such pairs, satisfying the condition vacuously.

References

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