Singleton (Mathematics)

For a sequence with one member, see 1-tuple.

In the grand tapestry of mathematics, where concepts often spiral into dizzying complexities, there exists a foundational entity of almost breathtaking simplicity: the singleton. Also rather uncreatively known as a unit set or a one-point set, it is precisely what its name implies—a set containing exactly one solitary element. It stands as the minimalist's ideal, a stark declaration of individuality in a universe often obsessed with multiplicity.

Consider, for instance, the set elegantly presented as:

{0}

This unassuming notation represents a singleton whose sole, singular inhabitant is the number 0. No more, no less. It’s a concept so fundamental, one might wonder why it even merits discussion, yet its implications ripple through nearly every branch of modern mathematical thought, much like the single, perfectly placed stone that initiates a complex wave pattern.

Properties

Within the meticulously constructed edifice of Zermelo–Fraenkel set theory (ZFC), the prevailing axiomatic framework for set theory, the very nature of a singleton is rigorously defined and safeguarded. A cornerstone of this framework is the axiom of regularity, a rather insistent rule that dictates no set can be an element of itself. This crucial axiom, like a cosmic bouncer, ensures that the boundaries between sets and their contents remain distinct, preventing paradoxical self-inclusion.

Consequently, this implies a rather obvious, yet profoundly important, distinction: a singleton is inherently and necessarily separate from the element it encapsulates. To illustrate this, consider the number 1. While it is a single entity, the set {1} is not the same thing. {1} is a container, a conceptual box, holding the number 1. The box itself is distinct from what it holds. This might seem like splitting hairs, but in the precise world of set theory, such distinctions are the bedrock of logical consistency. Similarly, the empty set (denoted ∅ or {}), which contains no elements at all, is fundamentally different from the set containing only the empty set, {{}}. The latter is, in fact, a singleton whose single element happens to be the empty set. It’s a set about nothing, contained within another set. How delightfully meta.

This principle extends to more complex structures. A set such as {{1,2,3}} might initially appear to be a collection of numbers, but upon closer inspection, it reveals itself as a quintessential singleton. Its single element is, in fact, another set: {1,2,3}. The internal complexity of that element does not detract from the fact that the outer set has only one member. It's a single, composite entity, much like a single, exquisitely wrapped gift containing multiple items.

A set is classified as a singleton if and only if its cardinality—the measure of the number of elements it contains—is precisely 1. This is, after all, the very definition of the term. In the elegant and influential von Neumann's set-theoretic construction of the natural numbers, which builds numbers purely from sets, the number 1 itself is defined as the singleton containing the number 0: {0}. This recursive definition forms a beautiful, if somewhat abstract, foundation for arithmetic, demonstrating how even the most basic numerical concepts can be rooted in the simplicity of singletons.

In the broader context of axiomatic set theory, the very existence of singletons is not a given but rather a direct consequence of more fundamental axioms. Specifically, the axiom of pairing is the workhorse here. For any given set A, applying the axiom to A and A (pairing a set with itself) asserts the existence of the set \{A, A\}, which, by the definition of a set (where element order and repetition do not matter), is precisely the same as the singleton {A}. It's a rather efficient way to conjure a single container from a single item.

The inherent simplicity of singletons also manifests in their functional properties. If A represents any arbitrary set and S is any singleton, there exists precisely one unique function mapping elements from A to S. This function, with an almost defiant lack of imagination, simply sends every element of A to the single, solitary element residing in S. This predictable behavior makes every singleton a terminal object within the category of sets, a concept we'll revisit later. They are the ultimate destination, the one true end point, making them exceptionally useful in category theory.

Furthermore, a singleton possesses the curious property that every function originating from it to any other arbitrary set is inherently injective. This means that distinct elements in the domain map to distinct elements in the codomain. Given that a singleton only has one element, this is trivially true—there are no two distinct elements to map. The only other non-singleton set that shares this rather convenient property is the empty set, which, having no elements at all, also cannot possibly map two distinct elements to the same place. It's a subtle point, highlighting the unique behaviors of the most basic sets.

In the realm of filters, every singleton set is considered an ultra prefilter. This delves into the more intricate structures used in topology and analysis. More specifically, if X is some ambient set and x is an element within X (i.e., x ∈ X), then the upward closure of the singleton {x} within X—which is formally defined as the set \{S ⊆ X: x ∈ S\} (the collection of all subsets of X that contain x)—constitutes a principal ultrafilter on X. Moreover, every principal ultrafilter on X is necessarily of this precise form, tied directly to a specific element x. This reveals a profound connection between individual elements and the most "maximal" forms of filters. The ultrafilter lemma, a non-constructive result, further implies that non-principal ultrafilters (often termed free ultrafilters) exist on every infinite set, suggesting that not all maximal filters are so neatly tied to single elements. It's a testament to the fact that even in the seemingly simple world of singletons, the universe finds ways to be complex.

Extending this, every net whose values are confined to a singleton subset X is, by definition, an ultranet within X. Nets are generalizations of sequences used to define convergence in topological spaces, and ultranets are their particularly well-behaved, "maximal" counterparts. A net restricted to a singleton is, effectively, always converging to that single point, making its behavior entirely predictable and, thus, "ultra" in its convergence properties.

Finally, in a more combinatorial context, the sequence of Bell number integers (OEIS: A000110) enumerates the total number of distinct partitions of a set. However, if one were to exclude partitions that consist solely of singletons (i.e., every element is in its own set), the resulting sequence of numbers is notably smaller (OEIS: A000296). This highlights the pervasive influence of singletons, even in counting problems, demonstrating that these minimal sets contribute significantly to combinatorial possibilities.

In category theory

Ah, category theory—the abstract language that seeks to unify mathematical structures. It's here that the humble singleton truly comes into its own, not through its complexity (of which it has none), but through its utter, undeniable simplicity. Structures built upon singletons frequently assume the distinguished roles of terminal objects or, in some cases, zero objects across a variety of categories. This isn't a coincidence; it's a fundamental consequence of their unique structure.

As previously noted, the statement that a singleton is a set with exactly one element implies that singleton sets are precisely the terminal objects within the fundamental category Set, which encompasses all sets and functions between them. A terminal object is essentially a unique "destination" object, meaning there is exactly one morphism (function) from any other object in the category to it. For a singleton, this is clear: every element from any other set simply maps to the singleton's one element. No other sets, with their pesky multiple elements or complete lack thereof, can claim such a universally accommodating role.
Moving to topology, any singleton admits only a single, unique topological space structure. This is because, in a set with only one point, the only possible subsets are the empty set and the set itself. Both of these are trivially considered "open" by definition. These singleton topological spaces then become the terminal objects in the category of topological spaces and continuous functions. Again, their simplicity makes them universal targets for continuous mappings. No other topological spaces manage to be quite so accommodating.
In the realm of algebra, any singleton can be endowed with a unique group structure. The single element within the set naturally serves as both the identity element and its own inverse. These singleton groups are not only terminal objects but also initial objects, making them zero objects in the category of groups and group homomorphisms. A zero object is simultaneously an initial object (there's a unique morphism from it to any other object) and a terminal object (there's a unique morphism from any other object to it). For groups, this means the trivial group (the singleton group) sits at the very center of the category, a testament to its fundamental role. No other groups can simultaneously be both the origin and the destination for all other groups.

Definition by indicator functions

The concept of a singleton can also be elegantly captured through the lens of indicator functions, a common tool in mathematical analysis and probability theory. Let S be a class (which, for most practical purposes here, can be considered a set) that is defined by an indicator function, typically denoted as b, which maps elements from a larger universe X to the binary values {0, 1}.

Formally, this function is expressed as:

b: X → {0,1}.

In this context, S is definitively called a singleton if and only if there exists one and only one specific element, let's call it y, within the universe X (i.e., y ∈ X) such that for every other element x also within X (i.e., x ∈ X), the indicator function b(x) evaluates to 1 if x is precisely y, and 0 otherwise.

This can be succinctly written as:

b(x) = (x=y).

This definition effectively states that the indicator function "points" to exactly one element in X by assigning it a 1 and leaving all other elements as 0. It's a precise way of identifying a set that has only one member, confirming its unique occupancy within a larger space.

Definition in Principia Mathematica

For those who enjoy a foray into the historical foundations of logic and mathematics, the definition of a singleton finds a prominent, if somewhat dense, place in the monumental work Principia Mathematica. This seminal text, authored by the indomitable duo of Alfred North Whitehead and Bertrand Russell in the early 20th century, meticulously attempted to derive all of mathematics from a set of logical axioms.

Within this work, they introduced a specific notation for the singleton. The symbol they employed was ι‘x. This rather archaic-looking glyph was designated to denote the singleton set {x}. It was a formal shorthand, a compact way to express "the set whose only member is x."

They further elaborated this by connecting it to the concept of a class. The expression ŷ(y=x) was used to denote "the class of objects identical with x," which is precisely equivalent to our modern understanding of {y : y=x}. This effectively means "the collection of all y such that y is equal to x," which, of course, can only ever be x itself.

This definition appears in the introduction of Principia Mathematica, serving to simplify the arguments presented in the main body of the text. There, it resurfaces as proposition 51.01 (found on page 357 of the work). This proposition is not merely an abstract curiosity; it serves a crucial role in their grand design.

Following this, the proposition is then leveraged to define the cardinal number 1. In their system, 1 is not just an abstract quantity but a class of sets. Specifically, it is defined as the class of all singletons:

1 = ᾱ((∃x)α = ι‘x) Df.

This means, in more accessible terms, that the number 1 is the collection (ᾱ) of all classes α such that there exists an x for which α is the singleton ι‘x. In essence, 1 is the property shared by all sets that contain exactly one element. This is presented as definition 52.01 (on page 363 of Principia Mathematica), cementing the singleton's role as a fundamental building block in the logical construction of numbers themselves. It's a testament to the profound reach of such a simple concept.

Singleton (Mathematics)

Properties

In category theory

Definition by indicator functions

Definition in Principia Mathematica

See also