nonsingleton

Nonsingleton is a mathematical term used to describe an object that is not a singleton. In standard usage a singleton is a set containing exactly one element, so a nonsingleton typically refers to a set with at least two elements. Some authors, however, use non-singleton to mean any object whose cardinality is not equal to one, which would also include the empty set.

Examples help illustrate the idea. The set {a, b} is a nonsingleton because it has two elements. The set ∅ (the empty set) is not a singleton by the usual definition; under the broader interpretation it can also be described as nonsingleton, though this depends on the adopted convention. Likewise, any set with three or more elements is clearly a nonsingleton.

The term appears across several areas of mathematics and related disciplines. In combinatorics, one might refer to a non-singleton subset of a given set to emphasize that it contains multiple elements. In topology, a non-singleton open set has more than one point. In group theory or dynamical systems, a non-singleton orbit or a non-singleton family of objects indicates multiplicity beyond a single element.

Because conventions vary on whether the empty set is included in nonsingleton, readers should check the specific definition used in a given text. In general, the term emphasizes multiplicity: a nonsingleton is an object that is not just a single element. See also: singleton, empty set, cardinality.