nonempty

Nonempty is a mathematical and logical term used to describe a collection that contains at least one element. In set theory, a set is nonempty if it is not the empty set; the empty set, denoted ∅, contains no elements and is the unique set that fails to be nonempty. Examples include {1}, {a,b}, and the set of natural numbers, while ∅ is not nonempty.

The concept of nonemptiness is often required to avoid vacuous statements in theorems and definitions. For instance, many results assume a nonempty collection to guarantee the existence of an element with a desired property. The intersection of a nonempty family of sets may be empty in general, but under additional conditions (such as the finite intersection property or compactness) it can be nonempty.

In topology and analysis, phrases like nonempty open set, nonempty interior, and nonempty compact set are common. Many results rely on the relevant set being nonempty; removing this assumption can change the outcome or require a different formulation.

In computer science and related fields, nonempty describes data structures with length at least one, such as nonempty strings or nonempty lists. In type theory and constructive mathematics, proving that a type is nonempty provides a concrete witness to its existence.

Overall, nonempty contrasts with empty, highlighting the presence of at least one element in a collection, string, list, or type.