closure

Closure is a concept used across disciplines to denote bringing something to a complete or settled state. In everyday usage it often refers to ending a process or resolving a concern. In mathematics and related fields, closure has precise definitions that describe how a set or a process can be completed within a given structure.

In algebra, a subset is closed under an operation if applying the operation to members yields a member of the same subset. For example, even integers are closed under addition and subtraction. A closure operator on a set assigns to each subset the smallest closed superset and satisfies extensivity (the subset is contained in its closure), idempotence (the closure of a closure is the same), and monotonicity (if A is contained in B, then the closure of A is contained in the closure of B). In field theory, the algebraic closure of a field is a minimal field extension in which every polynomial splits into linear factors; the notion also appears as the substructure generated by a given set.

In topology, the closure of a subset A is the intersection of all closed sets containing A; equivalently, it is A together with all its limit points. The closure contains A and is used to discuss convergence and continuity. The boundary of A can be described as the closure of A minus its interior.

In computer science, a closure is a function paired with its lexical environment, enabling the function to access non-local variables even when invoked outside their defining scope. In logic, closure properties describe what can be derived from axioms or rules of inference.

In psychology and everyday life, closure refers to achieving acceptance or resolution after an unresolved event, providing a sense of finality and coherence.