Closure (mathematics)

In mathematics, a subset of a given set is closed under an operation on the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are.

Similarly, a subset is said to be closed under a collection of operations if it is closed under each of the operations individually.

The closure of a subset is the result of a closure operator applied to the subset. The closure of a subset under some operations is the smallest superset that is closed under these operations. It is often called the span (for example linear span) or the generated set.

Let S be a set equipped with one or several methods for producing elements of S from other elements of S. A subset X of S is said to be closed under these methods if an input of purely elements of X always results in an element still in X. Sometimes, one may also say that X has the closure property.

The main property of closed sets, which results immediately from the definition, is that every intersection of closed sets is a closed set. It follows that for every subset Y of S, there is a smallest closed subset X of S such that ${\displaystyle Y\subseteq X}$ (it is the intersection of all closed subsets that contain Y). Depending on the context, X is called the closure of Y or the set generated or spanned by Y.

The concepts of closed sets and closure are often extended to any property of subsets that are stable under intersection; that is, every intersection of subsets that have the property has also the property. For example, in ${\displaystyle \mathbb {C} ^{n},}$ a Zariski-closed set, also known as an algebraic set, is the set of the common zeros of a family of polynomials, and the Zariski closure of a set V of points is the smallest algebraic set that contains V.

An algebraic structure is a set equipped with operations that satisfy some axioms. These axioms may be identities. Some axioms may contain existential quantifiers ${\displaystyle \exists ;}$ in this case it is worth to add some auxiliary operations in order that all axioms become identities or purely universally quantified formulas. See Algebraic structure for details. A set with a single binary operation that is closed is called a magma.

In this context, given an algebraic structure S, a substructure of S is a subset that is closed under all operations of S, including the auxiliary operations that are needed for avoiding existential quantifiers. A substructure is an algebraic structure of the same type as S. It follows that, in a specific example, when closeness is proved, there is no need to check the axioms for proving that a substructure is a structure of the same type.