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Concept
Closure (mathematics)
Summary
In mathematics, a subset of a given set is closed under an operation of 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, when all input elements are in X, then all possible results are also in X. Sometimes, one may also say that X has the .
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 (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 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 in this case it is worth to add some auxiliary operations in order that all axioms become identities or purely universally quantified formulas.
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