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: 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 be a set equipped with one or several methods for producing elements of from other elements of .[1] A subset of is said to be closed under these methods, if, when all input elements are in, then all possible results are also in . Sometimes, one may also say that 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 of, there is a smallest closed subset of such that
Y\subseteqX
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
\Complexn,
An algebraic structure is a set equipped with operations that satisfy some axioms. These axioms may be identities. Some axioms may contain existential quantifiers
\exists;
In this context, given an algebraic structure, a substructure of is a subset that is closed under all operations of, including the auxiliary operations that are needed for avoiding existential quantifiers. A substructure is an algebraic structure of the same type as . 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.
Given a subset of an algebraic structure, the closure of is the smallest substructure of that is closed under all operations of . In the context of algebraic structures, this closure is generally called the substructure generated or spanned by, and one says that is a generating set of the substructure.
For example, a group is a set with an associative operation, often called multiplication, with an identity element, such that every element has an inverse element. Here, the auxiliary operations are the nullary operation that results in the identity element and the unary operation of inversion. A subset of a group that is closed under multiplication and inversion is also closed under the nullary operation (that is, it contains the identity) if and only if it is non-empty. So, a non-empty subset of a group that is closed under multiplication and inversion is a group that is called a subgroup. The subgroup generated by a single element, that is, the closure of this element, is called a cyclic group.
In linear algebra, the closure of a non-empty subset of a vector space (under vector-space operations, that is, addition and scalar multiplication) is the linear span of this subset. It is a vector space by the preceding general result, and it can be proved easily that is the set of linear combinations of elements of the subset.
Similar examples can be given for almost every algebraic structures, with, sometimes some specific terminology. For example, in a commutative ring, the closure of a single element under ideal operations is called a principal ideal.
A binary relation on a set can be defined as a subset of
A x A,
xRy
(x,y)\inR.
(x,x)\inR
x\inA.
A x A
(x,y)
(y,x).
A x A
(x,y)
(y,z)
(x,z).
A preorder is a relation that is reflective and transitive. It follows that the reflexive transitive closure of a relation is the smallest preorder containing it. Similarly, the reflexive transitive symmetric closure or equivalence closure of a relation is the smallest equivalence relation that contains it.
See main article: Closure operator. In the preceding sections, closures are considered for subsets of a given set. The subsets of a set form a partially ordered set (poset) for inclusion. Closure operators allow generalizing the concept of closure to any partially ordered set.
C:S\toS
x\leC(x)
x\inS
C(C(x))=C(x)
x\ley\impliesC(x)\leC(y)
Equivalently, a function from to is a closure operator if
x\leC(y)\iffC(x)\leC(y)
x,y\inS.
An element of is closed if it is its own closure, that is, if
x=C(x).
An example is the topological closure operator; in Kuratowski's characterization, axioms K2, K3, K4' correspond to the above defining properties. An example not operating on subsets is the ceiling function, which maps every real number to the smallest integer that is not smaller than .
A closure on the subsets of a given set may be defined either by a closure operator or by a set of closed sets that is stable under intersection and includes the given set. These two definitions are equivalent.
Indeed, the defining properties of a closure operator implies that an intersection of closed sets is closed: if is an intersection of closed sets, then
C(X)
Xi.
C(X)=X
Conversely, if closed sets are given and every intersection of closed sets is closed, then one can define a closure operator such that
C(X)
This equivalence remains true for partially ordered sets with the greatest-lower-bound property, if one replace "closed sets" by "closed elements" and "intersection" by "greatest lower bound".