Closure (mathematics)
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In mathematics, a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of a given set is closed under an
operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Man ...
of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s 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 subset that is closed under these operations. It is often called the ''span'' (for example
linear span In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characteri ...
) or the ''generated set''.


Definitions

Let be a set equipped with one or several methods for producing elements of from other elements of .
Operations Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Man ...
and ( partial)
multivariate function In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a functio ...
are examples of such methods. If is a topological space, the limit of a sequence of elements of is an example, where there are an infinity of input elements and the result is not always defined. If is a field the roots in of a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
with coefficients in is another example where the result may be not unique.
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 say also 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\subseteq X (it is the intersection of all closed subsets that contain ). Depending on the context, is called the ''closure'' of or the set generated or spanned by . The concept 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 \Complex^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 of points is the smallest algebraic set that contains .


In algebraic structures

An algebraic structure is a set equipped with
operations Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Man ...
that satisfy some axioms. These axioms may be identities. Some axioms may contain existential quantifiers \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. 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 In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...
, 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 In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a 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 nonempty 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 group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bi ...
. In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matric ...
, the closure of a nonempty subset of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
(under vector-space operations, that is, addition and scalar multiplication) is the
linear span In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characteri ...
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.


In topology

In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
and related branches, the relevant operation is taking limits. The topological closure of a set is the corresponding closure operator. The Kuratowski closure axioms characterize this operator.


Binary relations

A
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
on a set can be defined as a subset of A\times A, the set of the ordered pairs of elements of . The notation xRy is commonly used for (x,y)\in R. Many properties or operations on relations can be used to define closures. Some of the most common ones follows. ; Reflexivity :A relation on the set is ''reflexive'' if (x,x)\in R for every x\in A. As every intersection of reflexive relations is reflexive, this defines a closure. The reflexive closure of a relation is thus R\cup \. ; Symmetry :Symmetry is the
unary operation In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation ...
on A\times A that maps (x,y) to (y,x). A relation is ''symmetric'' if it is closed under this operation, and the symmetric closure of a relation is its closure under this relation. ; Transitivity :Transitivity is defined by the partial binary operation on A\times A that maps (x,y) and (y,z) to (x,z). A relation is ''transitive'' if it is closed under this operation, and the transitive closure of a relation is its closure under this operation. 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 In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
that contains it.


Other examples

* In matroid theory, the closure of ''X'' is the largest superset of ''X'' that has the same rank as ''X''. * The transitive closure of a set. * The algebraic closure of a field. * The integral closure of an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
in a field that contains it. * The radical of an ideal in a commutative ring. * In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the convex hull of a set ''S'' of points is the smallest convex set of which ''S'' is a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
. * In
formal language In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of s ...
s, the Kleene closure of a language can be described as the set of strings that can be made by concatenating zero or more strings from that language. * In
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, the conjugate closure or normal closure of a set of group elements is the smallest normal subgroup containing the set. * In
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
and in
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, the closure of a collection of subsets of ''X'' under
countably many In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
set operations is called the σ-algebra generated by the collection.


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 In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binar ...
(poset) for inclusion. ''Closure operators'' allow generalizing the concept of closure to any partially ordered set. Given a poset whose partial order is denoted with , a ''closure operator'' on is a function C:S\to S that is ''increasing'' (x\le C(x) for all x\in S), idempotent (C(C(x))=C(x)), and monotonic (x\le y \implies C(x)\le C(y)). Equivalently, a function from to is a closure operator if x \le C(y) \iff C(x) \le C(y) for all x,y\in S. An element of if ''closed'' if it is its own closure, that is, if x=C(x). By idempotency, an element is closed
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...
it is the closure of some element of . An example of a closure operator that does not operate on subsets is given by the ceiling function, which maps every real number to the smallest integer that is not smaller than .


Closure operator vs. closed sets

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 X = \bigcap X_i is an intersection of closed sets, then C(X) must contain and be contained in every X_i. This implies C(X) = X by definition of the intersection. 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) is the intersection of the closed sets containing . This equivalence remains true for partially ordered sets with the
greatest-lower-bound property In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if eve ...
, if one replace "closet sets" by "closed elements" and "intersection" by "greatest lower bound".


Notes


References

* {{MathWorld , title=Algebraic Closure , urlname=AlgebraicClosure Set theory Closure operators Abstract algebra