closure (mathematics)
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In mathematics, a
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 ...
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 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 n ...
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 In mathematics, a closure operator on a Set (mathematics), set ''S'' is a Function (mathematics), function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X ...
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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
) or the ''generated set''.


Definitions

Let be a set equipped with one or several methods for producing elements of from other elements of . Operations and ( partial)
multivariate function 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, ...
are examples of such methods. If is a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, the
limit of a sequence As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit ...
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 (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
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 In mathematics, an algebraic structure consists of a nonempty Set (mathematics), set ''A'' (called the underlying set, carrier set or domain), a collection of operation (mathematics), operations on ''A'' (typically binary operations such as addit ...
is a set equipped with operations that satisfy some
axioms An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...
. These axioms may be identities. Some axioms may contain
existential quantifier In predicate logic, an existential quantification is a type of Quantifier (logic), quantifier, a logical constant which is interpretation (logic), interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by t ...
s \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 In mathematics, an algebraic structure consists of a nonempty Set (mathematics), set ''A'' (called the underlying set, carrier set or domain), a collection of operation (mathematics), operations on ''A'' (typically binary operations such as addit ...
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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
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 (mathematics), 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 alge ...
, such that every element has an
inverse element In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
. Here, the auxiliary operations are the
nullary Arity () is the number of argument of a function, arguments or operands taken by a function (mathematics), function, operation (mathematics), operation or relation (mathematics), relation in logic, mathematics, and computer science. In mathematics ...
operation that results in the identity element and the
unary operation In mathematics, an unary operation is an Operation (mathematics), 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 (mathematics), function , where ...
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 In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ...
. The subgroup generated by a single element, that is, the closure of this element, is called a
cyclic group 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 (mathemati ...
. 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 matrix (mat ...
, the closure of a nonempty subset of a
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
(under vector-space operations, that is, addition and scalar multiplication) is the
linear span In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
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 In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring proper ...
, the closure of a single element under ideal operations is called a principal ideal.


In topology

In
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
and related branches, the relevant operation is taking limits. The
topological closure In topology, the closure of a subset of points in a topological space consists of all Topology glossary#P, points in together with all limit points of . The closure of may equivalently be defined as the Union (set theory), union of and its Bo ...
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 pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s 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 In mathematics, the reflexive closure of a binary relation ''R'' on a Set (mathematics), set ''X'' is the smallest reflexive relation on ''X'' that contains ''R''. For example, if ''X'' is a set of distinct numbers and ''x R y'' means "''x'' is le ...
of a relation is thus R\cup \. ;
Symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
:Symmetry is the
unary operation In mathematics, an unary operation is an Operation (mathematics), 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 (mathematics), function , where ...
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 In mathematics, the transitive closure of a binary relation on a set (mathematics), set is the smallest Relation (mathematics), relation on that contains and is Transitive relation, transitive. For finite sets, "smallest" can be taken in its ...
of a relation is its closure under this operation. A
preorder In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
that contains it.


Other examples

* In
matroid In combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely relat ...
theory, the closure of ''X'' is the largest superset of ''X'' that has the same rank as ''X''. * The
transitive closure In mathematics, the transitive closure of a binary relation on a set (mathematics), set is the smallest Relation (mathematics), relation on that contains and is Transitive relation, transitive. For finite sets, "smallest" can be taken in its ...
of a set. * The
algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field (mathematics), field ''K'' is an algebraic extension of ''K'' that is algebraically closed field, algebraically closed. It is one of many closure (mathematics), closur ...
of a field. * The
integral closure In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' ...
of an
integral domain In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
in a field that contains it. * The
radical of an ideal In ring theory, a branch of mathematics, the radical of an ideal (ring theory), ideal I of a commutative ring is another ideal defined by the property that an element x is in the radical if and only if some power of x is in I. Taking the radical of ...
in a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring proper ...
. * 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 ca ...
, the
convex hull 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 o ...
of a set ''S'' of points is the smallest
convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the Real number, reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set o ...
of which ''S'' is a
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 ...
. * In
formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (formal languages), alphabet and are well-formedness, well-formed ...
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 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 (mathematics), set. A poset consists of a set toget ...
(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 Idempotence (, ) is the property of certain operation (mathematics), operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence ...
(C(C(x))=C(x)), and
monotonic In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that preserves or reverses the given order relation, order. This concept first aro ...
(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 bicondi ...
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 In mathematics and computer science, the floor function is the function (mathematics), function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, 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, 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