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 of ...

of a given set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...

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 In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S
:
Closure operators are de ...

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 characterized ...

) or the ''generated set''.
Definitions

Let be aset
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...

equipped with one or several methods for producing elements of from other elements of . Operations and (partial
Partial may refer to:
Mathematics
*Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant
** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial d ...

) multivariate function are examples of such methods. If is a topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...

, 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 limi ...

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
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...

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 exampl ...

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, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...

of closed sets is a closed set. It follows that for every subset of , there is a smallest closed subset of such that $Y\backslash subseteq\; X$ (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 $\backslash Complex^n,$ 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
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is ...

of a set of points is the smallest algebraic set that contains .
In algebraic structures

Analgebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...

is a set equipped with operations that satisfy some axioms. These axioms may be identities. Some axioms may contain existential quantifiers $\backslash exists;$ in this case it is worth to add some auxiliary operations in order that all axioms become identities or purely universally quantified
In mathematical logic, a universal quantification is a type of Quantification (logic), quantifier, a logical constant which is interpretation (logic), interpreted as "given any" or "for all". It expresses that a predicate (mathematical logic), pr ...

formulas. See Algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...

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, 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 o ...

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 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 matrice ...

, 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 can ...

(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 characterized ...

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 in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...

, the closure of a single element under ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...

operations is called a principal ideal
In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where ...

.
In topology

Intopology
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 ...

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

Abinary 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 sets and is a new set of ordered pairs consisting of elements in and in ...

on a set can be defined as a subset of $A\backslash times\; A,$ 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 $(x,y)\backslash in\; R.$ 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 $(x,x)\backslash in\; R$ for every $x\backslash in\; A.$ As every intersection of reflexive relations is reflexive, this defines a closure. The reflexive closure of a relation is thus $$R\backslash cup\; \backslash .$$
;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 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 o ...

on $A\backslash times\; A$ that maps $(x,y)$ to $(y,x).$ A relation is ''symmetric'' if it is closed under this operation, and the symmetric closure In mathematics, the symmetric closure of a binary relation R on a set X is the smallest symmetric relation on X that contains R.
For example, if X is a set of airports and xRy means "there is a direct flight from airport x to airport y", then the ...

of a relation is its closure under this relation.
; Transitivity
:Transitivity is defined by the partial binary operation on $A\backslash times\; A$ 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
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cas ...

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 aset
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...

.
* The algebraic closure 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'' is ...

of an integral domain in a field that contains it.
* The radical of an ideal in a commutative ring
In mathematics, a commutative ring is a 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 properties that are not ...

.
* 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
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

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 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 or a convex ...

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 of ...

.
* 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 sym ...

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 groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...

, the conjugate closure
In group theory, the normal closure of a subset S of a group G is the smallest normal subgroup of G containing S.
Properties and description
Formally, if G is a group and S is a subset of G, the normal closure \operatorname_G(S) of S is the ...

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, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...

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 ...

, 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 apartially 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 binary ...

(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 $C:S\backslash to\; S$ that is ''increasing'' ($x\backslash le\; C(x)$ for all $x\backslash in\; S$), idempotent ($C(C(x))=C(x)$), and monotonic
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...

($x\backslash le\; y\; \backslash implies\; C(x)\backslash le\; C(y)$).
Equivalently, a function from to is a closure operator if $x\; \backslash le\; C(y)\; \backslash iff\; C(x)\; \backslash le\; C(y)$ for all $x,y\backslash in\; S.$
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 bic ...

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 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 maps to the least int ...

, 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 $X\; =\; \backslash bigcap\; X\_i$ 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