Closure (mathematics)
Encyclopedia
In mathematics
, a set is said to be closed under some operation if performance of that operation
on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction
, but the natural numbers are not: 3 and 8 are both natural numbers, but the result of 3 − 8 is not. 0 is a closed set under multiplication.
Similarly, a set is said to be closed under a collection of operations if it is closed under each of the operations individually.
A set that is closed under an operation or collection of operations is said to satisfy a closure property. Often a closure property is introduced as an axiom, which is then usually called the axiom of closure. Note that modern set-theoretic definitions usually define operations as maps between sets, so adding closure to a structure as an axiom is superfluous, though it still makes sense to ask whether subsets are closed. For example, the set of real numbers is closed under subtraction, where (as mentioned above) its subset of natural numbers is not.
When a set S is not closed under some operations, one can usually find the smallest set containing S that is closed. This smallest closed set is called the closure of S (with respect to these operations). For example, the closure under subtraction of the set of natural numbers, viewed as a subset of the real numbers, is the set of integers. An important example is that of topological closure. The notion of closure is generalized by Galois connection
, and further by monad
s.
Note that the set S must be a subset of a closed set in order for the closure operator to be defined. In the preceding example, it is important that the reals are closed under subtraction; in the domain of the natural numbers subtraction is not always defined.
The two uses of the word "closure" should not be confused. The former usage refers to the property of being closed, and the latter refers to the smallest closed set containing one that isn't closed. In short, the closure of a set satisfies a closure property.
as a set with a binary product operator obeying several axioms, including an axiom that the product of any two elements of the group is again an element. However the modern definition of an operation makes this axiom superfluous; an n-ary operation
on S is just a subset of Sn+1. By its very definition, an operator on a set cannot have values outside the set.
Nevertheless, the closure property of an operator on a set still has some utility. Closure on a set does not necessarily imply closure on all subsets. Thus a subgroup
of a group is a subset on which the binary product and the unary operation
of inversion
satisfy the closure axiom.
An operation of a different sort is that of finding the limit point
s of a subset of a topological space
(if the space is first-countable
, it suffices to restrict consideration to the limits of sequences
but in general one must consider at least limits of nets
). A set that is closed under this operation is usually just referred to as a closed set
in the context of topology
. Without any further qualification, the phrase usually means closed in this sense. Closed intervals like [1,2] = {x: 1 ≤ x ≤ 2} are closed in this sense.
A partially ordered set is downward closed (and also called a lower set
) if for every element of the set all smaller elements are also in it; this applies for example for the real intervals (−∞, p) and (−∞, p], and for an ordinal number
p represented as interval [ 0, p); every downward closed set of ordinal numbers is itself an ordinal number.
Upward closed and upper set are defined similarly.
R ⊆ S×S, and an arbitrary property P in the following way: the P closure of R is the least relation Q ⊆ S×S that contains R (i.e. R ⊆ Q) and for which property P holds (i.e. P(Q) is true). For instance, one can define the symmetric closure
as the least symmetric relation
containing R. This generalization is often encountered in the theory of rewriting
systems, where one often uses more "wordy" notions such as the reflexive transitive closure R*—the smallest preorder
containing R, or the reflexive transitive symmetric closure R≡—the smallest equivalence relation
containing R, and therefore also known as the equivalence closure. For arbitrary P and R, the P closure of R need not exist. In the above examples, these exist because reflexivity, transitivity and symmetry are closed under arbitrary intersections. In such cases, the P closure can be directly defined as the intersection of all sets with property P containing R.
Given an operation on a set X, one can define the closure C(S) of a subset S in X to be the smallest subset closed under that operation that contains S as a subset. For example, the closure of a subset of a group is the subgroup generated
by that set.
The closure of sets with respect to some operation defines a closure operator on the subsets of X. The closed sets can be determined from the closure operator; a set is closed if it is equal to its own closure. Typical structural properties of all closure operations are:
An object that is its own closure is called closed. By idempotency, an object is closed if and only if
it is the closure of some object.
These three properties define an abstract closure operator. Typically, an abstract closure acts on the class of all subsets of a set.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a set is said to be closed under some operation if performance of that operation
Operation (mathematics)
The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....
on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction
Subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
, but the natural numbers are not: 3 and 8 are both natural numbers, but the result of 3 − 8 is not. 0 is a closed set under multiplication.
Similarly, a set is said to be closed under a collection of operations if it is closed under each of the operations individually.
A set that is closed under an operation or collection of operations is said to satisfy a closure property. Often a closure property is introduced as an axiom, which is then usually called the axiom of closure. Note that modern set-theoretic definitions usually define operations as maps between sets, so adding closure to a structure as an axiom is superfluous, though it still makes sense to ask whether subsets are closed. For example, the set of real numbers is closed under subtraction, where (as mentioned above) its subset of natural numbers is not.
When a set S is not closed under some operations, one can usually find the smallest set containing S that is closed. This smallest closed set is called the closure of S (with respect to these operations). For example, the closure under subtraction of the set of natural numbers, viewed as a subset of the real numbers, is the set of integers. An important example is that of topological closure. The notion of closure is generalized by Galois connection
Galois connection
In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets . The same notion can also be defined on preordered sets or classes; this article presents the common case of posets. Galois connections generalize the correspondence...
, and further by monad
Monad (category theory)
In category theory, a branch of mathematics, a monad, Kleisli triple, or triple is an functor, together with two natural transformations...
s.
Note that the set S must be a subset of a closed set in order for the closure operator to be defined. In the preceding example, it is important that the reals are closed under subtraction; in the domain of the natural numbers subtraction is not always defined.
The two uses of the word "closure" should not be confused. The former usage refers to the property of being closed, and the latter refers to the smallest closed set containing one that isn't closed. In short, the closure of a set satisfies a closure property.
Closed sets
A set is closed under an operation if that operation returns a member of the set when evaluated on members of the set. Sometimes the requirement that the operation be valued in a set is explicitly stated, in which case it is known as the axiom of closure. For example, one may define a groupGroup (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
as a set with a binary product operator obeying several axioms, including an axiom that the product of any two elements of the group is again an element. However the modern definition of an operation makes this axiom superfluous; an n-ary operation
Operation (mathematics)
The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....
on S is just a subset of Sn+1. By its very definition, an operator on a set cannot have values outside the set.
Nevertheless, the closure property of an operator on a set still has some utility. Closure on a set does not necessarily imply closure on all subsets. Thus a subgroup
Subgroup
In group theory, given a 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, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of a group is a subset on which the binary product and the unary operation
Unary operation
In mathematics, a unary operation is an operation with only one operand, i.e. a single input. Specifically, it is a functionf:\ A\to Awhere A is a set. In this case f is called a unary operation on A....
of inversion
Inverse element
In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...
satisfy the closure axiom.
An operation of a different sort is that of finding the limit point
Limit point
In mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S...
s of a subset of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
(if the space is first-countable
First-countable space
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis...
, it suffices to restrict consideration to the limits of sequences
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
but in general one must consider at least limits of nets
Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function with domain the natural numbers, and in the context of topology, the range of this function is...
). A set that is closed under this operation is usually just referred to as a closed set
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
in the context of topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. Without any further qualification, the phrase usually means closed in this sense. Closed intervals like [1,2] = {x: 1 ≤ x ≤ 2} are closed in this sense.
A partially ordered set is downward closed (and also called a lower set
Upper set
In mathematics, an upper set of a partially ordered set is a subset U with the property that x is in U and x≤y imply y is in U....
) if for every element of the set all smaller elements are also in it; this applies for example for the real intervals (−∞, p) and (−∞, p], and for an ordinal number
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
p represented as interval [ 0, p); every downward closed set of ordinal numbers is itself an ordinal number.
Upward closed and upper set are defined similarly.
P closures of binary relations
The notion of a closure can be generalized for an arbitrary binary relationBinary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
R ⊆ S×S, and an arbitrary property P in the following way: the P closure of R is the least relation Q ⊆ S×S that contains R (i.e. R ⊆ Q) and for which property P holds (i.e. P(Q) is true). For instance, one can define the symmetric closure
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....
as the least symmetric relation
Symmetric relation
In mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a.In mathematical notation, this is:...
containing R. This generalization is often encountered in the theory of rewriting
Rewriting
In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. What is considered are rewriting systems...
systems, where one often uses more "wordy" notions such as the reflexive transitive closure R*—the smallest preorder
Preorder
In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...
containing R, or the reflexive transitive symmetric closure R≡—the smallest equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
containing R, and therefore also known as the equivalence closure. For arbitrary P and R, the P closure of R need not exist. In the above examples, these exist because reflexivity, transitivity and symmetry are closed under arbitrary intersections. In such cases, the P closure can be directly defined as the intersection of all sets with property P containing R.
Closure operator
Given an operation on a set X, one can define the closure C(S) of a subset S in X to be the smallest subset closed under that operation that contains S as a subset. For example, the closure of a subset of a group is the subgroup generated
Generating set of a group
In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...
by that set.
The closure of sets with respect to some operation defines a closure operator on the subsets of X. The closed sets can be determined from the closure operator; a set is closed if it is equal to its own closure. Typical structural properties of all closure operations are:
- The closure is increasing or extensive: the closure of an object contains the object.
- The closure is idempotent: the closure of the closure equals the closure.
- The closure is monotone, that is, if X is contained in Y, then also C(X) is contained in C(Y).
An object that is its own closure is called closed. By idempotency, an object is closed if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it is the closure of some object.
These three properties define an abstract closure operator. Typically, an abstract closure acts on the class of all subsets of a set.
Examples
- In topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and related branches, the relevant operation is taking limits. The topological closure of a set is the corresponding closure operator. The Kuratowski closure axiomsKuratowski closure axiomsIn topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition...
characterize this operator. - In linear algebraLinear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, the linear spanLinear spanIn the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...
of a set X of vectors is the closure of that set; it is the smallest subset of the vector spaceVector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
that includes X and is closed under the operation of linear combinationLinear combinationIn mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
. This subset is a subspaceLinear subspaceThe concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
. - In matroidMatroidIn combinatorics, a branch of mathematics, a matroid or independence structure is a structure that captures the essence of a notion of "independence" that generalizes linear independence in vector spaces....
theory, the closure of X is the largest superset of X that has the same rank as X. - In set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, the transitive closureTransitive closureIn mathematics, the transitive closure of a binary relation R on a set X is the transitive relation R+ on set X such that R+ contains R and R+ is minimal . If the binary relation itself is transitive, then the transitive closure will be that same binary relation; otherwise, the transitive closure...
of a binary relationBinary relationIn mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
. - In algebraAbstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, the algebraic closureAlgebraic closureIn mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
of a field. - In commutative algebraCommutative algebraCommutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
, closure operations for ideals, as integral closure and tight closureTight closureIn mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic. It was introduced by Mel Hochster and Craig Huneke in the 1980s....
. - In geometryGeometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the convex hullConvex hullIn mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....
of a set S of points is the smallest convex setConvex setIn Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
of which S is a subsetSubsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
. - In the theory of formal languageFormal languageA formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
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 theoryGroup theoryIn mathematics and 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 as groups endowed with additional operations and...
, the normal closureGroup theoryIn mathematics and 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 as groups endowed with additional operations and...
of a set of groupGroup (mathematics)In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
elements is the smallest normal subgroup containing the set.