Closure (mathematics)

Template:Short description Template:About 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: Template:Nowrap 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.

Definitions

Let Template:Mvar be a set equipped with one or several methods for producing elements of Template:Mvar from other elements of Template:Mvar. A subset Template:Mvar of Template:Mvar is said to be closed under these methods, if, when all input elements are in Template:Mvar, then all possible results are also in Template:Mvar. Sometimes, one may also say that Template:Mvar has the Template:Vanchor.

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 Template:Mvar of Template:Mvar, there is a smallest closed subset Template:Mvar of Template:Mvar such that $Y\subseteq X$ (it is the intersection of all closed subsets that contain Template:Mvar). Depending on the context, Template:Mvar is called the closure of Template:Mvar or the set generated or spanned by Template:Mvar.

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 ${ℂ}^n,$ 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 Template:Mvar of points is the smallest algebraic set that contains Template:Mvar.

In algebraic structures

An algebraic structure is a set equipped with operations that satisfy some axioms. These axioms may be identities. Some axioms may contain existential quantifiers $\mathrm{\exists };$ 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. A set with a single binary operation that is closed is called a magma.

In this context, given an algebraic structure Template:Mvar, a substructure of Template:Mvar is a subset that is closed under all operations of Template:Mvar, including the auxiliary operations that are needed for avoiding existential quantifiers. A substructure is an algebraic structure of the same type as Template:Mvar. 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 Template:Mvar of an algebraic structure Template:Mvar, the closure of Template:Mvar is the smallest substructure of Template:Mvar that is closed under all operations of Template:Mvar. In the context of algebraic structures, this closure is generally called the substructure generated or spanned by Template:Mvar, and one says that Template:Mvar 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.

Binary relations

A binary relation on a set Template:Mvar can be defined as a subset Template:Mvar of $A\times A,$ the set of the ordered pairs of elements of Template:Mvar. 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 follow:

Reflexivity

A relation Template:Mvar on the set Template:Mvar 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 Template:Mvar is thus $$R\cup \{(x,x)\mid x\in A\}.$$

Symmetry

Symmetry is the 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 Template:Mvar 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 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 a field that contains it.
• The radical of an ideal in a commutative ring.
• In geometry, the convex hull of a set S of points is the smallest convex set of which S is a subset.
• In formal languages, 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, the conjugate closure or normal closure of a set of group elements is the smallest normal subgroup containing the set.
• In mathematical analysis and in probability theory, 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

Template:Main 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.

Given a poset Template:Mvar whose partial order is denoted with Template:Math, a closure operator on Template:Mvar 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 ($x\le y⟹C(x)\le C(y)$).

Equivalently, a function from Template:Mvar to Template:Mvar is a closure operator if $x\le C(y)⟺C(x)\le C(y)$ for all $x,y\in S.$

An element of Template:Mvar is closed if it is its own closure, that is, if $x=C(x).$ By idempotency, an element is closed if and only if it is the closure of some element of Template:Mvar.

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 Template:Mvar to the smallest integer that is not smaller than Template:Mvar.

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 Template:Mvar 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 Template:Mvar 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 Template:Mvar such that $C(X)$ is the intersection of the closed sets containing Template:Mvar.

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".

References

• Template:MathWorld