Closure (mathematics)

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: 1 − 2 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 $S$ be a set equipped with one or several methods for producing elements of $S$ from other elements of $S$ .^{[note 1]} A subset $X$ of $S$ is said to be closed under these methods, if, when all input elements are in $X$ , then all possible results are also in $X$ . Sometimes, one may also say that $X$ has the closure property.

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

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 $\mathbb {C} ^{n},$ 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 $V$ of points is the smallest algebraic set that contains $V$ .

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 $\exists ;$ 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 $S$ , a substructure of $S$ is a subset that is closed under all operations of $S$ , including the auxiliary operations that are needed for avoiding existential quantifiers. A substructure is an algebraic structure of the same type as $S$ . 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 $X$ of an algebraic structure $S$ , the closure of $X$ is the smallest substructure of $S$ that is closed under all operations of $S$ . In the context of algebraic structures, this closure is generally called the substructure generated or spanned by $X$ , and one says that $X$ 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 $A$ can be defined as a subset $R$ of $A\times A,$ the set of the ordered pairs of elements of $A$ . 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 $R$ on the set $A$ 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 $R$ 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 $R$ 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.^[1]
The algebraic closure of a field.^[2]
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.^[3]
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

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 $S$ whose partial order is denoted with $\leq$ , a closure operator on $S$ is a function $C:S\to S$ that is

increasing ( $x\leq C(x)$ for all $x\in S$ ),
idempotent ( $C(C(x))=C(x)$ ), and
monotonic ( $x\leq y\implies C(x)\leq C(y)$ ).^[4]

Equivalently, a function from $S$ to $S$ is a closure operator if $x\leq C(y)\iff C(x)\leq C(y)$ for all $x,y\in S.$

An element of $S$ 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 $S$ .

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 $x$ to the smallest integer that is not smaller than $x$ .

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 $C$ implies that an intersection of closed sets is closed: if ${\textstyle X=\bigcap X_{i}}$ is an intersection of closed sets, then $C(X)$ must contain $X$ 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 $C$ such that $C(X)$ is the intersection of the closed sets containing $X$ .

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".

Notes

^ Operations and (partial) multivariate function are examples of such methods. If $S$ is a topological space, the limit of a sequence of elements of $S$ is an example, where there are an infinity of input elements and the result is not always defined. If $S$ is a field the roots in $S$ of a polynomial with coefficients in $S$ is another example where the result may be not unique.

References

^ Weisstein, Eric W. "Transitive Closure". mathworld.wolfram.com. Retrieved 2020-07-25.
^ Weisstein, Eric W. "Algebraic Closure". mathworld.wolfram.com. Retrieved 2020-07-25.
^ Bernstein, Dennis S. (2005). Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory. Princeton University Press. p. 25. ISBN 978-0-691-11802-4. ...convex hull of S, denoted by coS, is the smallest convex set containing S.
^ Birkhoff, Garrett (1967). Lattice Theory. Colloquium Publications. Vol. 25. Am. Math. Soc. p. 111. ISBN 9780821889534.

Weisstein, Eric W. "Algebraic Closure". MathWorld.

[1] Operations and (partial) multivariate function are examples of such methods. If $S$ is a topological space, the limit of a sequence of elements of $S$ is an example, where there are an infinity of input elements and the result is not always defined. If $S$ is a field the roots in $S$ of a polynomial with coefficients in $S$ is another example where the result may be not unique.

[:0-2] Weisstein, Eric W. "Transitive Closure". mathworld.wolfram.com. Retrieved 2020-07-25.

[3] Weisstein, Eric W. "Algebraic Closure". mathworld.wolfram.com. Retrieved 2020-07-25.

[4] Bernstein, Dennis S. (2005). Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory. Princeton University Press. p. 25. ISBN 978-0-691-11802-4. ...convex hull of S, denoted by coS, is the smallest convex set containing S.

[5] Birkhoff, Garrett (1967). Lattice Theory. Colloquium Publications. Vol. 25. Am. Math. Soc. p. 111. ISBN 9780821889534.

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