Closure (mathematics): Difference between revisions

Content deleted Content added

Inline

Latest revision as of 22:36, 14 August 2024

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. A set with a single binary operation that is closed is called a magma.

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.

[note 1]

[1]

[2]

[3]

[4]

@@ Line 1: / Line 1: @@
-{{Short description|Operation on sets, or property of an operation}}
+{{Short description|Operation on the subsets of a set}}
+{{About|closures in general|the specific use in topology|Closure (topology)|the use in computer science|closure (computer science)}}
-{{cleanup|reason=The two first sections duplicate each other, and the article confuses two related but different concepts, the property of being closed under an operation, and the result of a [[closure operator]].|date=December 2021}}
+In mathematics, a [[subset]] of a given [[set (mathematics)|set]] is '''closed''' under an [[Operation (mathematics)|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]]s are closed under addition, but not under subtraction: {{nowrap|1 − 2}} is not a natural number, although both 1 and 2 are.
-{{About|closures in general|the specific use in topology|Closure (topology)}}
-In mathematics, a [[subset]] of a given [[set (mathematics)|set]] is '''closed''' under an [[Operation (mathematics)|operation]] of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the positive [[integer]]s are closed under addition, but not under subtraction: {{nowrap|1 − 2}} is not a positive integer even though both 1 and 2 are positive integers.
 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 subset that is closed under these operations. It is often called the ''span'' (for example [[linear span]]) or the ''generated set''.
+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==
-==Basic properties==
+Let {{mvar|S}} be a [[set (mathematics)|set]] equipped with one or several methods for producing elements of {{mvar|S}} from other elements of {{mvar|S}}.<ref group="note">[[Operation (mathematics)|Operations]] and ([[partial function|partial]]) [[multivariate function]] are examples of such methods. If {{mvar|S}} is a [[topological space]], the [[limit of a sequence]] of elements of {{mvar|S}} is an example, where there are an infinity of input elements and the result is not always defined. If {{mvar|S}} is a [[field (mathematics)|field]] the [[roots of a polynomial|roots]] in {{mvar|S}} of a [[polynomial]] with coefficients in {{mvar|S}} is another example where the result may be not unique.</ref>
-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'''.  Modern set-theoretic definitions usually define operations as maps between sets, so adding closure to a structure as an axiom is superfluous; however in practice operations are often defined initially on a superset of the set in question and a closure proof is required to establish that the operation applied to pairs from that set only produces members of that set. For example, the set of even integers is closed under addition, but the set of odd integers is not.
+A subset {{mvar|X}} of {{mvar|S}} is said to be ''closed'' under these methods, if, when all input elements are in {{mvar|X}}, then all possible results are also in {{mvar|X}}. Sometimes, one may also say that {{mvar|X}} has the '''{{vanchor|closure property}}'''.
+The main property of closed sets, which results immediately from the definition, is that every [[set intersection|intersection]] of closed sets is a closed set. It follows that for every subset {{mvar|Y}} of {{mvar|S}}, there is a smallest closed subset {{mvar|X}} of {{mvar|S}} such that <math>Y\subseteq X</math> (it is the intersection of all closed subsets that contain {{mvar|Y}}). Depending on the context, {{mvar|X}} is called the ''closure'' of {{mvar|Y}} or the set [[generating set|generated]] or [[span (linear algebra)|spanned]] by {{mvar|Y}}.
-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).<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Set Closure|url=https://mathworld.wolfram.com/SetClosure.html|access-date=2020-07-25|website=mathworld.wolfram.com|language=en|quote=The closure of a set A is the smallest closed set containing A}}</ref>  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 (category theory)|monad]]s.
+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 <math>\Complex^n,</math> 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 {{mvar|V}} of points is the smallest algebraic set that contains {{mvar|V}}.
-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.
+===In algebraic structures===
-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 may not be closed.  In short, the closure of a set satisfies a closure property.
+An [[algebraic structure]] is a set equipped with [[operation (mathematics)|operations]] that satisfy some [[axioms]]. These axioms may be [[identity (mathematics)|identities]]. Some axioms may contain [[existential quantifier]]s <math>\exists;</math> 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 (algebra)|magma]].
+In this context, given an algebraic structure {{mvar|S}}, a [[substructure (mathematics)|substructure]] of {{mvar|S}} is a subset that is closed under all operations of {{mvar|S}}, including the auxiliary operations that are needed for avoiding existential quantifiers. A substructure is an algebraic structure of the same type as {{mvar|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.
-==Closed sets==
-A set is closed under an operation if the operation returns a member of the set when evaluated on members of the set.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Set Closure|url=https://mathworld.wolfram.com/SetClosure.html|access-date=2020-07-25|website=mathworld.wolfram.com|language=en|quote=A set S and a binary operator * are said to exhibit closure if applying the binary operator to two elements S returns a value which is itself a member of S.}}</ref>  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 [[group (mathematics)|group]] 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 [[arity|''n''-ary]] [[operation (mathematics)|operation]] on ''S'' is just a subset of ''S''<sup>''n''+1</sup>.  By its very definition, an operator on a set cannot have values outside the set.
+Given a subset {{mvar|X}} of an algebraic structure {{mvar|S}}, the closure of {{mvar|X}} is the smallest substructure of {{mvar|S}} that is closed under all operations of {{mvar|S}}. In the context of algebraic structures, this closure is generally called the substructure ''generated'' or ''spanned'' by {{mvar|X}}, and one says that {{mvar|X}} is a [[generating set]] of the substructure.
-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 [[inverse element|inversion]] satisfy the closure axiom.
+For example, a [[group (mathematics)|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]].
-An operation of a different sort is that of finding the [[limit point]]s of a subset of a [[topological space]].  A set that is closed under this operation is usually referred to as a [[closed set]] in the context of [[topology]].  Without any further qualification, the phrase usually means closed in this sense.   [[Closed interval]]s like [1,2] = {''x'' : 1 ≤ ''x'' ≤ 2} are closed in this sense.
+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 combination]]s of elements of the subset.
-A subset of a partially ordered set is a '''downward closed set''' (also called a [[upper set|lower set]]) if for every element of the subset, all smaller elements are also in the subset. This applies for example to the real intervals {{open-open|−∞, ''p''}} and {{open-closed|−∞, ''p''}}, and for an [[ordinal number]] ''p'' represented as interval {{closed-open|0, ''p''}}. Every downward closed set of ordinal numbers is itself an ordinal number. '''Upward closed sets''' (also called upper sets) are defined similarly.
+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 (ring theory)|ideal]] operations is called a [[principal ideal]].
-==Examples==
-* In [[topology]] 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==
-* In [[linear algebra]], the [[linear span]] of a set ''X'' of vectors is the '''closure''' of that set; it is the smallest subset of the [[vector space]] that includes ''X'' and is closed under the operation of [[linear combination]].  This subset is a [[linear subspace|subspace]].
+A [[binary relation]] on a set {{mvar|A}} can be defined as a subset {{mvar|R}} of <math>A\times A,</math> the set of the [[ordered pair]]s of elements of {{mvar|A}}. The notation <math>xRy</math> is commonly used for <math>(x,y)\in R.</math> Many properties or operations on relations can be used to define closures. Some of the most common ones follow:
+;[[Reflexive relation|Reflexivity]]
+:A relation {{mvar|R}} on the set {{mvar|A}} is ''reflexive'' if <math>(x,x)\in R</math> for every <math>x\in A.</math> As every intersection of reflexive relations is reflexive, this defines a closure. The [[reflexive closure]] of a relation {{mvar|R}} is thus <math display = block>R\cup \{(x,x)\mid x\in A\}.</math>
+;[[Symmetric relation|Symmetry]]
+:Symmetry is the [[unary operation]] on <math>A\times A</math> that maps <math>(x,y)</math> to <math>(y,x).</math> A relation is ''symmetric'' if it is closed under this operation, and the [[symmetric closure]] of a relation {{mvar|R}} is its closure under this relation.
+;[[Transitive relation|Transitivity]]
+:Transitivity is defined by the [[partial operation|partial binary operation]] on <math>A\times A</math> that maps <math>(x,y)</math> and <math>(y,z)</math> to <math>(x,z).</math> 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''.
-* In [[set theory]], the [[Transitive_set#Transitive_closure|transitive closure]] of a [[set (mathematics)|set]].<ref name=":0" />
+* The [[Transitive set#Transitive closure|transitive closure]] of a [[set (mathematics)|set]].<ref name=":0">{{Cite web |last=Weisstein|first=Eric W. |title=Transitive Closure |url=https://mathworld.wolfram.com/TransitiveClosure.html | access-date=2020-07-25 |website=mathworld.wolfram.com | language=en}}</ref>
-* In [[set theory]], the [[transitive closure]] of a [[binary relation]].<ref name=":0">{{Cite web|last=Weisstein|first=Eric W.|title=Transitive Closure|url=https://mathworld.wolfram.com/TransitiveClosure.html|access-date=2020-07-25|website=mathworld.wolfram.com|language=en}}</ref>
+* The [[algebraic closure]] of a [[field (algebra)|field]].<ref>{{Cite web| last=Weisstein|first=Eric W.|title=Algebraic Closure| url=https://mathworld.wolfram.com/AlgebraicClosure.html|access-date=2020-07-25| website=mathworld.wolfram.com| language=en}}</ref>
+* The [[integral closure]] of an [[integral domain]] in a [[field (mathematics)|field]] that contains it.
-* In [[abstract algebra|algebra]], the [[algebraic closure]] of a [[field (algebra)|field]].<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Algebraic Closure|url=https://mathworld.wolfram.com/AlgebraicClosure.html|access-date=2020-07-25|website=mathworld.wolfram.com|language=en}}</ref>
+* The [[radical of an ideal]] in a [[commutative ring]].
-* In [[commutative algebra]], closure operations for ideals, as [[integral closure]] and [[tight closure]].
-* In [[geometry]], the [[convex hull]] of a set ''S'' of points is the smallest [[convex set]] of which ''S'' is a [[subset]].<ref>{{Cite book|last=Bernstein|first=Dennis S.|url=https://books.google.com/books?id=pmNRPwOFHKoC&q=smallest+convex+set&pg=PA25|title=Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory|date=2005|publisher=Princeton University Press|isbn=978-0-691-11802-4|pages=25|language=en|quote=...convex hull of S, denoted by coS, is the smallest convex set containing S.}}</ref>
+* In [[geometry]], the [[convex hull]] of a set ''S'' of points is the smallest [[convex set]] of which ''S'' is a [[subset]].<ref>{{Cite book| last=Bernstein|first=Dennis S.| url=https://books.google.com/books?id=pmNRPwOFHKoC&q=smallest+convex+set&pg=PA25|title=Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory| date=2005|publisher=Princeton University Press| isbn=978-0-691-11802-4|pages=25| language=en|quote=...convex hull of S, denoted by coS, is the smallest convex set containing S.}}</ref>
 * In [[formal language]]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]], the [[conjugate closure]] or normal closure of a set of [[group (mathematics)|group]] elements is the smallest normal subgroup containing the set.
@@ Line 40: / Line 52: @@
 ==Closure operator==
-:{{Main|closure operator}}
+{{Main|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 [[set inclusion|inclusion]]. ''Closure operators'' allow generalizing the concept of closure to any partially ordered set.
-Given an operation on a set ''X'', one can define the closure ''C''(''S'') of a subset ''S'' of ''X'' to be the smallest subset closed under that operation that contains ''S'' as a subset, if any such subsets exist.  Consequently, ''C''(''S'') is the intersection of all closed sets containing ''S''. For example, the closure of a subset of a group is the subgroup [[Generating set of a group|generated]] by that set.
+Given a poset {{mvar|S}} whose partial order is denoted with {{math|≤}}, a ''closure operator'' on {{mvar|S}} is a [[function (mathematics)|function]] <math>C:S\to S</math> that is
-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: <ref>{{cite book |last=Birkhoff |first=Garrett |author-link=Garrett Birkhoff |date=1967 |title=Lattice Theory |volume=25 |publisher=Am. Math. Soc. |series=Colloquium Publications |isbn=9780821889534 |page=111}}</ref>
+* ''increasing'' (<math>x\le C(x)</math> for all <math>x\in S</math>),
+* [[idempotent]] (<math>C(C(x))=C(x)</math>), and
+* [[monotonic]] (<math>x\le y \implies C(x)\le C(y)</math>).<ref>{{cite book |last=Birkhoff |first=Garrett |author-link=Garrett Birkhoff |date=1967 |title=Lattice Theory |volume=25 |publisher=Am. Math. Soc. |series=Colloquium Publications |isbn=9780821889534 |page=111}}</ref>
+Equivalently, a function from {{mvar|S}} to {{mvar|S}} is a closure operator if <math>x \le C(y) \iff C(x) \le C(y)</math> for all <math>x,y\in S.</math>
-*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]] it is the closure of some object.
+An element of {{mvar|S}} is ''closed'' if it is its own closure, that is, if <math>x=C(x).</math> By idempotency, an element is closed [[if and only if]] it is the closure of some element of {{mvar|S}}.
+An example is the [[topological closure]] operator; in [[Kuratowski closure axioms|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 {{mvar|x}} to the smallest integer that is not smaller than {{mvar|x}}.
-These three properties define an '''abstract closure operator'''.  Typically, an abstract closure acts on the class of all subsets of a set.
+===Closure operator vs. closed sets===
-If ''X'' is contained in a set closed under the operation then every subset of ''X'' has a closure.
+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 {{mvar|C}} implies that an intersection of closed sets is closed: if <math display = inline>X = \bigcap X_i</math> is an intersection of closed sets, then <math>C(X)</math> must contain {{mvar|X}} and be contained in every <math>X_i.</math> This implies <math>C(X) = X</math> by definition of the intersection.
-==Binary relation closures==
-Consider first [[homogeneous relation]]s ''R'' ⊆ ''A'' × ''A''. If a relation ''S'' satisfies ''aSb'' ⇒ ''bSa'', then it is a [[symmetric relation]]. An arbitrary homogeneous relation ''R'' may not be symmetric but it is always contained in some symmetric relation: ''R'' ⊆ ''S''. The operation of finding the ''smallest'' such ''S'' corresponds to a closure operator called [[symmetric closure]].
+Conversely, if closed sets are given and every intersection of closed sets is closed, then one can define a closure operator {{mvar|C}} such that <math>C(X)</math> is the intersection of the closed sets containing {{mvar|X}}.
-A [[transitive relation]] ''T'' satisfies ''aTb'' ∧ ''bTc'' ⇒ ''aTc''. An arbitrary homogeneous relation ''R'' may not be transitive but it is always contained in some transitive relation: ''R'' ⊆ ''T''. The operation of finding the ''smallest'' such ''T'' corresponds to a closure operator called [[transitive closure]].
+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".
-Among [[heterogeneous relation]]s there are properties of ''difunctionality'' and ''contact'' which lead to '''difunctional closure''' and '''contact closure'''.<ref>{{cite encyclopedia |first=Gunter |last=Schmidt |author-link=Gunther Schmidt |encyclopedia=Encyclopedia of Mathematics and its Applications |title=Relational Mathematics |publisher=[[Cambridge University Press]] |year=2011 |isbn=978-0-521-76268-7 |pages=169, 227 |volume=132 }}</ref> The presence of these closure operators in binary relations leads to [[topology]] since open-set axioms may be replaced by [[Kuratowski closure axioms]]. Thus each property ''P'', symmetry, transitivity, difunctionality, or contact corresponds to a relational topology.<ref>{{cite book |first1=Gunter |last1=Schmidt |first2=M. |last2=Winter |series=[[Lecture Notes in Mathematics]] |volume=2208 |title=Relational Topology |publisher=Springer Verlag |year=2018 |isbn=978-3-319-74451-3 }}</ref>
-In the theory of [[rewriting]] systems, one often uses more wordy notions such as the '''reflexive transitive closure''' ''R<sup>*</sup>''&mdash;the smallest [[preorder]] containing ''R'', or the '''reflexive transitive symmetric closure''' ''R''<sup>≡</sup>&mdash;the [[Equivalence_relation#Comparing_equivalence_relations|smallest equivalence relation]] containing ''R'', and therefore also known as the '''equivalence closure'''. When considering a particular [[term algebra]], an equivalence relation that is compatible with all operations of the algebra <ref group=note>that is, such that e.g. ''xRy'' implies ''f''(''x'',''x''<sub>2</sub>) ''R'' ''f''(''y'',''x''<sub>2</sub>) and ''f''(''x''<sub>1</sub>,''x'') ''R'' ''f''(''x''<sub>1</sub>,''y'') for any binary operation ''f'' and arbitrary ''x''<sub>1</sub>,''x''<sub>2</sub> ∈ ''S''</ref> is called a [[Congruence relation#Universal algebra|congruence relation]]. The '''congruence closure''' of ''R'' is defined as the smallest congruence relation containing ''R''.
-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''.<ref>{{cite book |last1=Baader |first1=Franz |author-link1=Franz Baader |last2=Nipkow |first2=Tobias |author-link2=Tobias Nipkow |date=1998 |title=Term Rewriting and All That |publisher=Cambridge University Press |pages=8–9 |url=https://books.google.com/books?id=N7BvXVUCQk8C&q=closure |isbn=9780521779203}}</ref>
-Some important particular closures can be constructively obtained as follows:
-* ''cl''<sub>ref</sub>(''R'') = ''R'' ∪ { ⟨''x'',''x''⟩ : ''x'' ∈ ''S'' } is the [[reflexive closure]] of ''R'',
-* ''cl''<sub>sym</sub>(''R'') = ''R'' ∪ { ⟨''y'',''x''⟩ : ⟨''x'',''y''⟩ ∈ ''R'' } is its symmetric closure,
-* ''cl''<sub>trn</sub>(''R'') = ''R'' ∪ { ⟨''x''<sub>1</sub>,''x''<sub>''n''</sub>⟩ : ''n'' >1 ∧ ⟨''x''<sub>1</sub>,''x''<sub>2</sub>⟩, ..., ⟨''x''<sub>''n''-1</sub>,''x''<sub>''n''</sub>⟩ ∈ ''R'' } is its [[transitive closure]],
-* ''cl''<sub>emb,Σ</sub>(''R'') = ''R'' ∪ { ⟨''f''(''x''<sub>1</sub>,…,''x''<sub>''i''-1</sub>,''x''<sub>''i''</sub>,''x''<sub>''i''+1</sub>,…,''x''<sub>''n''</sub>), ''f''(''x''<sub>1</sub>,…,''x''<sub>''i''-1</sub>,''y'',''x''<sub>''i''+1</sub>,…,''x''<sub>''n''</sub>)⟩ : ⟨''x''<sub>''i''</sub>,''y''⟩ ∈ ''R'' ∧ ''f'' ∈ Σ ''n''-ary ∧ 1 ≤ ''i'' ≤ ''n'' ∧ ''x''<sub>1</sub>,...,''x''<sub>''n''</sub> ∈ ''S'' } is its embedding closure with respect to a given set Σ of operations on ''S'', each with a fixed arity.
-The relation ''R'' is said to have closure under some ''cl''<sub>xxx</sub>, if ''R'' = ''cl''<sub>xxx</sub>(''R''); for example ''R'' is called symmetric if ''R'' = ''cl''<sub>sym</sub>(''R'').
-Any of these four closures preserves symmetry, i.e., if ''R'' is symmetric, so is any ''cl''<sub>xxx</sub>(''R''). <ref group=note>formally: if ''R'' = ''cl''<sub>sym</sub>(''R''), then ''cl''<sub>xxx</sub>(''R'') = ''cl''<sub>sym</sub>(''cl''<sub>xxx</sub>(''R''))</ref>
-Similarly, all four preserve reflexivity.
-Moreover, ''cl''<sub>trn</sub> preserves closure under ''cl''<sub>emb,Σ</sub> for arbitrary Σ.
-As a consequence, the equivalence closure of an arbitrary binary relation ''R'' can be obtained as ''cl''<sub>trn</sub>(''cl''<sub>sym</sub>(''cl''<sub>ref</sub>(''R''))), and the congruence closure with respect to some Σ can be obtained as ''cl''<sub>trn</sub>(''cl''<sub>emb,Σ</sub>(''cl''<sub>sym</sub>(''cl''<sub>ref</sub>(''R'')))). In the latter case, the nesting order does matter; e.g. if ''S'' is the set of terms over Σ = { ''a'', ''b'', ''c'', ''f'' } and ''R'' = { ⟨''a'',''b''⟩, ⟨''f''(''b''),''c''⟩ }, then the pair ⟨''f''(''a''),''c''⟩ is contained in the congruence closure ''cl''<sub>trn</sub>(''cl''<sub>emb,Σ</sub>(''cl''<sub>sym</sub>(''cl''<sub>ref</sub>(''R'')))) of ''R'', but not in the relation ''cl''<sub>emb,Σ</sub>(''cl''<sub>trn</sub>(''cl''<sub>sym</sub>(''cl''<sub>ref</sub>(''R'')))).
-==See also==
-*[[Open set]]
-*[[Clopen set]]
 ==Notes==
 {{Reflist|group=note}}
 ==References==
 {{Portal|Mathematics}}