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Encyclopedia > Closure (mathematics)

In mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 7 are both natural numbers, but the result of 3 − 7 is not. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In logic and mathematics, an operation ω is a function of the form ω : X1 × … × Xk → Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ... Please refer to Real vs. ... 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

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 real numbers are closed under subtraction, where (as mentioned above) the subset of natural numbers are 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 monads. The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ... In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets (posets). Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory. ... Look up monad in Wiktionary, the free dictionary. ...

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 group as a set with a binary product 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 operator on S is just a subset of Sn+1. By its very definition, an operator on a set cannot have values outside the set. This picture illustrates how the hours on a clock form a group under modular addition. ... In logic, mathematics, and computer science, the arity (synonyms include type, adicity, and rank) of a function or operation is the number of arguments or operands that the function takes. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ...

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. In group theory, given a group G under a binary operation *, we say that some subset H of G is 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 is a group... Unary may mean Unary numeral system Unary operator — a kind of operator that has only one operand This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ...

An operation of a different sort is that of finding the limit points 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. In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology, 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 local base. ... The limit of a sequence is one of the oldest concepts in mathematical analysis. ... In topology and related areas of mathematics a net or Moore-Smith sequence is a generalization of a sequence, intended to unify the various notions of limit and generalize them to arbitrary topological spaces. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ...

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. In mathematics, an upper set is a subset Y of a given set X such that, for all elements x and y, if x is less than or equal to y and x is an element of Y, then y is also in Y. More formally, An upper set is... In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ...

Upward closed and upper set are defined similarly.

Closure operator

Main article: 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 by that set. In mathematics, given a partially ordered set (P, ≤), a closure operator on P is a function C : P → P with the following properties: x ≤ C(x) for all x, i. ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ...

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 it is the closure of some object. In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... ↔ ⇔ ≡ logical symbols representing iff. ...

These three properties define an abstract closure operator. Typically, an abstract closure acts on the class of all subsets of a set.


  Results from FactBites:
Closure (mathematics) (1012 words)
For example, the closure of a subset of a group is the subgroup generated by that set.
The closure is idempotent: the closure of the closure equals the closure.
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 is closed under the operation of linear combination.
NationMaster - Encyclopedia: Closure (mathematics) (2699 words)
In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.
In combinatorial mathematics, a matroid is a structure that captures the essence of a notion of independence (hence independence structure) that generalizes linear independence in vector spaces.
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.
  More results at FactBites »



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