Closure (mathematics)
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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.
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.
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.
[edit] 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.
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 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.
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.
[edit] 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.
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.
These three properties define an abstract closure operator. Typically, an abstract closure acts on the class of all subsets of a set.
[edit] 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.
- 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. This subset is a subspace.
- 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 closure of a binary relation.
- In algebra, the algebraic closure of a field.
- 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.
- In the theory of 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 normal closure of a set of group elements is the smallest normal subgroup containing the set.
