Closure (mathematics)

From Freepedia

In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. (Thus, an object is, among other things, a set.) An object is closed if it is equal to its closure. Typical structural properties of all closure operations are:

• The closure is increasing: 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.

Examples

• In topology and related branches, the topological closure of a set.
• 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 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 algebra, the closure of a set S under a binary operation is the smallest set C(S) that includes S and is closed under the binary operation. To say that a set A is closed under an operation "×" means that for any members a, b of A, a×b is also a member of A. Examples: The set of all positive numbers is not closed under subtraction, since the difference of two positive numbers is in some cases not a positive number. The set of all positive numbers is closed under addition, since the sum of two positive numbers is in every case a positive number. The set of all integers is closed under subtraction.
• 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.