Closure (mathematics)
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. An object is closed if it is equal to its closure. Typical structural properties of all closure operations are:
The closure of an object contains the object.
The closure of an object is closed, so that the closure of the closure equals the closure.
The closure is monotonous, that is if X is contained in Y, then also CX is contained in CY.
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 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.