# 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.

- In topology and related branches, the topological closure of a set.
- In algebra, the algebraic closure of a field.
- 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
**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 real numbers is closed under subtraction.
- In geometry, the convex hull of a set
*S* of points is the smallest convex set of which *S* is a subset.