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# 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.

## Examples

• 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.

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