# Closure (mathematics)

## Operation on the subsets of a set / From Wikipedia, the free encyclopedia

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In mathematics, a subset of a given set is **closed** under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are.

Similarly, a subset is said to be closed under a *collection* of operations if it is closed under each of the operations individually.

The **closure** of a subset is the result of a closure operator applied to the subset. The *closure* of a subset under some operations is the smallest superset that is closed under these operations. It is often called the *span* (for example linear span) or the *generated set*.