Closure (mathematics)

Closure (mathematics)

A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but the result of 3 − 8 is not a natural number. Another example is the set containing only the number zero, which is a closed set under multiplication.

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

A set that is closed under an operation or collection of operations is said to satisfy a closure property. Often a closure property is introduced as an axiom, which is then usually called the axiom of closure. Note that modern set-theoretic definitions usually define operations as maps between sets, so adding closure to a structure as an axiom is superfluous, though it still makes sense to ask whether subsets are closed. For example, the set of real numbers is closed under subtraction, where (as mentioned above) its subset of natural numbers is not.

When a set S is not closed under some operations, one can usually find the smallest set containing S that is closed. This smallest closed set is called the closure of S (with respect to these operations). For example, the closure under subtraction of the set of natural numbers, viewed as a subset of the real numbers, is the set of integers. An important example is that of topological closure. The notion of closure is generalized by Galois connection, and further by monads.

Note that the set S must be a subset of a closed set in order for the closure operator to be defined. In the preceding example, it is important that the reals are closed under subtraction; in the domain of the natural numbers subtraction is not always defined.

The two uses of the word "closure" should not be confused. The former usage refers to the property of being closed, and the latter refers to the smallest closed set containing one that isn't closed. In short, the closure of a set satisfies a closure property.