Closure (mathematics) - Closure Operator

Closure Operator

Given an operation on a set X, one can define the closure C(S) of a subset S of X to be the smallest subset closed under that operation that contains S as a subset. For example, the closure of a subset of a group is the subgroup generated by that set.

The closure of sets with respect to some operation defines a closure operator on the subsets of X. The closed sets can be determined from the closure operator; a set is closed if it is equal to its own closure. Typical structural properties of all closure operations are:

• The closure is increasing or extensive: the closure of an object contains the object.
• The closure is idempotent: the closure of the closure equals the closure.
• The closure is monotone, that is, if X is contained in Y, then also C(X) is contained in C(Y).

An object that is its own closure is called closed. By idempotency, an object is closed if and only if it is the closure of some object.

These three properties define an abstract closure operator. Typically, an abstract closure acts on the class of all subsets of a set.