Closure (mathematics)

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

A closure on the subsets of a given set may be defined either by a closure operator or by a set of closed sets that is stable under intersection and includes the given set. These two definitions are equivalent.

This equivalence remains true for partially ordered sets with the greatest-lower-bound property, if one replace "closet sets" by "closed elements" and "intersection" by "greatest lower bound".