Closure

In topology, the closure of a set A is the intersection of all closed sets containing A. In metric spaces, this can be also defined as the set of all limit points of the set A.

A closure operator is an abstract (category theory) form of the topological notion of closure which can be applied to any set A. It is a function cl from A to the power set of A satisfying the following conditions:

• <math>A\in cl(A)</math> (augmentation)
• If <math>B\subset A</math> then <math>cl(B)\subset cl(A)</math> (monotonicity)
• <math>cl(A)\in cl(A)</math> (idempotence)

In the category of topological spaces, this operator is isomorphic to the standard topological one.