Closure operator

In mathematics a closure operator is a unary operator or function on subsets of a given set which maps a subset to a containing subset with a particular property.

A closure operator on a set X is a function C on the power set of X, $$C : \mathcal{P}X \rarr \mathcal{P}X$$, satisfying:


 * $$A \subseteq B \Rightarrow CA \subseteq CB ;\,$$
 * $$A \subseteq CA ;\,$$
 * $$CCA = CA .\,$$

A topological closure operator satisfies the further property


 * $$C(A \cup B) = CA \cup CB .\,$$