Definition:Closure Operator/Power Set

A closure operator on a set $$S$$ is a mapping $$\operatorname{cl}: \mathcal{P} \left({S}\right) \to \mathcal{P} \left({S}\right)$$ from the power set of $$S$$ to itself which satisfies the following conditions for all sets $$X, Y \subseteq S$$:


 * {| border="0"


 * $$X \subseteq \operatorname{cl} \left({X}\right)$$
 * i.e. $$\operatorname{cl}$$ is extensive
 * $$X \subseteq Y \implies \operatorname{cl} \left({X}\right) \subseteq \operatorname{cl} \left({Y}\right) \quad$$
 * i.e. $$\operatorname{cl}$$ is increasing
 * $$\operatorname{cl} \left({\operatorname{cl} \left({X}\right)}\right) = \operatorname{cl} \left({X}\right)$$
 * i.e. $$\operatorname{cl}$$ is idempotent
 * }
 * $$\operatorname{cl} \left({\operatorname{cl} \left({X}\right)}\right) = \operatorname{cl} \left({X}\right)$$
 * i.e. $$\operatorname{cl}$$ is idempotent
 * }