Definition:Closure Operator/Power Set

Definition
Let $S$ be a set and let $\mathcal P \left({S}\right)$ be the power set of $S$.

A closure operator on $S$ is a mapping:
 * $\operatorname{cl}: \mathcal P \left({S}\right) \to \mathcal P \left({S}\right)$

which satisfies the following conditions for all sets $X, Y \subseteq S$:

Remark
A closure operator on a set $S$ in this sense is a closure operator on the power set of that set under the order-theoretic definition. In the unlikely case that these senses of "on" lead to an ambiguity, it should be resolved in the text.