Definition:Closure Operator/Power Set

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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 $X, Y \subseteq S$:

\((1)\)   $:$   $\operatorname{cl}$ is inflationary      \(\displaystyle \forall X \subseteq S:\) \(\displaystyle X \subseteq \operatorname{cl} \left({X}\right) \)             
\((2)\)   $:$   $\operatorname{cl}$ is increasing      \(\displaystyle \forall X, Y \subseteq S:\) \(\displaystyle X \subseteq Y \implies \operatorname{cl} \left({X}\right) \subseteq \operatorname{cl} \left({Y}\right) \)             
\((3)\)   $:$   $\operatorname{cl}$ is idempotent      \(\displaystyle \forall X \subseteq S:\) \(\displaystyle \operatorname{cl} \left({\operatorname{cl} \left({X}\right)}\right) = \operatorname{cl} \left({X}\right) \)             


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.