# 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 $X, Y \subseteq S$:

 $(1)$ $:$ $\operatorname{cl}$ 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 $\displaystyle \forall X \subseteq S:$ $\displaystyle \operatorname{cl} \left({\operatorname{cl} \left({X}\right)}\right) = \operatorname{cl} \left({X}\right)$

## 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.