\begin{definition}[Definition:Closure Operator/Power Set]
Let $S$ be a set.
Let $\powerset S$ denote the power set of $S$.
A '''closure operator''' on $S$ is a mapping:
:$\cl: \powerset S \to \powerset S$
which satisfies the following conditions for all sets $X, Y \subseteq S$:
{{begin-axiom}}
{{axiom | n = 1
        | lc= $\cl$ is inflationary
        | q = \forall X \subseteq S
        | ml= X
        | mo= \subseteq
        | mr= \map \cl X
}}
{{axiom | n = 2
        | lc= $\cl$ is increasing
        | q = \forall X, Y \subseteq S
        | ml= X \subseteq Y
        | mo= \implies
        | mr= \map \cl X \subseteq \map \cl Y
}}
{{axiom | n = 3
        | lc= $\cl$ is idempotent
        | q = \forall X \subseteq S
        | ml= \map \cl {\map \cl X}
        | mo= =
        | mr= \map \cl X
}}
{{end-axiom}}
\end{definition}