Definition:Closure Operator/Power Set

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Definition

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 closure axioms as follows for all sets $X, Y \subseteq S$:

\((\text {cl} 1)\)   $:$   $\cl$ is inflationary:      \(\ds \forall X \subseteq S:\)    \(\ds X \)   \(\ds \subseteq \)   \(\ds \map \cl X \)      
\((\text {cl} 2)\)   $:$   $\cl$ is increasing:      \(\ds \forall X, Y \subseteq S:\)    \(\ds X \subseteq Y \)   \(\ds \implies \)   \(\ds \map \cl X \subseteq \map \cl Y \)      
\((\text {cl} 3)\)   $:$   $\cl$ is idempotent:      \(\ds \forall X \subseteq S:\)    \(\ds \map \cl {\map \cl X} \)   \(\ds = \)   \(\ds \map \cl X \)      


Sources

although at this point he does not name this operator, just describes it