Axiom:Closure Axioms/Power Set

From ProofWiki
Jump to navigation Jump to search


Let $S$ be a set.

Let $\powerset S$ denote the power set of $S$.

A closure operator on $S$ is a mapping $\cl: \powerset S \to \powerset S$ satisfying the following closure axioms 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:      \(\ds \forall X \subseteq S:\)    \(\ds \map \cl {\map \cl X} \)   \(\ds = \)   \(\ds \map \cl X \)      


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