Definition:Kuratowski Closure Operator/Definition 2

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Let $S$ be a set.

Let $\cl: \powerset S \to \powerset S$ be a mapping from the power set of $S$ to itself.

Then $\cl$ is a Kuratowski closure operator if and only if it satisfies the following axioms for all $A, B \subseteq X$:

\((1)\)   $:$   $\cl$ is a closure operator             
\((2)\)   $:$   $\map \cl {A \cup B} = \map \cl A \cup \map \cl B$             $\cl$ preserves binary unions
\((3)\)   $:$   $\map \cl \O = \O$             

Note that axioms $(2)$ and $(3)$ may be replaced by the single axiom that for any finite subset $\FF$ of $powerset S$:

$\displaystyle \map \cl {\bigcup \FF} = \map {\bigcup_{F \mathop \in \FF} } {\map \cl F}$

That is, the axiom that $\cl$ preserves finite unions.

Also see

Source of Name

This entry was named for Kazimierz Kuratowski.