Definition:Kuratowski Closure Operator/Definition 2

Definition
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 it satisfies the following axioms for all $A, B \subseteq X$:

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


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

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

Also see

 * Equivalence of Definitions of Kuratowski Closure Operator