Definition:Kuratowski Closure Operator/Definition 2

Definition
Let $S$ be a set.

Let $\operatorname {cl}: \mathcal P \left({S}\right) \to \mathcal P \left({S}\right)$ be a mapping from the power set of $S$ to itself.

Then $\operatorname{cl}$ 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 $\mathcal F$ of $\mathcal P \left({S}\right)$:


 * $\displaystyle \operatorname{cl} \left({\bigcup \mathcal F}\right) = \bigcup_{F \mathop \in \mathcal F} \left({\operatorname{cl} \left({F}\right)}\right)$

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

Also see

 * Equivalence of Definitions of Kuratowski Closure Operator