Definition:Kuratowski Closure Operator/Definition 2

Definition
Let $S$ be a set.

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

Then $\operatorname{cl}$ is a Kuratowski closure operator iff 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 $\mathcal P(X)$:


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

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