Definition:Kuratowski Closure Operator/Definition 1

Definition
Let $X$ 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 Kuratowski closure axioms for all $A, B \subseteq X$:

Note that axioms $(3)$ and $(4)$ may be replaced by the single axiom that for any finite subset, $\mathcal F$, of $\mathcal P(X)$:
 * $\displaystyle \operatorname{cl} \left({ \bigcup \mathcal F }\right) = \bigcup_{S \in \mathcal F} \left({ \operatorname{cl}(S) }\right)$