\begin{definition}[Definition:Kuratowski Closure Operator/Definition 2]
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''' {{iff}} it satisfies the following axioms for all $A, B \subseteq X$:
{{begin-axiom}}
{{axiom | n = 1
        | t = $\cl$ is a closure operator
}}
{{axiom | n = 2
        | t = $\map \cl {A \cup B} = \map \cl A \cup \map \cl B$
        | rc= $\cl$ preserves binary unions
}}
{{axiom | n = 3
        | t = $\map \cl \O = \O$
}}
{{end-axiom}}
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.
\end{definition}