\begin{definition}[Definition:Kuratowski Closure Operator/Definition 1]
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 '''Kuratowski closure axioms''' for all $A, B \subseteq S$:
{{begin-axiom}}
{{axiom | n =1
        | m = A \subseteq \map \cl A
        | rc= $\cl$ is inflationary
}}
{{axiom | n = 2
        | m = \map \cl {\map \cl A} = \map \cl A
        | rc= $\cl$ is idempotent
}}
{{axiom | n = 3
        | m = \map \cl {A \cup B} = \map \cl A \cup \map \cl B
        | rc= $\cl$ preserves binary unions
}}
{{axiom | n = 4
        | m = \map \cl \O = \O
}}
{{end-axiom}}
Note that axioms $(3)$ and $(4)$ may be replaced by the single axiom that for any finite subset $\FF$ of $\powerset S$:
:$\ds \map \cl {\bigcup \FF} = \map {\bigcup_{F \mathop \in \FF} } {\map \cl F}$
\end{definition}