# Definition:Kuratowski Closure Operator/Definition 2

## Definition

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 if and only if it satisfies the following axioms for all $A, B \subseteq X$:

 $(1)$ $:$ $\cl$ is a closure operator $(2)$ $:$ $\map \cl {A \cup B} = \map \cl A \cup \map \cl B$ $\cl$ preserves binary unions $(3)$ $:$ $\map \cl \O = \O$

Note that axioms $(2)$ and $(3)$ may be replaced by the single axiom that for any finite subset $\FF$ of $powerset S$:

$\displaystyle \map \cl {\bigcup \FF} = \map {\bigcup_{F \mathop \in \FF} } {\map \cl F}$

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

## Source of Name

This entry was named for Kazimierz Kuratowski.