# Definition:Kuratowski Closure Operator/Definition 2

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## 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.

## Also see

## Source of Name

This entry was named for Kazimierz Kuratowski.