# Definition:Kuratowski Closure Operator/Definition 2

## Definition

Let $S$ be a set.

Let $\operatorname {cl}: \mathcal P \left({S}\right) \to \mathcal P \left({S}\right)$ be a mapping from the power set of $S$ to itself.

Then $\operatorname{cl}$ is a Kuratowski closure operator if and only if it satisfies the following axioms for all $A, B \subseteq X$:

 $(1)$ $:$ $\operatorname{cl}$ is a closure operator $(2)$ $:$ $\operatorname{cl} \left({A \cup B}\right) = \operatorname{cl} \left({A}\right) \cup \operatorname{cl} \left({B}\right)$ $\operatorname{cl}$ preserves binary unions $(3)$ $:$ $\operatorname{cl} \left({\varnothing}\right) = \varnothing$

Note that axioms $(2)$ and $(3)$ may be replaced by the single axiom that for any finite subset $\mathcal F$ of $\mathcal P \left({S}\right)$:

$\displaystyle \operatorname{cl} \left({\bigcup \mathcal F}\right) = \bigcup_{F \mathop \in \mathcal F} \left({\operatorname{cl} \left({F}\right)}\right)$

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

## Source of Name

This entry was named for Kazimierz Kuratowski.