Definition:Kuratowski Closure Operator/Definition 1

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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 Kuratowski closure axioms for all $A, B \subseteq S$:

\((1)\)   $:$   \(\displaystyle A \subseteq \operatorname{cl} \left({A}\right) \)             $\operatorname{cl}$ is inflationary
\((2)\)   $:$   \(\displaystyle \operatorname{cl} \left({ \operatorname{cl} \left({A}\right)}\right) = \operatorname{cl} \left({A}\right) \)             $\operatorname{cl}$ is idempotent
\((3)\)   $:$   \(\displaystyle \operatorname{cl} \left({A \cup B}\right) = \operatorname{cl} \left({A}\right) \cup \operatorname{cl} \left({B}\right) \)             $\operatorname{cl}$ preserves binary unions
\((4)\)   $:$   \(\displaystyle \operatorname{cl} \left({\varnothing}\right) = \varnothing \)             

Note that axioms $(3)$ and $(4)$ may be replaced by the single axiom that for any finite subset $\mathcal F$ of $\mathcal P \left({S}\right)$:

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

Also see

Source of Name

This entry was named for Kazimierz Kuratowski.