Definition:Kuratowski Closure Operator/Definition 1

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

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

Note that axioms $(3)$ and $(4)$ may be replaced by the single axiom that for any finite subset $\FF$ of $\powerset S$:

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

Also see

Source of Name

This entry was named for Kazimierz Kuratowski.