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 $powerset S$:
- $\ds \map \cl {\bigcup \FF} = \map {\bigcup_{F \mathop \in \FF} } {\map \cl F}$
That is, the axiom that $\cl$ preserves finite unions.
Also see
Source of Name
This entry was named for Kazimierz Kuratowski.