Definition:Kuratowski Closure Operator
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Definition
Definition 1
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 $A, B \subseteq S$:
\((1)\) | $:$ | \(\ds A \subseteq \map \cl A \) | $\cl$ is inflationary | ||||||
\((2)\) | $:$ | \(\ds \map \cl {\map \cl A} = \map \cl A \) | $\cl$ is idempotent | ||||||
\((3)\) | $:$ | \(\ds \map \cl {A \cup B} = \map \cl A \cup \map \cl B \) | $\cl$ preserves binary unions | ||||||
\((4)\) | $:$ | \(\ds \map \cl \O = \O \) |
Definition 2
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 | ||||||
\((3)\) | $:$ | $\map \cl \O = \O$ |
Also see
Source of Name
This entry was named for Kazimierz Kuratowski.