# Definition:Kuratowski Closure Operator

## 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 $A, B \subseteq S$:

 $(1)$ $:$ $\displaystyle A \subseteq \map \cl A$ $\cl$ 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 $(4)$ $:$ $\displaystyle \map \cl \O = \O$

### 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 $(3)$ $:$ $\map \cl \O = \O$

## Source of Name

This entry was named for Kazimierz Kuratowski.