Equivalence of Definitions of Kuratowski Closure Operator

Theorem

The following definitions of the concept of Kuratowski Closure Operator are equivalent:

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

Proof

Definition 2 implies Definition 1

A closure operator, by definition, is inflationary and idempotent.

Thus it follows immediately that Definition 2 implies Definition 1.

Definition 1 implies Definition 2

Let $X$ be a set.

Let $\cl$ be a Kuratowski closure operator on $X$ by Definition 1.

By definition of closure operator, it remains to be proved that $\cl$ is increasing.

Let $A \subseteq B \subseteq X$.

Then by Definition 1 and Union with Superset is Superset:

$\map \cl B = \map \cl {A \cup B} = \map \cl A \cup \map \cl B$

By Set is Subset of Union:

$\map \cl A \subseteq \map \cl A \cup \map \cl B = \map \cl B$

$\blacksquare$