Equivalence of Definitions of Kuratowski Closure Operator

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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 $A, B \subseteq S$:

\((1)\)   $:$   \(\displaystyle A \subseteq \map \cl A \)             $\cl$ 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
\((3)\)   $:$   $\map \cl \O = \O$             


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:

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

$\blacksquare$