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 $\operatorname {cl}:\mathcal P \left({S}\right) \to \mathcal P \left({S}\right)$ be a mapping from the power set of $S$ to itself.

Then $\operatorname{cl}$ is a Kuratowski closure operator if and only if it satisfies the following Kuratowski closure axioms for all $A, B \subseteq S$:

\((1)\)   $:$   \(\displaystyle A \subseteq \operatorname{cl} \left({A}\right) \)             $\operatorname{cl}$ is inflationary
\((2)\)   $:$   \(\displaystyle \operatorname{cl} \left({ \operatorname{cl} \left({A}\right)}\right) = \operatorname{cl} \left({A}\right) \)             $\operatorname{cl}$ is idempotent
\((3)\)   $:$   \(\displaystyle \operatorname{cl} \left({A \cup B}\right) = \operatorname{cl} \left({A}\right) \cup \operatorname{cl} \left({B}\right) \)             $\operatorname{cl}$ preserves binary unions
\((4)\)   $:$   \(\displaystyle \operatorname{cl} \left({\varnothing}\right) = \varnothing \)             

Definition 2

Let $S$ be a set.

Let $\operatorname {cl}: \mathcal P \left({S}\right) \to \mathcal P \left({S}\right)$ be a mapping from the power set of $S$ to itself.

Then $\operatorname{cl}$ is a Kuratowski closure operator if and only if it satisfies the following axioms for all $A, B \subseteq X$:

\((1)\)   $:$   $\operatorname{cl}$ is a closure operator             
\((2)\)   $:$   $\operatorname{cl} \left({A \cup B}\right) = \operatorname{cl} \left({A}\right) \cup \operatorname{cl} \left({B}\right)$             $\operatorname{cl}$ preserves binary unions
\((3)\)   $:$   $\operatorname{cl} \left({\varnothing}\right) = \varnothing$             


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 $\operatorname{cl}$ be a Kuratowski closure operator on $X$ by Definition 1.

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


Let $A \subseteq B \subseteq X$.

Then by Definition 1 and Union with Superset is Superset:

$\operatorname{cl} \left({B}\right) = \operatorname{cl} \left({A \cup B}\right) = \operatorname{cl} \left({A}\right) \cup \operatorname{cl} \left({B}\right)$

By Set is Subset of Union:

$\operatorname{cl} \left({A}\right) \subseteq \operatorname{cl} \left({A}\right) \cup \operatorname{cl} \left({B}\right) = \operatorname{cl} \left({B}\right)$

$\blacksquare$