# 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 $\operatorname {cl}:\mathcal P \left({S}\right) \to \mathcal P \left({S}\right)$ 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:

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

$\blacksquare$