Kuratowski's Closure-Complement Problem

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$ be a subset of $T$.

By successive applications of the operations of complement relative to $S$ and the closure, there can be as many as $14$ distinct subsets of $S$ (including $A$ itself).

Example

 * Kuratowski-Closure-Complement-Theorem.png

Proof
That there can be as many as $14$ will be demonstrated by example.

Proof of Maximum
It remains to be shown that there can be no more than $14$.

Also known as
This result is also known as Kuratowski's Closure-Complement Theorem.