Kuratowski's closure-complement problem

In the mathematical subject of topology, Kuratowski's closure-complement problem is the question how many distinct sets can be obtained by repeatedly applying the set operations of closure and complement to a given starting subset of a topological space. The answer is: no more than 14. This result was first published by Kazimierz Kuratowski in 1922. It follows easily from the following facts that hold for any subsets S of a space, writing S^ for the closure of S, So for the interior of S, and S' for its complement:

(1) S^ ^ = S^

(2) S‘‘ = S

(3) S^ ' ^ ' ^ ' ^ = S^ ' ^

The first two are trivial. (3) follows easily from the fact that So ^ o ^ = So ^ (together with the triviality that So = S‘ ^ ‘ ).

Many variations have appeared since, especially after 1960.

A subset realizing the maximum of 14 is called a 14-set. The real numbers in their standard topology have subsets that are 14-sets. One such subset is:
 * $$(0,1) \cup(1,2)\cup\{3\}\cup\bigl([4,5]\cap\Q\bigr),$$

where $$(0,1)$$ and $$(1,2)$$ denote open intervals and $$[4,5]$$ denotes a closed interval.