In general topology, the Closure-Complement Problem is set as follows: From a given subset A of a topological space and counting A itself, how many sets can be constructed by applying complementation and closure successively? 

The polish mathematician Kazimierz Kuratowski answered this question in a paper published in 1922. He showed that at most 14 sets can be so constructed. In the same paper, he exhibited on the real line a subset, for which this construction yields 14 different sets. 

A proof of this Kuratowski’s Closure-Complement Theorem is the framework of the exercise proposed here:

A PDF file of the exercise