Let's define the operation b as b(X) = cl(X) - X. In other words, b(x) consists of all the points that are limit points of X, but are not elements of X. (There is probably a word for this but I couldn't find out what it's called, sorry.)
I'm interested what happens if you apply this recursively i.e. Xn+1 = b(Xn).
This can either lead to some kind of loop or not. There always exists at least one possible loop, which is the empty set, and many sets will eventually arrive at the empty set, where the chain will loop. But under which circumstances can there be a chain that never loops?
If this problem is too general to have a good answer, an answer for the special case of the usual topology of $\mathbb{R}$n would be satisfactory.
Here, $\mathbb{Q}$ leads to a loop of period 2. But I can't think of a chain that never loops.