I was pondering about the following question:
Does the statement $(\partial A)'=\partial(A'),A\subseteq\Bbb R^n$ hold?
$A'$ denotes the set of limit points of $A$ and $\partial A$ is the boundary of $A$.
It doesn't hold as I found a counterexample: take $A=\Bbb Q.$ $$(\partial \Bbb Q)'=\left(\overline{\Bbb Q}\cap\overline{\Bbb R\setminus\Bbb Q}\right)'=\Bbb R,$$ while $$\partial(\Bbb Q')=\partial\Bbb R=\overline{\Bbb R}\cap\overline{\Bbb R\setminus\Bbb R}=\Bbb R\cap\emptyset=\emptyset.$$
I think we can generalize this in higher dimensions and do some variations of this with a proper subset of $\Bbb Q$.
I was wondering how we could describe all sets for which the statement doesn't hold. Do they have to be dense in the ambient space?