Let A be a closed subset of the real numbers. It is always possible to find a subset B of the real numbers such that A is equal to the boundary of B? Prove if true, find a counterexample if not.
I think it's false but have no idea how to find a counterexample. Any help will be much appreciated.
Given $A$, let $B$ be any countable dense subset of $A$. (Such a $B$ exists; just take one point from each nonempty set of the form $A\cap(p,q)$ where $p<q$ are rational numbers.) So the closure of $B$ is $A$. The interior of $B$ is empty, because $B$ is countable. So the boundary of $B$ is $A$.