Closed set $F$ is the boundary of any subset of $\mathbb{R}^n$ I need show that any closed subset $F\subset\mathbb{R}^n$ is the boundary of some set $A$ in $\mathbb{R}^n$.
Intuition tells me to take $A=F\setminus(\mathbb{Q}^n\cap int(F))$ and $int(F)$ is the set of interior points of $F$ but I can't prove that boundary$(A)\subset(F)$

If x is in the boundary of A the it is an adherent point of A and thus an adherent point of F.  F is closed so all adherent points of F are in F.  So x is in F.
