If a sequence of closed subsets $\{F_k\}$ of $\mathbb{R}^n$ constitute a cover of $\mathbb{R}^n$ then the union of their interiors is dense in $\mathbb{R}^n$.
Let $x\in \mathbb{R}^n$, then$x\in\bigcup_{k\in K}F_k$ for $K$ in $\mathbb{N}$; if $x\notin\overline{\bigcup_{k\in K}int(F_k)}$ then there exist $r>0:B_r(x)$ is disjoint from $int(F_k)\;\forall k\in K$; which means $B_r(x)$ should be covered only by boundary type points. But I can tell (but not able to state mathematical) that there is no way a countable family of boundary point sets can cover an open subset of $\mathbb{R}^n$. How can I do?