If $U$ is an open set in $\mathbb R^n$, then there exists a sequence of open sets $\{U_i\}$, such that a.$U_i\subset \subset U_{i+1}$ (that is, ${\overline U _i}$ is compact and ${\overline U _i} \subset {U_{i + 1}}$), b.$U = \bigcup\limits_i {{U_i}}$.
My question is , can we choose all $U_i$ to have smooth boundary?
I come out this question as I was reading my PDE book, since we can in this way "approximate" a solution to an equation when $\partial U$ is rough.