Can we approximate any open set by sub-domains with smooth boundary?

Question

In some books, mainly about PDEs, I read that any open set can be approximated by sub-domain with smooth boundary (not just piecewise smooth). In 1 dimensional case, this seemly to be quite trivial: for any subdomain, use small open balls to cover its boundary and then mollify the connection parts. But in the higher dimensional case, I think this is not that obvious.

So the first question is: how can we approximate any open set by sub-domain with smooth boundary?

And the second question is: In what meaning the approximation is? Pointwise, i.e., we can find subdomain $D_n$ with smooth boundary such that $D_n\uparrow A$? uniformly pointwise? Or in the Lebesgue measure sense? etc.

Answer 1

The following proposition can be found in D. Daners, Domain Perturbation for Linear and Semi-Linear Boundary Value Problems, Handbook of differential equations, 2008.

Proposition 8.2.1. Let $\Omega\subset \mathbb{R}^n$ be an open set. Then there exists a sequence of bounded open sets $\Omega_n$ with $C^\infty$-boundary such that $\Omega_n\Subset \Omega_{n+1}\Subset \Omega$ for all $n\in \mathbb{N}$ and $\Omega=\bigcup_{n=1}^\infty \Omega_n$. If $\Omega$ is connected, we can choose $\Omega_n$ to be connected as well.

The proof uses Sard's lemma.