Approximation of sets by sets with regular border

Question

What kind of conditions on a (bounded) set $E \subset \mathbb{R}^{n}$ ensure that it can be approximated from outside/inside by sets with regular border (say Lipshitz or $C^{k}$ conditions) in the sense that there exist sets $F_{k} \subset \mathbb{R}^{n}$, $k \in \mathbb{N}$, (resp. $G_{k} \subset \mathbb{R}^{n}$) with regular border such that $E \subset F_{k+1} \subset F_{k}$ (resp. $G_{k} \subset G_{k+1} \subset E$) and $\cap_{k \in \mathbb{N}} F_{k} = E$ (resp. $\cup_{k \in \mathbb{N}} G_{k} = E$)?

In particular I am interested in knowing wether a compact set can be approximated from the outside.

Any help or reference for studying this problem would be greatly appreciated.

Answer 1

In the case of a compact set $E\subset\mathbb{R}^n$, we can arrange that the covers $F_k$ are each a finite union of open balls, with the closure of the next contained in the previous $\bar F_{k+1}\subset F_k$, and $\bigcap_k F_k=E$. To get these, simply cover $E$ with suitable tiny balls centered at each point of $E$, whose closure is contained in the previous cover, and apply compactness to find a finite subcover. By smoothing the edges where the balls in $F_k$ meet, we can make the boundary $C^\infty$.