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.



Approximation of sets by sets with regular border
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 $C^\infty$. 


