4
$\begingroup$

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.

$\endgroup$

1 Answer 1

3
$\begingroup$

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 $C^\infty$.

$\endgroup$
1
  • 1
    $\begingroup$ Just a remark to Joel's answer .We can think of E as sitting inside the n sphere and just take a smooth proper function on the complement of E on the sphere and look at its regular level sets .These can be used to approximate E from the outside $\endgroup$ Nov 10, 2014 at 22:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.