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I am trying to show that if U a subset of R^n is open, and K a subset of U is compact, then there is a compact set such that K is a subset of D^o and D is a subset of U.

I know that since K is compact and K is a subset of U, we can use the boundary of K (represented as delta(K) = K - K^o). I think I need to find a delta that can be used to find the biggest epsilon ball at any point on K and that would be the bound for D. I'm just stuck on how I find the value of delta. Can anyone help?

acme_2020
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1 Answers1

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Hint: Let $r=d(K,\mathbb R^{n}\setminus U)$. Since $K$ is compact and $\mathbb R^{n}\setminus U$ is closed we see that $r>0$. Let $D=\{x: d(x, K) \leq r/2\}$. Then $D$ is closed and bounded, hence compact. Also $K \subseteq D$. Now $K \subseteq ${x: d(x, K) < r/2}$ and $$D=\{x: d(x, K) < r/2\}$ is an open set conatined in $D$ It follows that $K$ is contained in the interior of $D$.

Kavi Rama Murthy
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