How to separate a nonempty open set in euclidean space into the union of countably compact sets so that $K_n$ lies in the interior of $K_{n+1}$.

Question

I am reading the functional analysis by rudin, and I don't know how to prove this conclusion: (Actually in one dimension I can make it.)

If $\Omega$ is a nonempty open set in euclidean space, then $ \Omega$ is the union of countably many compact sets $K_n \neq \emptyset$ which can be chosen so that $K_n$ lies in the interior of $K_{n+1}$.

Answer 1

Hint: Define $K_n$ as $\{ x\in \Omega: \|x|| \leq n, d(x,\Omega^{c}) \geq \frac 1 n\}$.

Note that $K_n \subseteq \{x\in \Omega: \|x||<n+\frac 1 2, d(x,\Omega^{c} )>\frac {\frac 1 n+\frac 1 {n+1}} 2 \}\subseteq K_{n+1}$ and $\{x\in \Omega: \|x||<n+\frac 1 2, d(x,\Omega^{c} )>\frac {\frac 1 n+\frac 1 {n+1}} 2 \}$ is an open set.