Let $K$ be a compact subset of an open set $\Omega$ in $\mathbb C$. Then is there a positive number $r$ depending only on $K$ such that each closed ball of radius $r$ centered at a point in $K$ is contained in $\Omega$? How to prove this?
My attempt: For each $z \in K$, find an open disk $D(z,\ 2r_z)$ contained in $\Omega$. Then all the disks $ D(z,\ r_z)$, $z \in K$, cover the compact set $K$ and we may choose $z_1$, $z_2$, $\cdots$, $z_n$ such that $D(z_i,\ r_i)$, ($i=1,\ \cdots,\ n$), covers $K$. Taking the minimum $r$ among $r_1$, $\cdots$, $r_n$ we cover $K$ with open disks $D(z_i,\ r)$ in $\Omega$ whose closures $\bar D(z_i,r)$ are also contained in $\Omega$.
But I know this $r$ is not the desired one because there still may exist a disk of radius $r$, centered in $K$, whose closure is not contained in $\Omega$. Is my attempt a bad way, or can I remedy this?
