= Solution
A <compact set> $K\subset\mathbb C$ has exactly one unbounded complementary component, since its complement contains the outside of a sufficiently large disk. The connected, unbounded open set $\Omega$, being disjoint from $K$, lies in that component. In particular $\zeta$ belongs to the unbounded component.
Fill all bounded components of $\mathbb C\setminus K$ to obtain the <polynomial hull> $\widehat K$. It is a <compact set> with connected complement, and $\zeta\notin\widehat K$. Therefore $1/(z-\zeta)$ is <holomorphic> on a neighborhood of $\widehat K$. The <Runge approximation theorem> gives <polynomials> $P_n$ converging uniformly to this function on $\widehat K$, hence on $K$. This proves the requested \b[uniform polynomial approximation on $K$], even when $K$ itself has holes. If $K$ is empty, the assertion is vacuous.
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