This question is to prove a different version for Runge's theorem. If $\Omega$ is open set, and $E \subset \mathbb{C}$ be such that $E$ has nonempty intersection with each bounded connected component of $\mathbb{C} - E$ which is relatively compact in $\Omega$. Let $R_E$ be the space of rational functions all whose poles are in $E$. Then $R_E$ is dense in $H(\Omega)$.
I know by Runge's theorem, for each holomorphic function $f$ in $\Omega$ we can find a sequence of rational functions with no poles in $\Omega$ which converges uniformly on compact subsets of $\Omega$, but not sure how to find a sequence of rational functions whose poles are in $E$ and $E$ is subset of $\Omega$.