 Uniform rational approximation algebra with prescribed poles

Let $K\subset \mathbb{C}$ be compact and permit polynomial terms and finite poles in $S\subset \mathbb{C}\setminus K$, meeting every bounded complementary component. In the uniform closure $A$ of these rational functions, the coordinate $u(z)=z$ has spectrum exactly $K$. Indeed the set of $\lambda \notin K$ with $(u-\lambda {)}^{-1}\in A$ is relatively open by invertibility and relatively closed by uniform continuity of the scalar resolvent; it meets each component by a prescribed pole, or by a Neumann expansion at infinity. It is therefore all of $\mathbb{C}\setminus K$. Holomorphic functional calculus then places every function holomorphic near $K$ in $A$, proving prescribed-pole Runge approximation theorem.