= 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>.
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