= Resolvent propagation across a complementary component
The polynomially approximable resolvent points form a subset that is both open and closed relative to $\mathbb C\setminus K$. Openness follows from the uniformly convergent expansion
$$
\frac1{z-\mu}
=\sum_{n=0}^{\infty}\frac{(\mu-\lambda)^n}{(z-\lambda)^{n+1}}
$$
when $|\mu-\lambda|<\operatorname{dist}(\lambda,K)$; relative closedness follows from the uniform continuity of the resolvent as its pole varies away from $K$. Hence approximability at one point propagates throughout its connected component of $\mathbb C\setminus K$.
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