 Character space of R(K)

For a compact set $K\subset \mathbb{C}$, let $\mathcal{R}(K)$ be the uniform closure on $K$ of the rational functions with no poles on $K$. Every character of $\mathcal{R}(K)$ is evaluation at a unique point of $K$, so ${\Phi }_{\mathcal{R}(K)}$ is naturally homeomorphic to $K$.