A condition for analytic structure

Abstract:

Let X be a compact Hausdorff space, A a uniform algebra on X, M the maximal ideal space of A. Let $f \in A$ and let W be a component of $C\backslash f(X)$. Suppose that, for all $\lambda \in W,{f^{ - 1}}(\lambda ) = \{ x \in M|f(x) = \lambda \}$ is at most countable. Then there is an open dense subset U of ${f^{ - 1}}(W)$ which can be given the structure of a one-dimensional complex analytic manifold so that for all $g \in A$, g is analytic on U.