An application of descriptive set theory to complex analysis

Abstract

The purpose of this paper is to prove a new general result about rings of complex analytic functions. Let $\Omega$ be an arbitrary nonempty open subset of the complex plane $\mathbb{C}, \mathcal{A}(\Omega)$ be the set of holomorphic functions on $\Omega$ viewed as a Polish ring (not a Polish algebra over $\mathbb{C}$ ) in the usual topology of uniform convergence on compact subsets of $\Omega$, let $R$ be a Polish ring and let $\varphi: R \rightarrow \mathcal{A}(\Omega)$ be an abstract algebraic isomorphism. The main goal of this paper is to prove Theorem 5.7 that $\varphi$ is a topological isomorphism. This result may be viewed as a strengthening of the assertion that there is only one Polish ring topology on $\mathcal{A}(\Omega)$. A special result of Bers is an easy corollary. Two additional items supplement these results, viz., that $B(\mathbb{D})$, the abstract ring of bounded analytic functions on the unit disk, cannot be made into a Polish ring (Theorem 7.5) and that $\mathcal{M}(\Omega)$, the abstract field of meromorphic functions on $\Omega$, cannot be made into a Polish field (Theorem 8.1).