An application of descriptive set theory to complex analysis - Caruvana, Christopher; Kallman, Robert (Topology Proceedings, 2024)
The purpose of this paper is to prove a new general result about rings of complex analytic functions. Let $Ω$ be an arbitrary nonempty open subset of the complex plane $\mathbb{C}, \mathcal{A}(Ω)$ be the set of holomorphic functions on $Ω$ viewed as a Polish ring (not a Polish algebra over $\mathbb{C}$ ) in the usual topology of uniform convergence on compact subsets of $Ω$, let $R$ be a Polish ring and let $φ: R → \mathcal{A}(Ω)$ be an abstract algebraic isomorphism. The main goal of this paper is to prove Theorem 5.7 that $φ$ is a topological isomorphism. This result may be viewed as a strengthening of the assertion that there is only one Polish ring topology on $\mathcal{A}(Ω)$. 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 $\mathcal{M}(Ω)$, the abstract field of meromorphic functions on $Ω$, cannot be made into a Polish field (Theorem 8.1).