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 C, A(Ω) be the set of holomorphic functions on Ω viewed as a Polish ring (not a Polish algebra over C ) in the usual topology of uniform convergence on compact subsets of Ω, let R be a Polish ring and let φ: R → 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 A(Ω). A special result of Bers is an easy corollary. Two additional items supplement these results, viz., that B(D), the abstract ring of bounded analytic functions on the unit disk, cannot be made into a Polish ring (Theorem 7.5) and that M(Ω), the abstract field of meromorphic functions on Ω, cannot be made into a Polish field (Theorem 8.1).