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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2101.07386 |
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Table of Contents:
- 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 compact open topology, let $R$ be a Polish ring and let $φ: R \to \mathcal{A}(Ω)$ be an abstract algebraic isomorphism. The main goal of this paper is to prove Theorem 36 that $φ$ is a topological isomorphism. 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 and that $\mathcal{M}(Ω)$, the abstract field of meromorphic functions on $Ω$, cannot be made into a Polish field.