Riemann surfaces and the Galois correspondence

TL;DR

This paper redefines Riemann surfaces using filter bases, establishes a Galois correspondence with deck transformation groups, and extends the monodromy theorem to include non-isolated singularities.

Contribution

It introduces a new topological framework for Riemann surfaces and generalizes the monodromy theorem to broader classes of singularities.

Findings

Established a Galois correspondence for Riemann surfaces and their automorphism groups.

Redefined Riemann surfaces with a topology based on filter bases.

Extended the monodromy theorem to functions with non-isolated singularities.

Abstract

In this paper we introduce a space with some additional topologies using filter bases and renew the definition of Riemann surfaces of algebraic functions. We then present a Galois correspondence between these Riemann surfaces and their deck transformation groups. We also extend the monodromy theorem to the case that the global analytic function possesses singularities, which can be non-isolated.