# The Golomb topology on a Dedekind domain and the group of units of its quotients

## TL;DR

This paper investigates the topological structure of Golomb spaces associated with Dedekind domains with torsion class groups, revealing invariants under homeomorphisms and characterizing self-homeomorphisms of the integers' Golomb space.

## Contribution

It establishes prime ideal preservation under homeomorphisms, introduces invariants from unit groups, and characterizes the automorphisms of the Golomb space of integers.

## Key findings

- Homeomorphisms map prime ideals to prime ideals.
- The $P$-adic topology is preserved under homeomorphisms.
- Self-homeomorphisms of $\\mathbb{Z}$'s Golomb space are only identity and multiplication by -1.

## Abstract

We study the Golomb spaces of Dedekind domains with torsion class group. In particular, we show that a homeomorphism between two such spaces sends prime ideals into prime ideals and preserves the $P$-adic topology on $R\setminus P$. Under certain hypothesis, we show that we can associate to a prime ideal $P$ of $R$ a partially ordered set, constructed from some subgroups of the group of units of $R/P^n$, which is invariant under homeomorphisms, and use this result to show that the unique self-homeomorphisms of the Golomb space of $\mathbb{Z}$ are the identity and the multiplication by $-1$. We also show that the Golomb space of any Dedekind domain contained in the algebraic closure of $\mathbb{Q}$ is not homeomorphic to the Golomb space of $\mathbb{Z}$.