The Golomb topology on a Dedekind domain and the group of units of its quotients (Q2310794)

Let \(R\) be a Dedekind domain. The sets \(a+R\), where \(I\neq (0)\) is an ideal of \(R\) and \(a\in R\) with \(\langle a,R\rangle=R\), form a base for a topology on \(R^*:=R\setminus\{0\}\). \(R^*\) with this topology is called the Golomb space \(G(R)\) of \(R\). The author proves that the only self-homeomorphisms of the Golomb space \(G(\mathbb{Z})\) are the identity and the multiplication by \(-1\). If \(R\) is a Dedekind domain contained in the algebraic closure of \(\mathbb{Q}\) such that \(G(\mathbb{Z})\simeq G(R)\), then \(R=\mathbb{Z}\). One of the results for general Dedekind domains \(R,S\) says: If \(G(R)\) and \(G(S)\) are homeomorphic, then the class group of \(R\) is torsion if and only if the class group of \(S\) is torsion.