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

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∖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/Pn, which is invariant under homeomorphisms, and use this result to show that the unique self-homeomorphisms of the Golomb space of 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 Q and different from Z is not homeomorphic to the Golomb space of Z.