Homeomorphisms of the space of non-zero integers with the Kirch topology

TL;DR

This paper proves that the space of nonzero integers with the Kirch topology has exactly two self-homeomorphisms, extending a known result from the Golomb topology to the Kirch topology.

Contribution

It establishes that the Kirch topology on nonzero integers admits only two self-homeomorphisms, similar to the Golomb topology, revealing a rigidity property.

Findings

The space with Kirch topology has exactly two self-homeomorphisms.

This result parallels the known property for the Golomb topology.

The paper extends the understanding of topological symmetries in arithmetic progressions.

Abstract

The $Golomb$ (resp. $Kirch$) topology on the set $\mathbb Z^\bullet$ of nonzero integers is generated by the base consisting of arithmetic progressions $a+b\mathbb Z=\{a+bn:n\in\mathbb Z\}$ where $a\in\mathbb Z^\bullet$ and $b$ is a (square-free) number, coprime with $a$. In 2019 Dario Spirito proved that the space of nonzero integers endowed with the Golomb topology admits only two self-homeomorphisms. In this paper we prove an analogous fact for the space of nonzero integers endowed with the Kirch topology: it also admits exactly two self-homeomorphisms.