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

Abstract

The Golomb (resp. Kirch ) topology on the set ℤ ∙ of nonzero integers is generated by the base consisting of arithmetic progressions a + b ℤ = a + b n : n ∈ ℤ where a ∈ ℤ ∙ 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.