On continuous self-maps and homeomorphisms of the Golomb space

Abstract

The Golomb space ${\mathbb N}_\tau$ is the set ${\mathbb N}$ of positive integers endowed with the topology $\tau$ generated by the base consisting of arithmetic progressions $\{a+ bn: n\ge 0\}$ with coprime $a,b$. We prove that the Golomb space ${\mathbb N}_\tau$ has continuum many continuous self-maps, contains a countable disjoint family of infinite closed connected subsets, the set $\Pi$ of prime numbers is a dense metrizable subspace of ${\mathbb N}_\tau$, and each homeomorphism $h$ of ${\mathbb N}_\tau$ has the following properties: $h(1)=1$, $h(\Pi)=\Pi$, $\Pi_{h(x)}=h(\Pi_x)$, and $h(x^{{\mathbb N}})=h(x)^{\,\mathbb N}$ for all $x\in{\mathbb N}$. Here $x^{\mathbb N}:=\{x^n\colon n\in{\mathbb N}\}$ and $\Pi_x$ denotes the set of prime divisors of $x$.