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 $\{a+bn{\}}_{n=0}^{\infty }$ 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: $h(1)=1$, $h(\Pi )=\Pi$ and ${\Pi }_{h(x)}=h({\Pi }_x)$ for all $x\in \mathbb{N}$. Here by ${\Pi }_x$ we denote the set of prime divisors of $x$.

12 pages