On continuous self-maps and homeomorphisms of the Golomb space

TL;DR

This paper investigates the topological properties of the Golomb space, revealing its rich structure of continuous maps, special subsets, and the behavior of its homeomorphisms, especially concerning prime divisors.

Contribution

It characterizes the structure of homeomorphisms and continuous maps of the Golomb space, highlighting the prime set's invariance and the space's complex mapping behavior.

Findings

Continuum many continuous self-maps of the Golomb space

Existence of a countable family of infinite closed connected subsets

Prime set is a dense metrizable subspace

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 $x$.