On continuous self-maps and homeomorphisms of the Golomb space
The Golomb space $\mathbb N_τ$ is the set $\mathbb N$ of positive integers endowed with the topology $τ$ generated by the base consisting of arithmetic progressions $\{a+bn\}_{n=0}^\infty$ with coprime $a,b$. We prove that the Golomb space $\mathbb N_τ$ has continuum many continuous self-maps, contains a countable disjoint family of infinite closed connected subsets, the set $Π$ of prime numbers is a dense metrizable subspace of $\mathbb N_τ$, and each homeomorphism $h$ of $\mathbb N_τ$ has the following properties: $h(1)=1$, $h(Π)=Π$ and $Π_{h(x)}=h(Π_x)$ for all $x\in\mathbb N$. Here by $Π_x$ we denote the set of prime divisors of $x$.
math.GN↗