  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 $x$.
