Relatively prime integer topology

Let $X=\mathbb{Z}^+$, the set of positive integers. For $a,b \in X$, let $U_a(b) = \{b+na \in X : n \in \mathbb{Z}\} = (b+a\mathbb{Z})\cap X$. Then $\{U_a(b) : a,b \in X, gcd(a,b)=1\}$ forms a basis for the relatively prime integer topology on $X$.

Defined as counterexample #60 ("Relatively Prime Integer Topology") in DOI 10.1007/978-1-4612-6290-9.