Connections between connected topological spaces on the set of positive integers (Q352735)

Recently the author proved [Demonstr. Math. 43, No. 4, 899--909 (2010; Zbl 1303.11021)] that in the topology with the base \(\{\{an+b\}: \gcd(a,b)=1\}\) the arithmetic progression \(\{an+b\}\) is connected if and only if \(\Theta(a)\subset\Theta(b)\), where \(\Theta(m)\) stands for the set of all prime factors of positive integer \(m\). Taking such arithmetic progressions as a base of a new topology, the author studies topological properties of this new topology. For instance, an arithmetic progression \(\{an+b\}\) is connected in this topology if and only if \(\gcd(a,b)=1\). Then the author examines the question of connectedness of arithmetic progressions in the divisor topology introduced by \textit{G. B. Rizza} [Riv. Mat. Univ. Parma, V. Ser. 2, 179--185 (1993; Zbl 0834.11006)] and the mutual relationship between these two topologies. Reviewer's remark: In studying topologies on the set of positive integers defined by arithmetic progressions \(\{an+b\}\) the author often forgot to impose the assumption \(b<a\) to clarify the definitions.