Increasing sequence of topologies on the set of positive integers

In this paper we deal with topologies defined on the set of positive integers. We introduce a sequence of topologies D-m that are subtopologies of Golomb's topology D. The sum of all topologies D-m is Golomb's topology. We show also that all topologies D-m are Haudsdorff but not regular on N. Further, we characterize connectedness of arithmetic progressions in each topology D-m. An immediate consequence of these characterizations is local connectedness of the space (N, D-m) for each m is an element of N. (C) 2022 Published by Elsevier B.V.