Increasing sequence of topologies on the set of positive integers

Abstract

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