Properties of arithmetics progressions in increasing sequence of T0-topologies on the set of positive integers

In this paper we continue researches deal with increasing sequence {Tm} of T0-topologies on the set of positive integers focusing on the properties of arithmetic progressions in topologies Tm. We characterize the closures of arithmetic progressions in all topologies Tm on N. Additionally, we present the characterization of the closures of arithmetic progressions in the common division topology T on N. Moreover, for each m∈N we characterize regular open arithmetic progressions in (N,Tm) and we examine which of these spaces are semiregular.