Abstract: A topological space X is totally Brown if for each n \in \mathbb{N} \setminus \{1\} and every nonempty open subsets U_1,U_2,\ldots,U_n of X we have {\rm cl}_X(U_1) \cap {\rm cl}_X(U_2) \cap~\cdots~\cap{\rm cl}_X(U_n) \ne \emptyset. Totally Brown spaces are connected. In this paper we consider a topology \tau_S on the set \mathbb{N} of natural numbers. We then present properties of the topological space (\mathbb{N},\tau_S), some of them involve the closure of a set with respect to this topology, while others describe subsets which are either totally Brown or totally separated. Our theorems generalize results proved by P. Szczuka in 2013, 2014, 2016 and by P. Szyszkowska and M. Szyszkowski in 2018.
Keywords: arithmetic progression; common division topology; totally Brown space; totally separated space
DOI: DOI 10.14712/1213-7243.2022.022
AMS Subject Classification: 11B25 54D05 11A41 11B05 54A05 54D10