José del Carmen Alberto-Domínguez, Gerardo Acosta, Gerardo Delgadillo-Piñón
Totally Brown subsets of the Golomb space and the Kirch space

Comment.Math.Univ.Carolin. 63,2 (2022) 189-219.

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 the Golomb topology \tau_G on the set \mathbb{N} of natural numbers, as well as the Kirch topology \tau_K on \mathbb{N}. Then we examine subsets of these spaces which are totally Brown. Among other results, we characterize the arithmetic progressions which are either totally Brown or totally separated in (\mathbb{N},\tau_G). We also show that (\mathbb{N},\tau_G) and (\mathbb{N},\tau_K) are aposyndetic. Our results generalize properties obtained by A. M. Kirch in 1969 and by P. Szczuka in 2010, 2013 and 2015.

Keywords: arithmetic progression; Golomb topology; Kirch topology; totally Brown space; totally separated space

DOI: DOI 10.14712/1213-7243.2022.017
AMS Subject Classification: 11B25 54D05 11A41 11B05 54A05 54D10