Totally Brown subsets of the Golomb space and the Kirch space

[摘要] 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.

[学科分类] 物理化学和理论化学

[关键词] arithmetic progression;Golomb topology;Kirch topology;totally Brown space;totally separated space