Article
MSC:
11A41,
11B05,
11B25,
54A05,
54D05,
54D10 | MR 4506132 | Zbl 07613030 | DOI: 10.14712/1213-7243.2022.017
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Keywords:
arithmetic progression; Golomb topology; Kirch topology; totally Brown space; totally separated space
arithmetic progression; Golomb topology; Kirch topology; totally Brown space; totally separated space
Summary:
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.
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