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Title: Totally Brown subsets of the Golomb space and the Kirch space (English)
Author: Alberto-Domínguez, José del Carmen
Author: Acosta, Gerardo
Author: Delgadillo-Piñón, Gerardo
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 63
Issue: 2
Year: 2022
Pages: 189-219
Summary lang: English
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Category: math
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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. (English)
Keyword: arithmetic progression
Keyword: Golomb topology
Keyword: Kirch topology
Keyword: totally Brown space
Keyword: totally separated space
MSC: 11A41
MSC: 11B05
MSC: 11B25
MSC: 54A05
MSC: 54D05
MSC: 54D10
idZBL: Zbl 07613030
idMR: MR4506132
DOI: 10.14712/1213-7243.2022.017
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Date available: 2022-11-02T09:16:56Z
Last updated: 2024-07-01
Stable URL: http://hdl.handle.net/10338.dmlcz/151085
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