Article
| 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: | 2023-03-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151085 |
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