Totally Brown subsets of the Golomb space and the Kirch space
José del Carmen Alberto-Domínguez; Gerardo Acosta; Gerardo Delgadillo-Piñón
Commentationes Mathematicae Universitatis Carolinae (2022)
- Volume: 62 63, Issue: 2, page 189-219
- ISSN: 0010-2628
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topAlberto-Domínguez, José del Carmen, Acosta, Gerardo, and Delgadillo-Piñón, Gerardo. "Totally Brown subsets of the Golomb space and the Kirch space." Commentationes Mathematicae Universitatis Carolinae 62 63.2 (2022): 189-219. <http://eudml.org/doc/298517>.
@article{Alberto2022,
abstract = {A topological space $X$ is totally Brown if for each $n \in \mathbb \{N\} \setminus \lbrace 1\rbrace $ 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.},
author = {Alberto-Domínguez, José del Carmen, Acosta, Gerardo, Delgadillo-Piñón, Gerardo},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {arithmetic progression; Golomb topology; Kirch topology; totally Brown space; totally separated space},
language = {eng},
number = {2},
pages = {189-219},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Totally Brown subsets of the Golomb space and the Kirch space},
url = {http://eudml.org/doc/298517},
volume = {62 63},
year = {2022},
}
TY - JOUR
AU - Alberto-Domínguez, José del Carmen
AU - Acosta, Gerardo
AU - Delgadillo-Piñón, Gerardo
TI - Totally Brown subsets of the Golomb space and the Kirch space
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2022
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62 63
IS - 2
SP - 189
EP - 219
AB - A topological space $X$ is totally Brown if for each $n \in \mathbb {N} \setminus \lbrace 1\rbrace $ 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.
LA - eng
KW - arithmetic progression; Golomb topology; Kirch topology; totally Brown space; totally separated space
UR - http://eudml.org/doc/298517
ER -
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