Geodesic
The common division topology on $\mathbb{N}$
Commentationes Mathematicae Universitatis Carolinae, Tome 63 (2022) no. 3, pp. 329-349.

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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 a topology $\tau_S$ on the set $\mathbb{N}$ of natural numbers. We then present properties of the topological space $(\mathbb{N},\tau_S)$, some of them involve the closure of a set with respect to this topology, while others describe subsets which are either totally Brown or totally separated. Our theorems generalize results proved by P. Szczuka in 2013, 2014, 2016 and by P. Szyszkowska and M. Szyszkowski in 2018.
DOI : 10.14712/1213-7243.2022.022
Classification : 11A41, 11B05, 11B25, 54A05, 54D05, 54D10
Keywords: arithmetic progression; common division topology; totally Brown space; totally separated space 
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Alberto-Domínguez, José del Carmen; Acosta, Gerardo; Madriz-Mendoza, Maira. The common division topology on $\mathbb{N}$. Commentationes Mathematicae Universitatis Carolinae, Tome 63 (2022) no. 3, pp. 329-349. doi : 10.14712/1213-7243.2022.022. http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2022.022/

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