The common division topology on ℕ

José del Carmen Alberto-Domínguez; Gerardo Acosta; Maira Madriz-Mendoza

Commentationes Mathematicae Universitatis Carolinae (2022)

  • Volume: 62 63, Issue: 3, page 329-349
  • ISSN: 0010-2628

Abstract

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A topological space X is totally Brown if for each n ∈ ℕ ∖ { 1 } and every nonempty open subsets U 1 , U 2 , ... , U n of X we have cl X ( U 1 ) ∩ cl X ( U 2 ) ∩ ⋯ ∩ cl X ( U n ) ≠ ∅ . Totally Brown spaces are connected. In this paper we consider a topology τ S on the set ℕ of natural numbers. We then present properties of the topological space ( ℕ , τ 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.

How to cite

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Alberto-Domínguez, José del Carmen, Acosta, Gerardo, and Madriz-Mendoza, Maira. "The common division topology on $\mathbb {N}$." Commentationes Mathematicae Universitatis Carolinae 62 63.3 (2022): 329-349. <http://eudml.org/doc/299032>.

@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 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.},
author = {Alberto-Domínguez, José del Carmen, Acosta, Gerardo, Madriz-Mendoza, Maira},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {arithmetic progression; common division topology; totally Brown space; totally separated space},
language = {eng},
number = {3},
pages = {329-349},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The common division topology on $\mathbb \{N\}$},
url = {http://eudml.org/doc/299032},
volume = {62 63},
year = {2022},
}

TY - JOUR
AU - Alberto-Domínguez, José del Carmen
AU - Acosta, Gerardo
AU - Madriz-Mendoza, Maira
TI - The common division topology on $\mathbb {N}$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2022
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62 63
IS - 3
SP - 329
EP - 349
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 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.
LA - eng
KW - arithmetic progression; common division topology; totally Brown space; totally separated space
UR - http://eudml.org/doc/299032
ER -

References

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