On continuous self-maps and homeomorphisms of the Golomb space
Taras O. Banakh; Jerzy Mioduszewski; Sławomir Turek
Commentationes Mathematicae Universitatis Carolinae (2018)
- Volume: 59, Issue: 4, page 423-442
- ISSN: 0010-2628
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topBanakh, Taras O., Mioduszewski, Jerzy, and Turek, Sławomir. "On continuous self-maps and homeomorphisms of the Golomb space." Commentationes Mathematicae Universitatis Carolinae 59.4 (2018): 423-442. <http://eudml.org/doc/294865>.
@article{Banakh2018,
abstract = {The Golomb space $\{\mathbb \{N\}\}_\tau $ is the set $\{\mathbb \{N\}\}$ of positive integers endowed with the topology $\tau $ generated by the base consisting of arithmetic progressions $\lbrace a+ bn: n\ge 0\rbrace $ with coprime $a,b$. We prove that the Golomb space $\{\mathbb \{N\}\}_\tau $ has continuum many continuous self-maps, contains a countable disjoint family of infinite closed connected subsets, the set $\Pi $ of prime numbers is a dense metrizable subspace of $\{\mathbb \{N\}\}_\tau $, and each homeomorphism $h$ of $\{\mathbb \{N\}\}_\tau $ has the following properties: $h(1)=1$, $h(\Pi )=\Pi $, $\Pi _\{h(x)\}=h(\Pi _x)$, and $h(x^\{\{\mathbb \{N\}\}\})=h(x)^\{\,\mathbb \{N\}\}$ for all $x\in \{\mathbb \{N\}\}$. Here $x^\{\mathbb \{N\}\}:=\lbrace x^n\colon n\in \{\mathbb \{N\}\}\rbrace $ and $\Pi _x$ denotes the set of prime divisors of $x$.},
author = {Banakh, Taras O., Mioduszewski, Jerzy, Turek, Sławomir},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Golomb space; arithmetic progression; superconnected space; homeomorphism},
language = {eng},
number = {4},
pages = {423-442},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On continuous self-maps and homeomorphisms of the Golomb space},
url = {http://eudml.org/doc/294865},
volume = {59},
year = {2018},
}
TY - JOUR
AU - Banakh, Taras O.
AU - Mioduszewski, Jerzy
AU - Turek, Sławomir
TI - On continuous self-maps and homeomorphisms of the Golomb space
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2018
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 59
IS - 4
SP - 423
EP - 442
AB - The Golomb space ${\mathbb {N}}_\tau $ is the set ${\mathbb {N}}$ of positive integers endowed with the topology $\tau $ generated by the base consisting of arithmetic progressions $\lbrace a+ bn: n\ge 0\rbrace $ with coprime $a,b$. We prove that the Golomb space ${\mathbb {N}}_\tau $ has continuum many continuous self-maps, contains a countable disjoint family of infinite closed connected subsets, the set $\Pi $ of prime numbers is a dense metrizable subspace of ${\mathbb {N}}_\tau $, and each homeomorphism $h$ of ${\mathbb {N}}_\tau $ has the following properties: $h(1)=1$, $h(\Pi )=\Pi $, $\Pi _{h(x)}=h(\Pi _x)$, and $h(x^{{\mathbb {N}}})=h(x)^{\,\mathbb {N}}$ for all $x\in {\mathbb {N}}$. Here $x^{\mathbb {N}}:=\lbrace x^n\colon n\in {\mathbb {N}}\rbrace $ and $\Pi _x$ denotes the set of prime divisors of $x$.
LA - eng
KW - Golomb space; arithmetic progression; superconnected space; homeomorphism
UR - http://eudml.org/doc/294865
ER -
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Citations in EuDML Documents
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- José del Carmen Alberto-Domínguez, Gerardo Acosta, Maira Madriz-Mendoza, Aposyndesis in
- José del Carmen Alberto-Domínguez, Gerardo Acosta, Gerardo Delgadillo-Piñón, Totally Brown subsets of the Golomb space and the Kirch space
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