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

Abstract

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The Golomb space ℕ τ is the set ℕ of positive integers endowed with the topology τ generated by the base consisting of arithmetic progressions { a + b n : n ≥ 0 } with coprime a , b . We prove that the Golomb space ℕ τ has continuum many continuous self-maps, contains a countable disjoint family of infinite closed connected subsets, the set Π of prime numbers is a dense metrizable subspace of ℕ τ , and each homeomorphism h of ℕ τ has the following properties: h ( 1 ) = 1 , h ( Π ) = Π , Π h ( x ) = h ( Π x ) , and h ( x ℕ ) = h ( x ) ℕ for all x ∈ ℕ . Here x ℕ : = { x n : n ∈ ℕ } and Π x denotes the set of prime divisors of x .

How to cite

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Banakh, 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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