The Golomb space is topologically rigid

Taras O. Banakh; Dario Spirito; Sławomir Turek

Commentationes Mathematicae Universitatis Carolinae (2021)

  • Volume: 62, Issue: 3, page 347-360
  • 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 is topologically rigid in the sense that its homeomorphism group is trivial. This resolves a problem posed by T. Banakh at Mathoverflow in 2017.

How to cite

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Banakh, Taras O., Spirito, Dario, and Turek, Sławomir. "The Golomb space is topologically rigid." Commentationes Mathematicae Universitatis Carolinae 62.3 (2021): 347-360. <http://eudml.org/doc/297537>.

@article{Banakh2021,
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 $ is topologically rigid in the sense that its homeomorphism group is trivial. This resolves a problem posed by T. Banakh at Mathoverflow in 2017.},
author = {Banakh, Taras O., Spirito, Dario, Turek, Sławomir},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Golomb topology; topologically rigid space},
language = {eng},
number = {3},
pages = {347-360},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The Golomb space is topologically rigid},
url = {http://eudml.org/doc/297537},
volume = {62},
year = {2021},
}

TY - JOUR
AU - Banakh, Taras O.
AU - Spirito, Dario
AU - Turek, Sławomir
TI - The Golomb space is topologically rigid
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2021
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62
IS - 3
SP - 347
EP - 360
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 $ is topologically rigid in the sense that its homeomorphism group is trivial. This resolves a problem posed by T. Banakh at Mathoverflow in 2017.
LA - eng
KW - Golomb topology; topologically rigid space
UR - http://eudml.org/doc/297537
ER -

References

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  13. Spirito D., 10.1016/j.topol.2020.107101, Topology Appl. 273 (2020), 107101, 20 pages. DOI10.1016/j.topol.2020.107101
  14. Spirito D., 10.2989/16073606.2019.1704904, Quaest. Math. 44 (2021), no. 4, 447–468. DOI10.2989/16073606.2019.1704904
  15. Steen L. A., Seebach J. A., Jr., Counterexamples in Topology, Dover Publications, Mineola, New York, 1995. Zbl0386.54001
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  17. Szczuka P., The Darboux property for polynomials in Golomb's and Kirch's topologies, Demonstratio Math. 46 (2013), no. 2, 429–435. 

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