| Title:
|
The Golomb space is topologically rigid (English) |
| Author:
|
Banakh, Taras |
| Author:
|
Spirito, Dario |
| Author:
|
Turek, Sławomir |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
62 |
| Issue:
|
3 |
| Year:
|
2021 |
| Pages:
|
347-360 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $\{a+bn: n\ge 0\}$ 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. (English) |
| Keyword:
|
Golomb topology |
| Keyword:
|
topologically rigid space |
| MSC:
|
11A99 |
| MSC:
|
54G15 |
| idMR:
|
MR4331287 |
| DOI:
|
10.14712/1213-7243.2021.023 |
| . |
| Date available:
|
2021-10-20T09:23:11Z |
| Last updated:
|
2023-10-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149148 |
| . |
| Reference:
|
[1] Banakh T.: Is the Golomb countable connected space topologically rigid?.https://mathoverflow.net/questions/285557. |
| Reference:
|
[2] Banakh T., Mioduszewski J., Turek S.: On continuous self-maps and homeomorphisms of the Golomb space.Comment. Math. Univ. Carolin. 59 (2018), no. 4, 423–442. |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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[13] Spirito D.: The Golomb topology on a Dedekind domain and the group of units of its quotients.Topology Appl. 273 (2020), 107101, 20 pages. 10.1016/j.topol.2020.107101 |
| Reference:
|
[14] Spirito D.: The Golomb topology of polynomial rings.Quaest. Math. 44 (2021), no. 4, 447–468. 10.2989/16073606.2019.1704904 |
| Reference:
|
[15] Steen L. A., Seebach J. A., Jr.: Counterexamples in Topology.Dover Publications, Mineola, New York, 1995. Zbl 0386.54001 |
| Reference:
|
[16] Szczuka P.: The connectedness of arithmetic progressions in Furstenberg's, Golomb's, and Kirch's topologies.Demonstratio Math. 43 (2010), no. 4, 899–909. 10.1515/dema-2010-0416 |
| Reference:
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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. |
| . |
