Article
| Title: | A countably cellular topological group all of whose countable subsets are closed need not be $\mathbb{R}$-factorizable (English) |
| Author: | Tkachenko, Mikhail |
| Language: | English |
| Journal: | Commentationes Mathematicae Universitatis Carolinae |
| ISSN: | 0010-2628 (print) |
| ISSN: | 1213-7243 (online) |
| Volume: | 64 |
| Issue: | 1 |
| Year: | 2023 |
| Pages: | 127-135 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We construct a Hausdorff topological group $G$ such that $\aleph_1$ is a precalibre of $G$ (hence, $G$ has countable cellularity), all countable subsets of $G$ are closed and $C$-embedded in $G$, but $G$ is not $\mathbb{R}$-factorizable. This solves Problem 8.6.3 from the book ``Topological Groups and Related Structures" (2008) in the negative. (English) |
| Keyword: | $\mathbb{R}$-factorizable |
| Keyword: | cellularity |
| Keyword: | $C$-embedded |
| Keyword: | Sorgenfrey line |
| Keyword: | $P$-group |
| Keyword: | Dieudonné completion |
| Keyword: | Hewitt--Nachbin completion |
| Keyword: | Bohr topology |
| MSC: | 22A05 |
| MSC: | 54D30 |
| MSC: | 54G20 |
| MSC: | 54H11 |
| idZBL: | Zbl 07790587 |
| idMR: | MR4631595 |
| DOI: | 10.14712/1213-7243.2023.016 |
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| Date available: | 2023-08-28T09:52:12Z |
| Last updated: | 2024-02-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151801 |
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| Reference: | [1] Arhangel'skiĭ A. V., Tkachenko M. G.: Topological Groups and Related Structures.Atlantis Stud. Math., 1, Atlantis Press, Paris World Scientific Publishing Co., Hackensack, 2008. MR 2433295 |
| Reference: | [2] Engelking R.: General Topology.Sigma Ser. Pure Math., 6, Heldermann, Berlin, 1989. Zbl 0684.54001, MR 1039321 |
| Reference: | [3] Noble N., Ulmer M.: Factoring functions on Cartesian products.Trans. Amer. Math. Soc. 163 (1972), 329–339. MR 0288721, 10.1090/S0002-9947-1972-0288721-2 |
| Reference: | [4] Reznichenko E. A., Sipacheva O. V.: The free topological group on the Sorgenfrey line is not $\mathbb{R}$-factorizable.Topology Appl. 160 (2013), no. 11, 1184–1187. MR 3062768, 10.1016/j.topol.2013.04.010 |
| Reference: | [5] Sipacheva O. V.: Free Boolean topological groups.Axioms 4 (2015), no. 4, 492–517. 10.3390/axioms4040492 |
| Reference: | [6] Tkachenko M. G.: Some results on inverse spectra. II.Comment. Math. Univ. Carolin. 22 (1981), no. 4, 819–841. MR 0647029 |
| Reference: | [7] Tkachenko M. G.: Cellularity in quotient spaces of topological groups.Forum Math. 31 (2019), no. 2, 351–359. MR 3918445, 10.1515/forum-2018-0195 |
| Reference: | [8] Xie L.-H., Yan P.-F.: The continuous $d$-open homomorphism images and subgroups of $\mathbb{R}$-factorizable paratopological groups.Topology Appl. 300 (2021), Paper No. 107627, 7 pages. MR 4281998 |
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