A countably cellular topological group all of whose countable subsets are closed need not be \(\mathbb{R}\)-factorizable. (English) Zbl 07790587

Tkachenko, Mikhail

Commentat. Math. Univ. Carol. 64, No. 1, 127-135 (2023).

The author constructs a Hausdorff topological group \(G\) such that \(\aleph_1\) is a precalibre of \(G\), and 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 gives negative answer on Problem 8.6.3 from the book of A. Arhangel’skii and M. Tkachenko [Topological groups and related structures. Hackensack, NJ: World Scientific; Paris: Atlantis Press (2008; Zbl 1323.22001)].

Reviewer: Oleg V. Gutik (Lviv)

MSC:

22A05	Structure of general topological groups
54H11	Topological groups (topological aspects)
54D30	Compactness
54G20	Counterexamples in general topology

Keywords:

\(\mathbb{R}\)-factorizable; cellularity; \(C\)-embedded; Sorgenfrey line; \(P\)-group; Dieudonné completion; Hewitt-Nachbin completion; Bohr topology

Citations:

Zbl 1323.22001

References:

[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 · Zbl 1323.22001
[2]	Engelking R., General Topology, Sigma Ser. Pure Math., 6, Heldermann, Berlin, 1989 · Zbl 0684.54001
[3]	Noble N.; Ulmer M., Factoring functions on Cartesian products, Trans. Amer. Math. Soc. 163 (1972), 329-339 · Zbl 0207.21302 · doi:10.1090/S0002-9947-1972-0288721-2
[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 · Zbl 1311.54026 · doi:10.1016/j.topol.2013.04.010
[5]	Sipacheva O. V., Free Boolean topological groups, Axioms 4 (2015), no. 4, 492-517 · Zbl 1372.54028 · doi:10.3390/axioms4040492
[6]	Tkachenko M. G., Some results on inverse spectra. II, Comment. Math. Univ. Carolin. 22 (1981), no. 4, 819-841 · Zbl 0494.54007
[7]	Tkachenko M. G., Cellularity in quotient spaces of topological groups, Forum Math. 31 (2019), no. 2, 351-359 · Zbl 1415.43003 · doi:10.1515/forum-2018-0195
[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 · Zbl 1479.54064