Geodesic
A countably cellular topological group all of whose countable subsets are closed need not be $\mathbb{R}$-factorizable
Commentationes Mathematicae Universitatis Carolinae, Tome 64 (2023) no. 1, pp. 127-135.

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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.
DOI : 10.14712/1213-7243.2023.016
Classification : 22A05, 54D30, 54G20, 54H11
Keywords: $\mathbb{R}$-factorizable; cellularity; $C$-embedded; Sorgenfrey line; $P$-group; Dieudonné completion; Hewitt--Nachbin completion; Bohr topology 
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Tkachenko, Mikhail. A countably cellular topological group all of whose countable subsets are closed need not be $\mathbb{R}$-factorizable. Commentationes Mathematicae Universitatis Carolinae, Tome 64 (2023) no. 1, pp. 127-135. doi : 10.14712/1213-7243.2023.016. http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2023.016/

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