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Keywords:
$\mathbb{R}$-factorizable; cellularity; $C$-embedded; Sorgenfrey line; $P$-group; Dieudonné completion; Hewitt--Nachbin completion; Bohr topology
$\mathbb{R}$-factorizable; cellularity; $C$-embedded; Sorgenfrey line; $P$-group; Dieudonné completion; Hewitt--Nachbin completion; Bohr topology
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.
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