A countably cellular topological group all of whose countable subsets are closed need not be $\mathbb{R}$-factorizable

Abstract

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.