R-factorizability of products with countable groups

Establish whether for every R-factorizable topological group G and every countable topological group H, the product group G × H is R-factorizable as a topological group.

Background

A topological group is R-factorizable if every continuous map f: G → ℝ factors through a continuous homomorphism to a second-countable group. Prior work cited in the paper shows that products of Lindelöf (hence R-factorizable) groups can fail to be R-factorizable, and raises broader questions about when R-factorizability is preserved under taking products.

The specific case of taking the product with a countable group is singled out as an unresolved question. A positive resolution would imply strong consequences (e.g., pseudo-ω1-compactness of R-factorizable groups), while a negative resolution would illuminate the non-preservation of R-factorizability under continuous homomorphisms.