Hereditarily -factorizable groups

We show that every subgroup of the σ-product of a family { G i : i ∈ I } of regular paratopological groups satisfying N a g ( G i ) ⩽ ω has countable cellularity, is perfectly κ-normal and R 3 -factorizable. For topological groups, we prove a more general result as follows. Let C be the minimal class of topological groups that contains all Lindelöf Σ-groups and is closed under taking arbitrary subgroups, countable products, continuous homomorphic images, and forming σ-products. Then every group in C has countable cellularity, is hereditarily R -factorizable and perfectly κ-normal.