R-factorizable paratopological groups

For i = 1 , 2 , 3 , 3.5 , we define the class of R i -factorizable paratopological groups G by the condition that every continuous real-valued function on G can be factorized through a continuous homomorphism p : G → H onto a second countable paratopological group H satisfying the T i -separation axiom. We show that the Sorgenfrey line is a Lindelöf paratopological group that fails to be R 1 -factorizable. However, every Lindelöf totally ω-narrow regular (Hausdorff) paratopological group is R 3 -factorizable (resp. R 2 -factorizable). We also prove that a Lindelöf totally ω-narrow regular paratopological group is topologically isomorphic to a closed subgroup of a product of separable metrizable paratopological groups.