R-factorizability and ω-uniform continuity in topological groups

In this paper the concept of property ω-U is introduced in topological groups. The main results are that (1) every Lindelöf topological group and every totally bounded topological group have property ω-U; (2) a topological group is R-factorizable if and only if it is an ω-narrow group with property ω-U; (3) M-factorizable groups are preserved by open continuous homomorphisms, which gives a positive answer to a problem posed by A.V. Arhangelʼskiı̌ and M. Tkachenko.