The fineness index of topological groups and M-factorizable groups

A topological group G is called M-factorizable (resp. R-factorizable) if every continuous real-valued function on G admits a factorization via a continuous homomorphism onto a first-countable (resp. second-countable) group. The purpose of this paper is a further study of M-factorizability in topological groups. It is shown that a topological group G is M-factorizable if and only if G has property ω-U, i.e., every continuous real-valued function on G is ω-uniform continuous. To achieve this goal, we introduce a new cardinal function of topological groups, which is called the fineness index. The cardinal invariance is also used to characterize Mm-factorizable groups which are generalization of m-factorizable groups introduced in [2].