On Ψω-factorizable groups

A topological group G is called ${\Psi }_{\omega }$-factorizable (resp. $M$-factorizable) if every continuous real-valued function on G admits a factorization via a continuous homomorphism onto a topological group H with $\psi \left(H\right)\le \omega$ (resp. a first-countable group). The first purpose of this article is to discuss some characterizations of ${\Psi }_{\omega }$-factorizable groups. It is shown that a topological group G is ${\Psi }_{\omega }$-factorizable if and only if every continuous real-valued function on G is $G_{\delta }$-uniformly continuous, if and only if for every cozero-set U of G, there exists a $G_{\delta }$-subgroup N of G such that $UN=U$. Sufficient conditions on the ${\Psi }_{\omega }$-factorizable group G to be $M$-factorizable are that G is τ-fine and τ-steady for a cardinal τ.