Characterization of $\Bbb R$-factorizable $G$-spaces

In this paper we characterize $\mathbb R$-factorizability of $G$-spaces and prove the equivalence of $\mathbb R$-factorizability and $\omega$-$U$ property for $G$-spaces with $\mathrm{d}$-open actions of $\omega$-narrow groups. It is shown that the $\mathbb R$-factorizability characterizes those compact coset spaces which are coset spaces of $\omega$-narrow groups. The notion of $m$- and $M$-factorizable $G$-spaces is introduced, which generalizes the corresponding notions for topological groups.