Conservation of factorizability of $G$-spaces by equivariant mappings

In this paper we prove the $\mathbb R$-factorizability of an equivariant image of an $\mathbb R$-factorizable $G$-space with a $\mathrm{d}$-open action of an $\omega$-narrow $P$-group. It is shown that the $\mathbb R$-factorizability, $m$-factorizability, and $M$-factorizability of $G$-spaces hold in the case of $\mathrm{d}$-open equivariant images. It is proved that the $\mathbb R$-factorizability of topological groups holds under $\mathrm{d}$-open homomorphisms.