On MM-ω-balancedness and FR(Fm)-factorizable semi(para)topological groups

In the second part of this article, we introduce a notion which is called MM-ω-balancedness in the class of semitopological groups. We show that if G is a semitopological (paratopological) group, then G is topologically isomorphic to a subgroup of the product of a family of metacompact Moore semitopological (paratopological) groups if and only if G is regular MM-ω-balanced and $Ir\left(G\right)\le \omega$. If G is a $T_0$ bM-ω-balanced semitopological group and $f:G\to H$ is an open continuous homomorphism of G onto a first-countable semitopological group H such that $\ker \left(f\right)$ is a countably compact subgroup of G, then H is a metacompact developable space.

In the third part of this article, we introduce notions of $FR$-factorizability and $Fm$-factorizability. We give some equivalent conditions that a semitopological (paratopological) group is $FR$-factorizable or $Fm$-factorizable. If G is a Tychonoff $FR$ ($Fm$)-factorizable semitopological group and $f:G\to H$ is a continuous open homomorphism of G onto a semitopological group H, then H is $FR$ ($Fm$)-factorizable. If G is a $FR$ ($Fm$)-factorizable paratopological group and $f:G\to H$ is a continuous d-open homomorphism of G onto a paratopological group H, then H is $FR$ ($Fm$)-factorizable.