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-omega-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-omega-balanced and Ir(G) <= omega. If G is a T-0 bM-omega-balanced semitopological group and f : G -> H is an open continuous homomorphism of G onto a first-countable semitopological group H such that ker(f) 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 -> 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 -> H is a continuous d-open homomorphism of G onto a paratopological group H, then H is FR (Fm)-factorizable.