On some properties of PR (Pm)-factorizable semitopological groups

In this article, we introduce the notions of (R?,FM1SG) ((MS,FM1SG))-factorizability and (R?, SCSG) ((MS, SCSG))-factorizability. In the second part of this article, we study the relationship between (R?,PSG)-factorizability and (MS,PSG)-factorizability. In the third part of this article, we discuss some properties of certain continuous homomorphic images of (R?, SCSG)- and (MS, SCSG)-factorizable semitopological groups. In the fourth part of this article, (R?, SCSG)-factorizability and (MS, SCSG)-factorizability are discussed. We prove the following results. Let G and H be semitopological groups. If G ? H is a continuous surjective closed and open homomorphism such that G is a Tychonoff (R?,FM1SG) ((MS,FM1SG))-factorizable semitopological group with a q-point and H satisfies Sm(H) ? ?, then H is (R?,FM1SG) ((MS,FM1SG))-factorizable. If G ? H is a continuous open surjective homomorphism such that G is a Tychonoff (R?, SCSG) ((MS,SCSG))-factorizable semitopological group, then H is (R?, SCSG) ((MS, SCSG))-factorizable. If G is a regular semitopological group with a q-point, property (***) and satisfying Sm(G) ? ?, then G is topologically isomorphic to a subgroup of the product of a family of separable metrizable semitopological groups.