On bM-ω-balancedness and [formula omitted]-factorizability of para(semi)topological groups

Abstract

In this article, we introduce a notion which is called bM-ω-balancedness for semitopological groups. We show that a semitopological group G is topologically isomorphic to a subgroup of the product of a family of metrizable semitopological groups if and only if G is T0bM-ω-balanced. Thus a semitopological group G is T0bM-ω-balanced if and only if G is regular, has property (⁎) and countable index of regularity. The notion of property (⁎) is in the sense of I. Sánchez in [16].We show that if G is a T0bM-ω-balanced weakly Lindelöf paratopological group with a q-point, then for every continuous real-valued function f:G→R, there exist a continuous clopen homomorphism π:G→H onto a separable metrizable paratopological group H such that ker(π) is countably compact and a continuous real-valued function g:H→R such that f=g∘π.We introduce the notions of M (DE)-factorizability for semitopological groups. We show that a semitopological group G is Tychonoff and M-factorizable if and only if G is T0bM-ω-balanced and has property ω-QU. If G is a Tychonoff M-factorizable semitopological group and f:G→H is an open continuous homomorphism onto a semitopological group H such that ker(f) is countably compact, then H is DE-factorizable.