On some kinds of ω-balancedness in semitopological groups

Abstract

In this article, we introduce two notions which are called M-ω-balancedness and C-ω-balancedness, respectively. We get the following conclusions. A regular semitopological group G is topologically isomorphic to a subgroup of the product of a family of Moore semitopological groups if and only if G is M-ω-balanced and Ir(G)≤ω. A T1 semitopological group G is topologically isomorphic to a subgroup of the product of a family of T1 semitopological groups with a point-countable base if and only if G is C-ω-balanced and Sm(G)≤ω. If G is a Hausdorff countably compact semitopological group with Hs(G)≤ω, then the M-ω-balancedness (C-ω-balancedness) of G implies that G is a topological group.Let G be an ω-balanced paratopological group and let e be the identity of G. We show that if for each U∈N(e) there exist ω-good set V∈N(e) with V⊂U and A⊂G such that ⋃{g(V∩V−1):g∈A}=G and {g(V∩V−1)V:g∈A} is star-countable, then G is completely ω-balanced, where N(e) is the family of open neighborhoods of e in G.