On some kinds of ω-balancedness and (*) properties in certain semitopological groups

Abstract

In this article, we discuss some relationships of ω-balancedness and (⁎) properties which were introduced for giving characterizations of subgroups of topological products of certain para(semi)topological groups. We mainly get the following results.If G is a regular ω-balanced locally ω-good semitopological group with a q-point, then Ir(G)≤ω if and only if Sm(G)≤ω. If G is a regular strongly paracompact semitopological group with a q-point and Sm(G)≤ω, then G is completely ω-balanced if and only if G has property (⁎). If G is a regular paracompact ω-balanced locally good semitopological group with a q-point and Sm(G)≤ω, then G has property (w⁎) if and only if G has property (**). If G is a regular metacompact semitopological group with a q-point and Sm(G)≤ω, then G is MM-ω-balanced if and only if G is M-ω-balanced.We show that a semitopological group G admits a homeomorphic embedding as a subgroup of a product of metrizable semitopological groups if and only if G is topologically isomorphic to a subgroup of a product of semitopological groups which are first-countable paracompact regular σ-spaces and is topologically isomorphic to a subgroup of a product of Moore semitopological groups.