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

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 $Ir\left(G\right)\le \omega$ if and only if $Sm\left(G\right)\le \omega$. If G is a regular strongly paracompact semitopological group with a q-point and $Sm\left(G\right)\le \omega$, then G is completely ω-balanced if and only if G has property $\left({}^{⁎}\right)$. If G is a regular paracompact ω-balanced locally good semitopological group with a q-point and $Sm\left(G\right)\le \omega$, 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 $Sm\left(G\right)\le \omega$, 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.