On some kinds of \(\omega \)-balancedness and (\(^*\)) properties in certain semitopological groups (Q6581833)

The authors investigate some relationships of \(\omega\)-balancedness and \((*)\) properties. In particular, the following results are shown: If \(G\) is a regular \(\omega\)-balanced locally \(\omega\)-good semitopological group with a \(q\)-point, then \(Ir(G) \leq \omega\) if and only if \(Sm(G) \leq \omega\). If \(G\) is a regular strongly paracompact semitopological group with a \(q\)-point and \(Sm(G) \leq \omega\), then \(G\) is completely \(\omega\)-balanced if and only if \(G\) has property \((*)\). If \(G\) is a regular paracompact \(\omega\)-balanced locally good semitopological group with a \(q\)-point and \(Sm(G) \leq \omega\), 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 \(Sm(G) \leq \omega\), then \(G\) is \(MM\)-\(\omega\)-balanced if and only if \(G\) is \(M\)-\(\omega\)-balanced. Furthermore, the authors prove 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 that are first-countable paracompact regular \(\sigma\)-spaces and is topologically isomorphic to a subgroup of a product of Moore semitopological groups.