How many \(\omega\)-bounded subgroups? (Q1356953)

A topological space is said to be \(\omega\)-bounded if each of its countable subsets has compact closure. For a topological group \(G\) let \(\Omega(G)\) be the set of all dense \(\omega\)-bounded subgroups of \(G\). The authors show the following interesting result: Let \(G\) be a compact group such that \(w(G)=w(G)^\omega\). If \(G\) is Abelian or connected, then \(|\Omega (G)|= 2^{| G|}\). This gives a partial answer to a question of G. Itzkowitz and D. Shakhmatov. It remains open if \(|\Omega (G)|= 2^{| G|}\) holds for every compact group \(G\) of uncountable weight.