Cardinality constraints for pseudocompact and for totally dense subgroups of compact topological groups (Q1075444)

For a compact Hausdorff group K, let m(K) be the minimum cardinality of a dense pseudocompact subgroup of K. Then: m(K) is a function of \(w(K)=\alpha\); \(m(K)<| K|\) if \((\log\alpha)^{\omega} < 2^{\alpha}\); and, assuming SCH, \(m(K)=(\log \alpha)^{\omega}\), whence \(m(K)=| K|\) if \((\log\alpha)^{\omega}= 2^{\alpha}\). Cardinals \(\alpha\) such that \((\log\alpha)^{\omega}\neq 2^{\alpha}\) are characterized and shown to abound in all models of ZFC. A subgroup G of K is totally dense if, for each closed subgroup H of K, \(G\cap H\) is dense in H. If K is connected, every totally dense subgroup D of K satisfies \(| D| =| K|\). Some K with \(w(K)=\alpha\) has a totally dense subgroup D with \(| D| <| K|\) if and only if \(cf(\alpha)=\omega\) and \(\alpha =\log (2^{\alpha})\); w.l.o.g., K is totally disconnected. The characterization of these two classes of cardinals are incompatible in some models of ZFC, compatible in others. The existence of a compact Hausdorff group K with a totally dense pseudocompact subgroup D with \(| D| <| K|\) is therefore undecidable in ZFC.