Cardinal numbers associated with dense pseudocompact, countably compact, and ω-bounded subgroups

A brief survey is offered in Section 1 on earlier progress in the solution of two related problems concerning dense proper pseudocompact (countably compact, ω-bounded) subgroups of compact nonmetrizable groups. These are: Does every nonmetrizable compact group contain such a subgroup and if a compact group has such a subgroup how large may a distinguished family of such subgroups be? Section 2 contains new results and considers the second question in detail. It is shown that each compact nonmetrizable group G that is product-like contains a family of 2¦G¦ distinct dense pseudocompact subgroups. In the special case where L is a Cartesian product of more than ω1 compact simply connected simple Lie groups, L even contains 2¦L¦ free subgroups that are dense and pseudocompact. In the final section, it is shown that if 2ω < 2ω1 then each nonmetrizable compact group contains at least 2ω1 distinct dense countably compact subgroups. Conditions are given under which a compact group has a large collection of dense ω-bounded subgroups. Finally, this section gives an example due to Comfort, poses some additional questions, and records a recent development.