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

Non-Abelian Pseudocompact Groups

“…For emphasis: * every infinite compact group K admits a proper dense subgroup G with |G| < |K|. Similarly it follows easily, as in [4] …”

“…And κ is λ-admissible if there is a pseudocompact group such that |G| = κ and wG = λ. (i) * Given a topological group G, we write P(G) := {H : |H| is a dense pseudocompact subgroup of G}, and, following [4], for a compact group K we write m(K) := min{|H| : H ∈ P(K)}.…”

“…(a) Every pseudocompact space is a Baire space [1] (3.10.F(e)), so in particular * every pseudocompact group is a Baire space (alternatively one may argue as in [4] (2.4(b)): a G δ -dense subspace of a Baire space is itself a Baire space, so a pseudocompact group G, being G δ -dense in the compact space G, is necessarily a Baire space).…”

“…(b) Using (a), several workers (e.g., [4,23], [24] (2.5), [25]) made elementary cardinality observations like these, valid for infinite pseudocompact groups G.…”

“…In [76] it is proved that if K is a connected, compact group with center Z, then K/Z is a ∏-group. Comfort and Robertson [4] (4.2) showed that (for non-trivial K) the group K/Z(K) is a product of compact, connected, non-abelian Lie groups which are algebraically simple.…”