Let K be a compact, Hausdorff
topological group, 𝒢(K) the set of dense, pseudocompact subgroups of K, and
m(K) =min{|G| : G ∈𝒢(K)}. We show: (1) m(K) is a function of the weight of K
(in the sense that if K′ is another such group with w(K) = w(K′), then
m(K) = m(K′)); and (2) if K is connected then every totally dense subgroup D of K
satisfies |D| = |K|. With these results in hand we classify (a) those cardinals α such
that m(K) < |K| when w(K) = α and (b) those cardinals α such that some compact
K with w(K) = α admits a totally dense subgroup D with |D| < |K|. The
conditions of (a) and (b) are incompatible in some models of ZFC (e.g.,
under GCH) and are compatible in others. Thus the following question, the
origin of this work, is undecidable in ZFC: Is there a compact, Hausdorff,
topological group K with a totally dense, pseudocompact subgroup G such that
|G| < |K|?
