Abstract
Here are three recently-established theorems from the literature. (A) (2006) Every non-metrizable compact abelian group K has 2|K| -many proper dense pseudocompact subgroups. (B) (2003) Every non-metrizable compact abelian group K admits 22|K| -many strictly finer pseudocompact topological group refinements. (C) (2007) Every non-metrizable pseudocompact abelian group has a proper dense pseudocompact subgroup and a strictly finer pseudocompact topological group refinement. (Theorems (A), (B) and (C) become false if the non-metrizable hypothesis is omitted.) With a detailed view toward the relevant literature, the present authors ask: What happens to (A), (B), (C) and to similar known facts about pseudocompact abelian groups if the abelian hypothesis is omitted? Are the resulting statements true, false, true under certain natural additional hypotheses, etc.? Several new results responding in part to these questions are given, and several specific additional questions are posed.
Highlights
Specific references to the literature concerning Theorems (A), (B) and (C) of the Abstract are given in 5.7(d), 8.2.2 and 4(l), respectively
(b) It is a fundamental theorem of Weil [7] that ∗ the totally bounded groups are exactly the topological groups G which embed as a dense topological subgroup of a compact group
(f) Negating the tempting conjecture that parallel results might hold for locally compact groups, Rajagopalan and Soundrarajan [32] show that for each infinite cardinal κ there is on the group Tκ a locally compact group topology which admits no proper dense subgroup
Summary
Specific references to the literature concerning Theorems (A), (B) and (C) of the Abstract are given in 5.7(d), 8.2.2 and 4(l), respectively. Abelian or not, is compact, admits neither a proper dense pseudocompact subgroup nor a proper pseudocompact group refinement (see 4(a)); (A), (B) and (C) all become false when the non-metrizability hypothesis is omitted. All hypothesized topological spaces and topological groups in this paper are assumed to be Tychonoff spaces
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