Pseudocompact totally dense subgroups

Abstract

Abstract. It was shown by Dikranjan and Shakhmatov in 1992 that if a compact abelian group K admits a proper totally dense pseudocompact subgroup, then K cannot have a torsion closed G δ -subgroup; moreover this condition was shown to be also sufficient under LH. We prove in ZFC that this condition actually ensures the existence of a proper totally dense subgroup H of K that contains an ω-bounded dense subgroup of K (such an H is necessarily pseudocompact). This answers two questions posed by Dikranjan and Shakhmatov (Proc.

Non-Abelian Pseudocompact Groups

“…Since this topic is a bit removed from our central focus here, for details in this direction we simply refer the reader to the relevant papers known to us: [50,51], [36] [52][53][54][55]. We note explicitly that, building upon and extending results from her thesis [56], Giordano Bruno and Dikranjan [57] characterized those compact abelian groups with a proper totally dense pseudocompact subgroup as those with no closed torsion G δ -subgroup.…”

“…Prior to the appearance of [44,45], researchers in Udine, Italy, considered conditions weaker than metrizability which suffice to guarantee that a pseudocompact abelian group G is both r-and s-extremal [56,57,60,61]. Here is a sample result.…”

Pseudocompact groups: progress and problems

“…Concatenating definitions and results from [31,23,22,32], let us say that a pseudocompact abelian group G is c-extremal [resp., singular] if every dense, pseudocompact subgroup H of G has r 0 (G/H ) < c [resp., if there is an integer m > 0 such that mG is metrizable]. The utility of these ideas in the study of extremal pseudocompact groups is evident from this theorem, shown in [31,23]: If some N ∈ Λ(G) is not c-extremal then G itself is not c-extremal (hence, is neither r-nor s-extremal).…”

“…The utility of these ideas in the study of extremal pseudocompact groups is evident from this theorem, shown in [31,23]: If some N ∈ Λ(G) is not c-extremal then G itself is not c-extremal (hence, is neither r-nor s-extremal). This prompted Giordano Bruno to raise these questions (see also [22] ( §1) for additional motivational discussion). [32].)…”

“…Parts (a) and (b) of Question 5 have been solved completely when G is compact, in [22] and [32], respectively.…”