Weakly metrizable pseudocompact groups

Abstract

We study various weaker versions of metrizability for pseudocompact abelian groups G: singularity (G possesses a compact metrizable subgroup of the form mG, m>0), almost connectedness (G is metrizable modulo the connected component) and various versions of extremality in the sense of Comfort and co-authors (s-extremal, if G has no proper dense pseudocompact subgroups, r-extremal, if G admits no proper pseudocompact refinement). We introduce also weaklyextremal pseudocompact groups (weakening simultaneously s-extremal and r-extremal). It turns out that this “symmetric” version of ex-tremality has nice properties that restore the symmetry, to a certain extent, in the theory of extremal pseudocompact groups obtaining simpler uniform proofs of most of the known results. We characterize doubly extremal pseudocompact groups within the class of s-extremal pseudocompact groups. We give also a criterion for r-extremality for connected pseudocompact groups

Pseudocompact groups: progress and problems

“…(One should note that this result, essential to our approach, was found also by Dikranjan, Giordano Bruno, and Milan [23] (4.11).) Suppose not only that some N ∈ Λ(G) has a proper, dense pseudocompact subgroup D, but also that D may be chosen so that r 0 (N/D) c. Then, there is a selection set X for the coset space G/N , constructed recursively with some care, such that the pseudocompact subgroup H := D ∪ X of G is proper, indeed it is proper for the good reason that r 0 (G/H ) c. This fulfills two purposes at once: It obviously gives a proper, dense, pseudocompact subgroup of G, thus responding to Question I.…”

“…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). This prompted Giordano Bruno to raise these questions (see also [22] ( §1) for additional motivational discussion).…”

Non-Abelian Pseudocompact Groups

“…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.…”

Extremal pseudocompact Abelian groups are compact metrizable

“…(Note added September 15, 2006. The referee has pointed out that a proof of Lemma 3.1 is also available in the preprint [10]. )…”

“…This question has generated much attention during the last two decades. See [1], [12] and [10] for more information. An affirmative answer was given in [7] for zero-dimensional Abelian groups.…”