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Axioms 2016, 5(1), 2; doi:10.3390/axioms5010002

Non-Abelian Pseudocompact Groups

1,†,* and 2,†,*
1
Department of Mathematics, Wesleyan University, Middletown, CT 06457, USA
2
Institut für Mathematik, Universität Paderborn, Warburger Str. 100, Paderborn D-33095, Germany
These authors contributed equally to this work.
*
Authors to whom correspondence should be addressed.
Academic Editor: Sidney A. Morris
Received: 29 September 2015 / Revised: 24 November 2015 / Accepted: 23 December 2015 / Published: 12 January 2016
(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)
View Full-Text   |   Download PDF [296 KB, uploaded 12 January 2016]

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. View Full-Text
Keywords: topological group; group topology; pseudocompact topological group; Gδ-dense subgroup; countably compact group; ω-bounded group; refinement of topology; extremal pseudocompact group topological group; group topology; pseudocompact topological group; Gδ-dense subgroup; countably compact group; ω-bounded group; refinement of topology; extremal pseudocompact group
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Comfort, W.W.; Remus, D. Non-Abelian Pseudocompact Groups. Axioms 2016, 5, 2.

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