Dense minimal pseudocompact, subgroups of compact Abelian groups

摘要

Motivated by a recent theorem of Comfort and van Mill, we study when a pseudocompact Abelian group admits proper dense minimal pseudocompact subgroups and give a complete answer in the case of compact Abelian groups. Moreover we characterize compact Abelian groups that admit proper dense minimal subgroups. Throughout this paper all topological groups are Hausdorff. A topological group G is pseudocompact if every continuous real-valued function of G is bounded [22]. Moreover a pseudocompact group G is s-extremal if it has no proper dense pseudocompact subgroup and r-extremal if there exists no strictly finer pseudocompact group topology on G [2]. Recently in [8] Comfort and van Mill proved the following relevant result about pseudocompact Abelian groups, which solves a problem raised in 1982 [3,7] and studied intensively since then.