Extremal pseudocompact topological groups
A pseudocompact group \(G\) is said to be \(r\)-\textit{extremal} [resp., \(s\)-\textit{extremal}] if \(G\) admits no strictly finer pseudocompact group topology [resp., \(G\) has no dense pseudocompact subgroup]. A pseudocompact group is called \textit{doubly extremal} if it is both \(r\)- and \(s\)-extremal. The starting point of this paper is the following conjecture which remains open: No nonmetrizable pseudocompact Abelian group is neither \(r\)- nor \noindent\(s\)-extremal. The authors develop a technique of strengthening a pseudocompact group topology (by using homomorphisms in the circle group \(\mathbb{T}\)) (Theorem 3.10). It is proved that the following classes of pseudocompact groups are neither \(r\)-extremal nor \(s\)-extremal: i) non-divisible connected pseudocompact Abelian groups (Theorem 4.5); ii) totally disconnected Abelian groups of uncountable weight (Corollary 5.8): iii) pseudocompact Abelian groups for which \(\omega < w(G)\leq 2^{\omega}\) (here \(w(G)\) is the weight of \(G\)) (Theorem 5.10(a)). If in a pseudocompact Abelian group \(G\) of uncountable weight every closed subgroup is pseudocompact and \(G\) contains a nonmetrizable finitely generated subgroup, then \(G\) is not \(s\)-extremal (Theorem 6.9). It is proved that under Luzin's hypothesis \([2^\omega=2^{\omega_1}]\) if in a topological Abelian group \(G\) of uncountable weight every closed subgroup is pseudocompact, then \(G\) is not \(s\)-extremal (Theorem 6.10). Furthermore, it is proved that if \(G\) is a pseudocompact group, \(w(G)>\omega,\) then: i) If every closed subgroup of \(G\) is pseudocompact, then \(G\) is not doubly extremal; ii) if \(G\) is countably compact or torsion-free, then \(G\) is not doubly extremal (Corollary 7.3). The paper contains other interesting results and open questions.
scientific article; zbMATH DE number 2157520
scientific article