On the supremum of the pseudocompact group topologies (Q2469558)

A topological space on which each real valued continuous function is bounded is called pseudocompact. Let \textbf{P} be the class of pseudocompact Hausdorff topological groups, and \({\mathbf P}'\) the class of groups \(G\) admitting a topology \({\mathcal T}\) such that \((G,{\mathcal T})\in{\mathbf P}\). It is known that every \((G,{\mathcal T})\in{\mathbf P}\) is totally bounded, i.e., for every nonempty \(U\in{\mathcal T}\) there is a finite subset \(F\) of \(G\) such that \(G=FU\). Thus, if for \(G\in{\mathbf P}'\) one denotes by \({\mathcal T}^{\vee}(G)\) the supremum of all pseudocompact group topologies on \(G\), and by \({\mathcal T}^{\#}(G)\) the supremum of all totally bounded group topologies on \(G\), then the inclusion \({\mathcal T}^{\vee}(G)\subseteq {\mathcal T}^{\#}(G)\) holds. The present paper is devoted to the following conjecture: Every abelian group \(G\in{\mathbf P}'\) satisfies the equality \({\mathcal T}^{\vee}(G)= {\mathcal T}^{\#}(G)\). It is proved that this conjecture is true for abelian \(G\in{\mathbf P}'\) with any of the following (overlapping) properties: (a) \(G\) is a torsion group; (b) \(| G| \leq 2^{\mathfrak c}\); (c) \(r_0(G)=| G| =| G| ^{\omega}\), where \(r_0(G)\) denotes the torsion-free rank of \(G\); (d) \(| G| \) is a strong limit cardinal, and \(r_0(G)=| G| \); (e) some topology \({\mathcal T}\) with \((G,{\mathcal T})\in\) \textbf{P} satisfies \(w(G, {\mathcal T})\leq {\mathfrak c}\); (f) some pseudocompact group topology on \(G\) is metrizable; (g) \(G\) admits a compact group topology, and \(r_0(G)=| G| \).