Nonexistence of countable extremally disconnected groups with many open subgroups (Q465862)

A topological space is called extremally disconnected if the closure of an open subset is open. A. V. Arhangel'skii posed the following question (still unsolved): Does there exist in ZFC a nondiscrete Hausdorff topological group? V. I. Malykhin has proved that any nondiscrete extremally disconnected group contains an open Boolean subgroup (an abelian group written additively is called Boolean if \(2x=0\) for all \(x\)) (see the Section 4.5 in [\textit{A. Arhangel'skii} and \textit{M. Tkachenko}, Topological groups and related structures. Hackensack, NJ: World Scientific; Paris: Atlantis Press (2008; Zbl 1323.22001)]).NEWLINENEWLINEIn this paper it is proved that if there exists in ZFC a countable nondiscrete extremally disconnected group, then there exists a countable nondiscrete extremally disconnected Boolean group without open proper subgroups.