ИСТИНА

The problem of the existence in ZFC of a nondiscrete Hausdorff extremally disconnected topological group was posed by Arhangel'skii in 1967 and has been extensively studied since then. Several consistent examples have been constructed, but Arhangel'skii's problem remains unsolved. All (consistent) examples of extremally disconnected groups known to the author have a base of neighborhoods of the identity element consisting of subgroups. The main result of this paper is that a countable ZFC example cannot be constructed in this way; to be more precise, we prove that the existence of a countable extremally disconnected topological group containing a family of open subgroups whose intersection has empty interior (or, equivalently, admitting a continuous isomorphism onto the direct sum $\bigoplus_\omega\mathbb Z_2$ of countably many copies of $\mathbb Z_2$ with the product topology) implies the existence of a rapid ultrafilter. Thus, if there exists in ZFC a countable nondiscrete extremally disconnected group, then there must exist such a group without open subgroups.