Discrete subsets in topological groups and countable extremally disconnected groupsстатьяИсследовательская статья
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Аннотация:In 1967 Arhangel’skii posed the problem of the existence in ZFCof a nondiscrete extremally disconnected topological group. The general caseis still open, but we solve Arhangel’skii’s problem for the class of countablegroups. Namely, we prove that the existence of a countable nondiscrete ex-tremally disconnected group implies the existence of a rapid ultrafilter; hence,such a group cannot be constructed in ZFC. We also prove that any countabletopological group in which the filter of neighborhoods of the identity elementis not rapid contains a discrete set with precisely one limit point, which gives anegative answer to Protasov’s question on the existence in ZFC of a countablenondiscrete group in which all discrete subsets are closed.