  It is proved that any countable topological group in which the filter of
neighborhoods of the identity element is not rapid contains a discrete set with
precisely one nonisolated point. This gives a negative answer to Protasov's
question on the existence in ZFC of a countable nondiscrete group in which all
discrete subsets are closed. It is also proved that the existence of a
countable nondiscrete extremally disconnected group implies the existence of a
rapid ultrafilter and, hence, a countable nondiscrete extremally disconnected
group cannot be constructed in ZFC.
