Discrete Subsets in Topological Groups and Countable Extremally Disconnected Groups

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