Abstract:In 1967 Arhangel'skii posed the problem of the existence in ZFC of a nondiscrete extremally disconnected topological group. The general case is still open, but we solve Arhangel'skii's problem for the class of countable groups. Namely, we prove that the existence of a countable nondiscrete extremally disconnected group implies the existence of a rapid ultrafilter; hence, such a group cannot be constructed in ZFC. We also prove 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 limit point, which 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.

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中文翻译：

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拓扑组和可数极端断开组中的离散子集

摘要：1967年，Arhangel'skii提出了ZFC中存在一个非离散的，极端断开的拓扑群的问题。一般情况仍未解决，但我们为可数组的类解决了Arhangel'skii问题。就是说，我们证明了一个可数的非离散的极端断开的群的存在暗示了一个快速超滤器的存在。因此，不能在ZFC中构造这样的组。我们还证明，其中身份元素的邻域过滤器不快速的任何可数拓扑组都包含一个恰好具有一个极限点的离散集，这给普罗塔索夫关于ZFC中存在可数非离散组的问题的否定答案。所有离散的子集都是封闭的。

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