On topological and algebraic structure of extremally disconnected semitopological groups
On topological and algebraic structure of extremally disconnected semitopological groups
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发表时间：
2000
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通讯作者：
A. Arhangel'skii
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作者：
A. Arhangel'skii
从一个关于极端不连通空间同胚的Frol'\i k定理的非常简单的证明开始，我们证明了该定理如何隐含Malychin的一个著名结果：每一个极端不连通拓扑群都包含一个开子群和一个闭子群，这些子群由$2阶的元素组成。应用Frol'\i k定理，进一步得到了关于极端不连通拓扑群和具有连续逆的半拓扑群结构的定理。特别地，每一个具有连续逆和平方根的Lindelof极断半拓扑群都是可数的，每一个极断拓扑域都是离散的。
Starting with a very simple proof of Frol'\i k's theorem on homeomorphisms of extremally disconnected spaces, we show how this theorem implies a well known result of Malychin: that every extremally disconnected topological group contains an open and closed subgroup, consisting of elements of order $2$. We also apply Frol'\i k's theorem to obtain some further theorems on the structure of extremally disconnected topological groups and of semitopological groups with continuous inverse. In particular, every Lindelof extremally disconnected semitopological group with continuous inverse and with square roots is countable, and every extremally disconnected topological field is discrete.