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Ellis–Numakura lemma : ウィキペディア英語版
Ellis–Numakura lemma
In mathematics, the Ellis–Numakura lemma states that if ''S'' is a non-empty semigroup with a topology such that ''S'' is compact and the product is semi-continuous, then ''S'' has an idempotent element ''p'', (that is, with ''pp'' = ''p''). The lemma is named after Robert Ellis and Katsui Numakura.
==Applications==
Applying this lemma to the Stone–Čech compactification ''βN'' of the natural numbers shows that there are idempotent elements in ''βN''. The product on ''βN'' is not continuous, but is only semi-continuous (right or left, depending on the preferred construction, but never both).

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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