Ellis–Numakura lemma explained

In mathematics, the Ellis–Numakura lemma states that if S is a non-empty semigroup with a topology such that S is a compact space 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.

The compact topological semigroups appearing in this lemma should be distinguished with compact semigroups, in which "compact" is not used with its topological meaning.

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).

Proof

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