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.

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).

By compactness, there is a minimal non-empty compact sub semigroup of *S*, so replacing *S* by this sub semi group we can assume *S* is minimal.
Choose *p* in *S*. The set *Sp* is a non-empty compact subsemigroup, so by minimality it is *S* and in particular contains *p*, so the set of elements *q* with *qp* = *p* is non-empty.
The set of all elements *q* with *qp* = *p* is a compact semigroup, and is nonempty by the previous step, so by minimality it is the whole of *S* and therefore contains *p*. So *pp* = *p*.