 Ellis–Numakura lemma

Every nonempty compact Hausdorff space carrying a left-topological semigroup structure contains an idempotent element of a semigroup.

Choose an inclusion-minimal nonempty compact subsemigroup $K$, using Zorn's lemma and the finite intersection property. For $a\in K$, the set $aK$ is a nonempty compact subsemigroup, so minimality gives $aK=K$. Hence $F=\{x\in K:ax=a\}$ is nonempty. It is a compact closed subset of $K$ by continuity and the Hausdorff space assumption, and is a subsemigroup because $a(xy)=(ax)y=ay=a$ for $x,y\in F$. Minimality gives $F=K$, so $a^2=a$.^[ref]

Reversing multiplication gives the same conclusion for a right-topological semigroup. Joint continuity of multiplication is not needed.

The Ellis-Numakura lemma is a result in the field of dynamical systems, particularly in the study of topological dynamics and the behavior of semigroups. It is named after mathematicians John Ellis and Kōji Numakura. The lemma deals with the connection between a compact space and the continuous semigroups acting on it, providing conditions under which certain properties hold for the invariant measures of these semigroups.