= Ellis–Numakura lemma
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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 $a^2=a$.<Ergodic Ramsey Theory - an update by Vitaly Bergelson>[§3, Theorem 3.3 and its minimal compact subsemigroup proof, p. 28]{ref}
Reversing multiplication gives the same conclusion for a <right-topological semigroup>. Joint continuity of multiplication is not needed.
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