Theorem 47.4.11 (Ellis).label Let $S$ be a non-empty compact Hausdorff right topological semigroup and $I \subset S$ be a closed left ideal, then there exists a minimal left ideal $I \subset S$ and a minimal idempotent $e \in I$.
Proof. By Zorn’s Lemma, $I$ admits a minimal non-empty closed right topological subsemigroup $K \subset S$. For each $a \in K$, $Ka$ is closed by continuity of right multiplication and Proposition 5.16.3. By minimality of $K$, $K = Ka$, so there exists $e \in K$ such that $a = ea$. Let $E = \bracsn{b \in K|ba = a}$, then $E$ is also a closed and right topological subsemigroup. By minimality of $K$, $E = K$, and in particular $a \in E$.$\square$
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