Ellis Lemma on idempotent elements asserts that:



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> **Lemma (Ellis).** Every compact semigroup has an idempotent.
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The proof below is excerpted from Todorcevic's *Introduction to Ramsey Spaces*, [Lemma 2.1](https://books.google.com/books?id=Yhyi8vRfPNAC&pg=PA27).



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> Let $S$ be a compact semigroup. Pick by Zorn’s Lemma a minimal compact subsemigroup $R \subseteq S$ and an arbitrary $s \in R$. Then $Rs$ is also a compact subsemigroup and $Rs \subseteq R$. Hence $Rs = R$. Let $P = \{x \in R : xs = s\}$. Then $P \neq \emptyset$, since $s \in Rs$. Note that $P$ is also a compact subsemigroup of $S$. Hence $P = R$ and therefore
> $s^2 = s$.
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It appears that Zorn's Lemma is required to obtain a minimal compact subsemigroup $R \subseteq S$. I do not understand why, as it appears to me that we can simply let:
$$
R := \bigcap \{Q \subseteq S : Q \text{ is a compact subsemigroup of $S$}\}
$$
What is the error in reasoning here?



Why is choice needed in Ellis' Lemma?
The issue is that $R$ could be empty. When you apply Zorn's lemma, a nested intersection of non-empty compact sets is non-empty, guaranteeing minimal non-empty elements.


