TITLE: Why is choice needed in Ellis' Lemma?
QUESTION [9 upvotes]: Ellis Lemma on idempotent elements asserts that:

Lemma (Ellis). Every compact semigroup has an idempotent.

The proof below is excerpted from Todorcevic's Introduction to Ramsey Spaces, Lemma 2.1.

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
$s^2 = s$.

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?

REPLY [9 votes]: 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.