Encounters with Zorn's lemma early in study of mathematics

Question

Many mathematics students encounter the axiom of choice relatively early in their studies. For example, they see the claim that if we have a surjection $g\colon B\to A$, then there is one sided inverse $f\colon A\to B$ satisfying $g\circ f=id_B$. (This is equivalent to the axiom of choice.¹) Another proof which involves Axiom of Choice is the proof that, for real functions, sequential continuity is equivalent to the $\varepsilon$-$\delta$ definition of continuity.²

In more advanced courses, Axiom of Choice is often encountered in the form of Zorn's lemma. However, some preparation is probably needed before students can be shown such proofs. (At the very least, they should have good grasp of partially ordered sets.)

Question. Which applications of Zorn's lemma are among the earliest ones which appear in the typical curriculum for mathematics students.

¹See: The equivalence of “Every surjection has a right inverse” and the Axiom of Choice. (Also related: Why don't you need the Axiom of Choice when constructing the “inverse” of an injection? and There exists an injection from $X$ to $Y$ if and only if there exists a surjection from $Y$ to $X$.)

²For the role of AC in this statement, see: Continuity and the Axiom of Choice.

Answer 1

I think that the first encounter I had was in linear algebra I or II. After proving that every finite dimensional vector space has a basis, some people wondered what happens if we omit "finite dimensional", and the answer is positive with Zorn's lemma being mentioned.

But maybe this doesn't count, since there is no discussion about the lemma or the proof of the statement. It's just a remark to satisfy the immediate curiosity of more advanced students.

During the second year, however, at least in mathematics programs in Israel, Zorn's lemma makes a lot of appearances. I would often hear from my second year students that "this week we saw Zorn's lemma in all the courses", so it sort of appeared out of nowhere.

1. Algebraic structures and ring theory: every commutative ring with a unit has a maximal ideal.

2. Topology: Tychonoff's theorem, which may depend on how you prove it may already required you to talk about Zorn's lemma (e.g. talking about ultrafilters).

3. Logic: the completeness theorem (or compactness theorem, whichever is done first).

4. Set theory: Zorn's lemma itself. Obviously.

Which one is the first one encountered depends a lot on the program, the course, the syllabus, and the professor. Some of these proofs may lend themselves a bit more naturally to the Teichmüller–Tukey lemma, or the Hausdorff Maximality Principle, or maybe something else. It also depends on the prerequisites of each course, in my first first semester I already learned about partial orders and chains, so Zorn's lemma was easier to explain compared to the well-ordering theorem or transfinite recursion.

With the exception of the vector spaces example, which perhaps some students also see in their early linear algebra courses already, I think those are the natural examples.

Answer 2

We were hit with Zorn's lemma in the first group theory course before any other (this is in India). I never had the pleasure of taking courses in logic and set theory, and ring theory naturally came only later, so this answer is technically different from Asaf's.

Exercise. Let $G$ be a non-trivial finitely generated group, and $H$ a proper subgroup of $G$. Use Zorn's lemma to show that there is a maximal proper subgroup of $G$ that contains $H$.

This was our first experience getting punched in the gut by the damn thing (assignment 3 - week 5), and by the end of the course we were all bent over crying for mercy. It took several years, including courses in ring theory and topology, before we were acclimatized to it.