Why is the axiom of choice not taught from the start to mathematics undergraduates? I've recently discovered that the following theorems require the axiom of choice to be proven:


*

*every surjective function has a right inverse.

*a real-valued function that is sequentially continuous at a point is necessarily continuous in the neighbourhood sense at that point.

*every vector space has a basis.


When I revisited the proofs I was taught in first year, I was surprised that my lecturers had used the axiom of choice without declaring so. 
It seems strange that so much effort was dedicated to establishing that mathematics is a rigorous subject [indeed much time was spent on learning the field axioms, well-ordering axioms, Archimedean principle and (later) the completeness principle] but to ignore the axiom of choice.
I am interested if there are reasons for omitting to mention the axiom of choice. Are there pedagogical reasons? Is it deemed too complicated? Is it more contentious than the other axioms?
Question also asked at Mathematics Educators S.E.
 A: I talked about this literally today with my students, since we finally arrived to the discussion where the axiom of choice is openly assumed.
First of all, in courses which are not set theory, the axiom of choice is usually chucked aside. Who didn't construct a sequence by recursion in calculus? Or proved that $f$ is continuous at $x$ if and only every sequence $x_n\to x$ satisfies $f(x_n)\to f(x)$?
Those things, in their generality require choice. If you get to courses about logic, the compactness theorem needs and uses choice, and in algebraic structure the proof that every unital ring has a maximal ideal requires choice as well. Many other proofs use choice without any consequence. And nothing bad happens.
But in set theory the axiom of choice has a special place, from two accounts:

*

*Historically the axiom of choice was controversial. The reason is that its implications are vast and many of them are counterintuitive. Of course, the negation of the axiom of choice is also strange and counterintuitive, how can you have non-empty sets without being able to choose from them at once?
The reason, of course, is that it is not the axiom of choice or its negation which are counterintuitive. It is infinite sets, and infinite objects which baffles our finitary minds.


*Pedagogically, until the axiom of choice enters the discussion openly, you usually avoid explicit axioms, or that you do your best to allow some naiveté in the proofs (even when the axioms are mentioned). But everything that you prove to exist comes from specific formula.
You want to show that $\Bbb Q$ is countable? Find a surjection from $\Bbb N$ onto $\Bbb Q$, and show that this suffices to establish that result.
You want to show that $\Bbb R$ and $\mathcal P(\Bbb N)$ are equipotent? Write injections for both sides, and prove the Cantor-Bernstein theorem which essentially gives you a recipe for a bijection, given the two injections.
But with the axiom of choice it becomes different. You are taking for granted some choice function exists. How does it work? What does it do? Nobody knows. Not only that, different choice functions will often give out different result in many constructions, which is fine, but the non-canonicity is deep-seated in the proofs here.
And that is both an important thing to point out to students, as well an issue for confusion when you expect the student to be able and identify what sort of arguments were used at each point and turn.
So the axiom of choice can be easily assumed, and in fact in many places it is assumed, even implicitly, from the start. But if you want to properly train people to understand set theoretic subtleties then separating the axiom of choice is usually a good thing.
As a result, in some universities, and in some courses, the use of the axiom of choice is minimized when it is not totally necessary.
