Mathematical questions whose answer depends on the Axiom of Choice Question inspired by the following surprising claim:
The chromatic number of the $R^n$ hyperplane may depend on whether the Axiom of Choice is available or not.
https://shelah.logic.at/papers/E33/
See more on the chromatic number (Hadwiger-Nelson) problem:
https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem
Are there any interesting non-artificial claims out there (like well known theorems) whose veracity critically depends on the Axiom of Choice - they comletely fall apart (or the answer changes) if the AC is removed?
 A: Of course:

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*There exists a free ultrafilter.

*There exists a Lebesgue non-measurable set.

*There exists a non-Borel set.

*Countable unions of countable sets are countable.

*The real numbers are not a countable union of countable sets.

*Every complete metric space is a Baire space.

*Every vector space has a Hamel basis.

*Every field has an algebraic closure, and it is unique up to isomorphism.

*Urysohn's lemma.

*Every infinite set has a countably infinite subset.

*Product of compact spaces are compact.

*Every tree of height $\omega$ in which every level is finite has a branch.

*Continuity of a real function at $x$ is equivalent to sequential continuity.

*Banach-Tarski theorem.

And so on and so forth. There's about infinity of them.
Also relevant:

*

*Advantage of accepting the axiom of choice

*Advantage of accepting non-measurable sets
And books:

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*Herrlich, H. Axiom of Choice. Lecture Notes in Mathematics, Springer, 2006.


*Jech, T. The Axiom of Choice. North-Holland (1973).


*Howard, P. and Rubin, J.E. Consequences of the Axiom of Choice. American Mathematical Soc. (1998). Also see the online database for the book.


*Moore, G. H. Zermelo's Axiom of Choice. Springer-Verlag (1982).
A: I'd say the Banach-Alaoglu theorem is another example (BA : the unit ball of the dual of a Banach space is weakly-* compact).
In the very common case where the space is also separable, the Axiom of choice is not needed, but we there are spaces which are not separable (for example, $L^\infty(\mu)$). So (even though it is a bit far fetched), if you have a probability space $(\Omega, \mu)$ with no natural locally compact topology, and complicated enough so that $L^\infty(\mu)$ is not separable, then you'd need the axiom of choice to extract weak-* convergent subsequences of a sequence of bounded linear forms on $L^\infty(\mu)$.
