A common reply to the question "why should nonlogicians take note when using the Axiom of Choice (AC)" seems to be "AC fails in many categories, and we want to know that our results continue to hold internal to those categories". This is a great reason (as far as I'm concerned) and is my default answer when I am posed this question. After all, it is entirely natural to consider topological objects (groups, vector spaces, etc.) and AC fails in $\mathsf{Top}$ (any continuous bijection which is not open is a surjection that isn't split).
Unfortunately, the only examples that I have found which exhibit this failure have been rather complicated (cf. this famous MO question).
I would love to see a "simple" theorem which fails in this way, or if there isn't one, I would love to know why.
As an example of the kind of theorem I am interested in, here is one which doesn't quite work:
Is there a finitely generated topological group $G$ which has no maximal subgroup?
The classical proof that every (finitely generated) group has a maximal subgroup requires AC in the form of Zorn's Lemma. It seems reasonable, then, that this is a result which might fail in $\mathsf{Top}$.
Unfortunately, this theorem doesn't fail in $\mathsf{Top}$ after all. We can do our dirty work in $\mathsf{Set}$, finding a maximal subgroup with AC (ignoring the topology), and then endow it with the subspace topology afterwards.
Are there any (preferably algebraic) theorems whose proofs in $\mathsf{Set}$ require AC, and which are false in $\mathsf{Top}$? Ideally they would be as simple as the one above, though they would actually fail...
Edit: I am equally interested in the failure of theorems relying on AC in categories of sheaves, etc. if it is easier to find an example there. I would like the theorems to be algebraic, though.
Thanks in advance ^_^
Let $F$ be a field object in the category $\mathbf{Set}$. Every $F$-vector-space object in $\mathbf{Set}$ is isomorphic to a direct sum of copies of $F$. This is equivalent to the fact that every vector space has a basis, which requires the Axiom of Choice.
We consider the following generalization: let $F$ be a field object in the category $\mathbf{Top}$. Is every $F$-vector-space object in $\mathbf{Top}$ isomorphic to a direct sum of copies of $F$?
Let $F= \mathbb{R}$ equipped with the usual Euclidean topology. Take $1 \leq p < q$. Assume for a contradiction that $\ell^p$ and $\ell^q$ are both direct sums of copies of $\mathbb{R}$. Both spaces have cardinality $|\mathbb{R}|$, and have bases of that same cardinality. Since all bases of a vector space are equipotent, this would mean that they're both isomorphic to the direct sum of $|\mathbb{R}|$-many copies of $\mathbb{R}$. But by Pitt's theorem $\ell^p$ and $\ell^q$ are not linearly homeomorphic, a contradiction.
We should note that the counterexample given above is honest in the sense that Zermelo-Fraenkel set theory without Choice does not prove the existence of a basis for $\ell^p$. Dishonest counterexamples abound (and their existence should not surprise us, given that the failure of Choice is by far not the only property distinguishing $\mathbf{Set}$ from $\mathbf{Top}$).
Let $S$ be a non-initial object in the category $\mathbf{Set}$. Then we can find arrows $e: 1 \rightarrow S$, $i: S \rightarrow S$ and $m: S \times S \rightarrow S$ making $S$ into a group object. Given Choice, this follows immediately from the existence of cyclic groups (for the finite case) and the upward Löwenheim–Skolem theorem (for the infinite case). Moreover, there's an easy argument showing that this result strictly requires Choice.
Again, the corresponding statement fails for objects $S$ in $\mathbf{Top}$, even if we restrict our attention to very nice spaces. For example, the bouquet of $n$ circles does not admit a topological group structure for any $n \geq 2$.
Now take your favorite non-trivial group $G$, and consider the category $G$-$\mathbf{Set}$. An object of this category is a set equipped with a left-action of $G$, and a morphism is just an equivariant map between two such sets.
Unlike $\mathbf{Top}$, $G$-$\mathbf{Set}$ has very similar properties to $\mathbf{Set}$: it forms what is known as a Boolean topos. However, the analogue of the statement above fails even for objects of the category $G$-$\mathbf{Set}$. For consider the underlying set of the group $G$ equipped with the left multiplication action of $G$, and denote it $G^\star$. To make $G^\star$ into a group object, one has to construct a morphism from the terminal object $e: 1 \rightarrow G^\star$. The terminal object $1$ of $G$-$\mathbf{Set}$ is the one-element set $\{0\}$ equipped with the trivial $G$-action, so one would have $\forall g \in G. e(0) = e(g \cdot 0) = g e(0)$. Therefore, if $G$ is non-trivial, no suitable $e$ exists.
This relates directly to the failure of the Axiom of Choice in $G$-$\mathbf{Set}$: the argument above proves that the unique map $G^\star \rightarrow 1$ does not have a section.