Example of surjective function without right-inverse (without AoC)

Question

As I understand from this question, without the axiom of choice, we can have surjective functions without right-inverse. Is that correct? Is there any such example of a surjective function where we need to use the AoC to prove the existence of the right-inverse?

Answer 1

From the comments: Let $S$ be the set of all sequences $f: \mathbb{N} \to \mathbb{R}$ and $T = \{Im(f)| f\in S\}$. Then the function $\phi: S \to T$ with $\phi(f)=Im(f)$ is surjective by definition. In order to define the right-inverse of $\phi$ for a given $t \in T$, you need to assign an $n \in \mathbb{N}$ to every real number in $t$. For this, you need the axiom of choice.