Let $f : N → Y$ be a map having right inverse $g$. Prove that $Y$ is at most countable?

Question

I know this implies $F$ is surjective, but not sure if this helps.

Answer 1

In this case, $g$ is an injection from $Y$ to $\mathbb N$, and then there is a bijection between $Y$ and $\operatorname{im} g \subseteq \mathbb N$. Since every subset of $\mathbb N$ is at most countable, the result follows.