We say an infinite set $X$ is *splittable* if there are $X_1, X_2\subseteq X$ with $X_1\cap X_2 = \emptyset$, $X_1\cup X_2 = X$ and there are bijections $\varphi:X_1\to X_2$ and $\psi:X_1\to X$.

Does the statement "Every infinite set is splittable" imply $\mathsf{AC}$?

youasked this question not too long ago. $\endgroup$ – Asaf Karagila Apr 19 '16 at 14:43