# Axiom of choice variants [duplicate]

Is there any good book where the equivalence of AC to the statement "Any surjection has a right inverse" is proved (and maybe other equivalences)? I could do $AC \Rightarrow$ "Any surjection has a right inverse" by myself but I have no idea for the other direction.

## marked as duplicate by Asaf Karagila♦ set-theory StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Aug 10 '14 at 12:40

Given any family of sets $A_i$ indexed by $i\in I$, consider the map $\phi:\mathcal{A}\to I$ from the disjoint union of the family to $I$. The existence of a one-sided inverse to $\phi$ is precisely a choice function. Hence the converse is immediate.