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do you need to invoke the axiom of choice to choose a point from a collection of sets if the sets are single-point sets?

for example, suppose f:A->B is injective. to create a left inverse g:f(A)->A, we need to "choose" a point from the preimage of b for all b in f(A) and send b to this point by g. but because f is injective, the preimage of b is just one point. do we have to use the axiom of choice to create the left inverse g? after all, the new set being created by "choosing" these points from the preimages simply give the set A, which is not a new set.

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# Axiom of choice with single-point sets

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