= Right-inverse characterization of the axiom of choice
The assertion that every <surjective function> has a <right inverse> is equivalent to the <axiom of choice>. Choice selects one member of each nonempty <fiber of a function>, giving a right inverse. Conversely, for a family $(X_i)_{i\in I}$ of nonempty sets, the projection from the disjoint union $\{(i,x):i\in I,\ x\in X_i\}$ onto $I$ is surjective. A right inverse selects one element from each $X_i$, giving a choice function.
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