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The nature of the individual nonempty sets in the collection may make it possible to avoid the axiom of choice even for certain infinite collections.
For example, suppose that each member of the collection X is a nonempty subset of the natural numbers.
Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set.
This gives us a definite choice of an element from each set, and makes it unnecessary to apply the axiom of choice.

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