Abstract.
The axiom of choice is equivalent to the shrinking principle: every indexed cover of a set has a refinement with the same index set which is a partition. A simple and direct proof of this equivalence is given within an elementary fragment of constructive Zermelo–Fraenkel set theory. Variants of the shrinking principle are discussed, and it is related to a similar but different principle formulated by Vaught.
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Banaschewski, B., Schuster, P. The shrinking principle and the axiom of choice. Mh Math 151, 263–270 (2007). https://doi.org/10.1007/s00605-007-0468-2
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DOI: https://doi.org/10.1007/s00605-007-0468-2