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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Cohomology detects failures of the axiom of choice
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by Andreas Blass PDF
Trans. Amer. Math. Soc. 279 (1983), 257-269 Request permission

Abstract:

We propose that failures of the axiom of choice, that is, surjective functions admitting no sections, can be reasonably classified by means of invariants borrowed from algebraic topology. We show that cohomology, when defined so that its usual exactness properties hold even in the absence of the axiom of choice, is adequate for detecting failures of this axiom in the following sense. If a set $X$, viewed as a discrete space, has trivial first cohomology for all coefficient groups, then every $X$-indexed family of nonempty sets has a choice function. We also obtain related results when the coefficient groups are required to be abelian or well-orderable. In particular, we show that, if all discrete spaces have trivial first cohomology for all abelian coefficient groups, then the axiom of choice holds.
References
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 279 (1983), 257-269
  • MSC: Primary 03E25; Secondary 03G30, 55N25, 55N99
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0704615-7
  • MathSciNet review: 704615