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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Cohomology detects failures of the axiom of choice


Author: Andreas Blass
Journal: Trans. Amer. Math. Soc. 279 (1983), 257-269
MSC: Primary 03E25; Secondary 03G30, 55N25, 55N99
MathSciNet review: 704615
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0704615-7
PII: S 0002-9947(1983)0704615-7
Keywords: Axiom of choice, cohomology, torsor, permutation model
Article copyright: © Copyright 1983 American Mathematical Society