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Specker's theorem for Nöbeling's group

Author: Andreas Blass
Journal: Proc. Amer. Math. Soc. 130 (2002), 1581-1587
MSC (2000): Primary 20K20; Secondary 03E25, 03E35, 03E60, 03E75, 20K25, 20K30, 20K45
Published electronically: October 23, 2001
MathSciNet review: 1887001
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Abstract: Specker proved that the group $\mathbb{Z} ^{\aleph_0}$ of integer-valued sequences is far from free; all its homomorphisms to $\mathbb{Z} $ factor through finite subproducts. Nöbeling proved that the subgroup $\mathcal{B}$ consisting of the bounded sequences is free and therefore has many homomorphisms to $\mathbb{Z} $. We prove that all ``reasonable'' homomorphisms $\mathcal{B}\to\mathbb{Z} $ factor through finite subproducts. Among the reasonable homomorphisms are all those that are Borel with respect to a natural topology on $\mathcal{B}$. In the absence of the axiom of choice, it is consistent that all homomorphisms are reasonable and therefore that Specker's theorem applies to $\mathcal{B}$as well as to $\mathbb{Z} ^{\aleph_0}$.

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Additional Information

Andreas Blass
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109–1109

Received by editor(s): October 13, 2000
Received by editor(s) in revised form: December 18, 2000
Published electronically: October 23, 2001
Additional Notes: This work was partially supported by NSF grant DMS–0070723. The author thanks the Mittag-Leffler Institute for supporting a visit in October 2000, during which this paper was written.
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2001 American Mathematical Society

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