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A countable free closed non-reflexive subgroup of $ \mathbb{Z}^{\mathfrak{c}}$


Authors: Maria Vicenta Ferrer, Salvador Hernández and Dmitri Shakhmatov
Journal: Proc. Amer. Math. Soc. 145 (2017), 3599-3605
MSC (2010): Primary 22A25; Secondary 20C15, 20K30, 22A05, 54B10, 54D30, 54H11
DOI: https://doi.org/10.1090/proc/13532
Published electronically: April 12, 2017
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Abstract: We prove that the group $ G=\mathrm {Hom}(\mathbb{Z}^{\mathbb{N}}, \mathbb{Z})$ of all homomorphisms from the Baer-Specker group $ \mathbb{Z}^{\mathbb{N}}$ to the group $ \mathbb{Z}$ of integer numbers endowed with the topology of pointwise convergence contains no infinite compact subsets. We deduce from this fact that the second Pontryagin dual of $ G$ is discrete. As $ G$ is non-discrete, it is not reflexive. Since $ G$ can be viewed as a closed subgroup of the Tychonoff product $ \mathbb{Z}^{\mathfrak{c}}$ of continuum many copies of the integers $ \mathbb{Z}$, this provides an example of a group described in the title, thereby resolving a problem by Galindo, Recoder-Núñez and Tkachenko. It follows that an inverse limit of finitely generated (torsion-)free discrete abelian groups need not be reflexive.


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Additional Information

Maria Vicenta Ferrer
Affiliation: Instituto de Matemáticas de Castellón, Universitat Jaume I, Campus de Riu Sec, 12071 Castellón, Spain
Email: mferrer@mat.uji.es

Salvador Hernández
Affiliation: Departamento de Matemáticas, Universitat Jaume I, Campus de Riu Sec, 12071 Cast- ellón, Spain
Email: hernande@mat.uji.es

Dmitri Shakhmatov
Affiliation: Division of Mathematics, Physics and Earth Sciences, Graduate School of Science and Engineering, Ehime University, Matsuyama 790-8577, Japan
Email: dmitri.shakhmatov@ehime-u.ac.jp

DOI: https://doi.org/10.1090/proc/13532
Keywords: Pontryagin duality, reflexive group, Baer-Specker group, integer-valued homomorphism group, prodiscrete group, compact set
Received by editor(s): February 9, 2016
Received by editor(s) in revised form: September 13, 2016
Published electronically: April 12, 2017
Additional Notes: The first and second authors acknowledge partial support by the Spanish Ministerio de Economía y Competitividad, grant MTM2016-77143-P, and the Universitat Jaume I, grant P1$⋅$1B2015-77. The second author also acknowledges partial support by Generalitat Valenciana, grant code: PROMETEO/2014/062.
The third author was partially supported by the Grant-in-Aid for Scientific Research (C) No. 26400091 by the Japan Society for the Promotion of Science (JSPS)
Communicated by: Ken Bromberg
Article copyright: © Copyright 2017 American Mathematical Society