A countable free closed non-reflexive subgroup of $\mathbb {Z}^{\mathfrak {c}}$
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- by Maria Vicenta Ferrer, Salvador Hernández and Dmitri Shakhmatov PDF
- Proc. Amer. Math. Soc. 145 (2017), 3599-3605 Request permission
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.References
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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
- MR Author ID: 898177
- Email: mferrer@mat.uji.es
- Salvador Hernández
- Affiliation: Departamento de Matemáticas, Universitat Jaume I, Campus de Riu Sec, 12071 Cast-,ellón, Spain
- MR Author ID: 84870
- 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
- MR Author ID: 196690
- Email: dmitri.shakhmatov@ehime-u.ac.jp
- 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$\cdot$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
- © Copyright 2017 American Mathematical Society
- 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
- MathSciNet review: 3652811