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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Pseudocompact totally dense subgroups


Authors: Dikran Dikranjan and Anna Giordano Bruno
Journal: Proc. Amer. Math. Soc. 136 (2008), 1093-1103
MSC (2000): Primary 22A05, 54H11; Secondary 22C05, 54D25.
Published electronically: November 30, 2007
MathSciNet review: 2361886
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Abstract: It was shown by Dikranjan and Shakhmatov in 1992 that if a compact abelian group $ K$ admits a proper totally dense pseudocompact subgroup, then $ K$ cannot have a torsion closed $ G_\delta$-subgroup; moreover this condition was shown to be also sufficient under LH. We prove in ZFC that this condition actually ensures the existence of a proper totally dense subgroup $ H$ of $ K$ that contains an $ \omega$-bounded dense subgroup of $ K$ (such an $ H$ is necessarily pseudocompact). This answers two questions posed by Dikranjan and Shakhmatov (Proc. Amer. Math. Soc. 114 (1992), 1119-1129).


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Additional Information

Dikran Dikranjan
Affiliation: Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy
Email: dikranja@dimi.uniud.it

Anna Giordano Bruno
Affiliation: Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy
Email: giordano@dimi.uniud.it

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09099-5
PII: S 0002-9939(07)09099-5
Received by editor(s): May 8, 2006
Received by editor(s) in revised form: October 23, 2006
Published electronically: November 30, 2007
Additional Notes: This work was partially supported by a PRIN2005 grant of the Italian MIUR and by funds of the PhD program at the Department of Mathematics of the University of Udine.
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.