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On zero-dimensionality and the connected component of locally pseudocompact groups

Authors: D. Dikranjan and Gábor Lukács
Journal: Proc. Amer. Math. Soc. 139 (2011), 2995-3008
MSC (2010): Primary 22A05, 54D25, 54H11; Secondary 22D05, 54D05, 54D30
Published electronically: March 23, 2011
MathSciNet review: 2801639
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Abstract: A topological group is locally pseudocompact if it contains a non-empty open set with pseudocompact closure. In this paper, we prove that if $ G$ is a group with the property that every closed subgroup of $ G$ is locally pseudocompact, then $ G_0$ is dense in the component of the completion of $ G$, and $ G/G_0$ is zero-dimensional. We also provide examples of hereditarily disconnected pseudocompact groups with strong minimality properties of arbitrarily large dimension, and thus show that $ G/G_0$ may fail to be zero-dimensional even for totally minimal pseudocompact groups.

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Additional Information

D. Dikranjan
Affiliation: Department of Mathematics and Computer Science, University of Udine, Via delle Scienze, 208 – Loc. Rizzi, 33100 Udine, Italy

Gábor Lukács
Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada

Received by editor(s): February 15, 2010
Received by editor(s) in revised form: May 30, 2010
Published electronically: March 23, 2011
Additional Notes: The first author acknowledges the financial aid received from SRA, grants P1-0292-0101, J1-9643-0101, and MTM2009-14409-C02-01
The second author gratefully acknowledges the generous financial support received from NSERC and the University of Manitoba, which enabled him to do this research
Dedicated: Dedicated to Wis Comfort on the occasion of his 78th birthday
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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