On zero-dimensionality and the connected component of locally pseudocompact groups
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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.References
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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
- Email: dikranja@dimi.uniud.it
- Gábor Lukács
- Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada
- Email: lukacs@cc.umanitoba.ca
- 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 - Communicated by: Alexander N. Dranishnikov
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 2995-3008
- MSC (2010): Primary 22A05, 54D25, 54H11; Secondary 22D05, 54D05, 54D30
- DOI: https://doi.org/10.1090/S0002-9939-2011-10626-9
- MathSciNet review: 2801639
Dedicated: Dedicated to Wis Comfort on the occasion of his 78th birthday