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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Cardinal restrictions on some homogeneous compacta
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by István Juhász, Peter Nyikos and Zoltán Szentmiklóssy PDF
Proc. Amer. Math. Soc. 133 (2005), 2741-2750 Request permission

Abstract:

We give restrictions on the cardinality of compact Hausdorff homogeneous spaces that do not use other cardinal invariants, but rather covering and separation properties. In particular, we show that it is consistent that every hereditarily normal homogeneous compactum is of cardinality $\mathfrak {c}$. We introduce property wD($\kappa$), intermediate between the properties of being weakly $\kappa$-collectionwise Hausdorff and strongly $\kappa$-collectionwise Hausdorff, and show that if $X$ is a compact Hausdorff homogeneous space in which every subspace has property wD($\aleph _{1}$), then $X$ is countably tight and hence of cardinality $\le 2^{\mathfrak {c}}$. As a corollary, it is consistent that such a space $X$ is first countable and hence of cardinality $\mathfrak {c}$. A number of related results are shown and open problems presented.
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Additional Information
  • István Juhász
  • Affiliation: Alfred Rényi Institute, P.O. Box 127, 1364 Budapest, Hungary
  • Peter Nyikos
  • Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
  • Zoltán Szentmiklóssy
  • Affiliation: Department of Mathematics, Eötvös Loránd University, Pázmány sétány 1/C, Budapest, H-1117 Hungary
  • Received by editor(s): January 1, 2004
  • Received by editor(s) in revised form: May 27, 2004
  • Published electronically: March 29, 2005
  • Additional Notes: Research of the first and third authors partially supported by OTKA grant no. 37758.
    Research of the second author partially supported by a grant from the Erdős Center of the János Bolyai Mathematical Society
  • Communicated by: Alan Dow
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 2741-2750
  • MSC (2000): Primary 03E35, 54A25, 54D15, 54D30, 54F99; Secondary 03E50, 54D45
  • DOI: https://doi.org/10.1090/S0002-9939-05-07861-5
  • MathSciNet review: 2146223