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Cardinal restrictions on some homogeneous compacta

Authors: István Juhász, Peter Nyikos and Zoltán Szentmiklóssy
Journal: Proc. Amer. Math. Soc. 133 (2005), 2741-2750
MSC (2000): Primary 03E35, 54A25, 54D15, 54D30, 54F99; Secondary 03E50, 54D45
Published electronically: March 29, 2005
MathSciNet review: 2146223
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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

Keywords: Compactum, homogeneous, hereditarily, [weakly] [strongly] $\kappa$-cwH, property wD($\kappa$), T${}_5$, tightness, $\pi$-character, $\pi$-base, first countable, $G_{\delta}$
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
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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