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

DOI:
https://doi.org/10.1090/S0002-9939-05-07861-5

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 . We introduce property wD(), intermediate between the properties of being weakly -collectionwise Hausdorff and strongly -collectionwise Hausdorff, and show that if is a compact Hausdorff homogeneous space in which every subspace has property wD( ), then is countably tight and hence of cardinality . As a corollary, it is consistent that such a space is first countable and hence of cardinality . 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

DOI:
https://doi.org/10.1090/S0002-9939-05-07861-5

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.