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 | References | Similar Articles | Additional Information

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.

- A. V. Arkhangel′skiĭ,
*Topological homogeneity. Topological groups and their continuous images*, Uspekhi Mat. Nauk**42**(1987), no. 2(254), 69–105, 287 (Russian). MR**898622** - Z. Balogh,
*Locally nice spaces and axiom R*, Topology Appl.**125**(2002), no. 2, 335–341. MR**1933581**, DOI https://doi.org/10.1016/S0166-8641%2801%2900286-3 - Z. Balogh and M. E. Rudin,
*Monotone normality*, Topology Appl.**47**(1992), no. 2, 115–127. MR**1193194**, DOI https://doi.org/10.1016/0166-8641%2892%2990066-9 - Eric K. van Douwen,
*The integers and topology*, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR**776622** - Alan Dow,
*An introduction to applications of elementary submodels to topology*, Topology Proc.**13**(1988), no. 1, 17–72. MR**1031969** - Alan Dow,
*Compact spaces of countable tightness in the Cohen model*, Set theory and its applications (Toronto, ON, 1987) Lecture Notes in Math., vol. 1401, Springer, Berlin, 1989, pp. 55–67. MR**1031765**, DOI https://doi.org/10.1007/BFb0097331 - Alan Dow, Franklin D. Tall, and William A. R. Weiss,
*New proofs of the consistency of the normal Moore space conjecture. I*, Topology Appl.**37**(1990), no. 1, 33–51. MR**1075372**, DOI https://doi.org/10.1016/0166-8641%2890%2990013-R - V. V. Fedorchuk, “Bicompacta in which each infinite closed subset is n-dimensional,”
*Math. USSR Sbornik***25**(1975) 37–57. - István Juhász,
*A weakening of $\clubsuit $, with applications to topology*, Comment. Math. Univ. Carolin.**29**(1988), no. 4, 767–773. MR**982796** - I. Juhász,
*On the minimum character of points in compact spaces*, Topology. Theory and applications, II (Pécs, 1989) Colloq. Math. Soc. János Bolyai, vol. 55, North-Holland, Amsterdam, 1993, pp. 365–371. MR**1244377** - I. Juhász,
*Cardinal functions*, Recent progress in general topology (Prague, 1991) North-Holland, Amsterdam, 1992, pp. 417–441. MR**1229134** - I. Juhász and Z. Szentmiklóssy,
*Convergent free sequences in compact spaces*, Proc. Amer. Math. Soc.**116**(1992), no. 4, 1153–1160. MR**1137223**, DOI https://doi.org/10.1090/S0002-9939-1992-1137223-8 - J. van Mill, “On the cardinality of power homogeneous compacta,” preprint.
- Peter J. Nyikos,
*Applications of some strong set-theoretic axioms to locally compact $T_5$ and hereditarily scwH spaces*, Fund. Math.**176**(2003), no. 1, 25–45. MR**1971471**, DOI https://doi.org/10.4064/fm176-1-3 - P. Nyikos and J.E. Porter, “Hereditarily strongly cwH and separation axioms," in preparation. Preliminary draft: www.math.sc.edu/$\sim$nyikos/preprints.html
- Judy Roitman,
*Basic $S$ and $L$*, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 295–326. MR**776626** - Z. Szentmiklóssy,
*$S$-spaces and $L$-spaces under Martin’s axiom*, Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978) Colloq. Math. Soc. János Bolyai, vol. 23, North-Holland, Amsterdam-New York, 1980, pp. 1139–1145. MR**588860** - W. Stephen Watson,
*Locally compact normal spaces in the constructible universe*, Canadian J. Math.**34**(1982), no. 5, 1091–1096. MR**675681**, DOI https://doi.org/10.4153/CJM-1982-078-8 - Scott W. Williams and Haoxuan Zhou,
*Order-like structure of monotonically normal spaces*, Comment. Math. Univ. Carolin.**39**(1998), no. 1, 207–217. MR**1623026**

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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.