Cardinal restrictions on some homogeneous compacta
Authors:
István Juhász, Peter Nyikos and Zoltán Szentmiklóssy
Journal:
Proc. Amer. Math. Soc. 133 (2005), 27412750
MSC (2000):
Primary 03E35, 54A25, 54D15, 54D30, 54F99; Secondary 03E50, 54D45
Published electronically:
March 29, 2005
MathSciNet review:
2146223
Fulltext PDF Free Access
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 . 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.
 [A]
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
(89b:54004)
 [B]
Z.
Balogh, Locally nice spaces and axiom R, Topology Appl.
125 (2002), no. 2, 335–341. MR 1933581
(2003i:54029), http://dx.doi.org/10.1016/S01668641(01)002863
 [BR]
Z.
Balogh and M.
E. Rudin, Monotone normality, Topology Appl.
47 (1992), no. 2, 115–127. MR 1193194
(94b:54065), http://dx.doi.org/10.1016/01668641(92)900669
 [vD]
Eric
K. van Douwen, The integers and topology, Handbook of
settheoretic topology, NorthHolland, Amsterdam, 1984,
pp. 111–167. MR 776622
(87f:54008)
 [D1]
Alan
Dow, An introduction to applications of elementary submodels to
topology, Topology Proc. 13 (1988), no. 1,
17–72. MR
1031969 (91a:54003)
 [D2]
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
(91a:54004), http://dx.doi.org/10.1007/BFb0097331
 [DTW]
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
(92b:54008a), http://dx.doi.org/10.1016/01668641(90)90013R
 [F]
V. V. Fedorchuk, ``Bicompacta in which each infinite closed subset is ndimensional,'' Math. USSR Sbornik 25 (1975) 3757.
 [J1]
István
Juhász, A weakening of ♣, with applications to
topology, Comment. Math. Univ. Carolin. 29 (1988),
no. 4, 767–773. MR 982796
(90d:54005)
 [J2]
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, NorthHolland,
Amsterdam, 1993, pp. 365–371. MR 1244377
(94k:54004)
 [J3]
I.
Juhász, Cardinal functions, Recent progress in general
topology (Prague, 1991) NorthHolland, Amsterdam, 1992,
pp. 417–441. MR
1229134
 [JSz]
I.
Juhász and Z.
Szentmiklóssy, Convergent free sequences in compact
spaces, Proc. Amer. Math. Soc.
116 (1992), no. 4,
1153–1160. MR 1137223
(93b:54024), http://dx.doi.org/10.1090/S00029939199211372238
 [vM]
J. van Mill, ``On the cardinality of power homogeneous compacta,'' preprint.
 [Ny]
Peter
J. Nyikos, Applications of some strong settheoretic axioms to
locally compact 𝑇₅ and hereditarily scwH spaces, Fund.
Math. 176 (2003), no. 1, 25–45. MR 1971471
(2004k:54008), http://dx.doi.org/10.4064/fm17613
 [NyP]
P. Nyikos and J.E. Porter, ``Hereditarily strongly cwH and separation axioms," in preparation. Preliminary draft: www.math.sc.edu/nyikos/preprints.html
 [R]
Judy
Roitman, Basic 𝑆 and 𝐿, Handbook of
settheoretic topology, NorthHolland, Amsterdam, 1984,
pp. 295–326. MR 776626
(87a:54043)
 [Sz]
Z.
Szentmiklóssy, 𝑆spaces and 𝐿spaces under
Martin’s axiom, Topology, Vol. II (Proc. Fourth Colloq.,
Budapest, 1978) Colloq. Math. Soc. János Bolyai, vol. 23,
NorthHolland, AmsterdamNew York, 1980, pp. 1139–1145. MR 588860
(81k:54032)
 [W]
W.
Stephen Watson, Locally compact normal spaces in the constructible
universe, Canad. J. Math. 34 (1982), no. 5,
1091–1096. MR 675681
(83k:54021), http://dx.doi.org/10.4153/CJM19820788
 [WZ]
Scott
W. Williams and Haoxuan
Zhou, Orderlike structure of monotonically normal spaces,
Comment. Math. Univ. Carolin. 39 (1998), no. 1,
207–217. MR 1623026
(99c:54029)
 [A]
 A. V. Arhangel'skii, ``Topological homogeneity. Topological groups and their continuous maps,'' Uspekhi Mat. Nauk 42 (2)(254) (1987) 69105, 287. MR 0898622 (89b:54004)
 [B]
 Z. Balogh, ``Locally nice spaces and axiom R,'' Topology Appl. 125 (2) (2002) 335341. MR 1933581 (2003i:54029)
 [BR]
 Z. Balogh and M.E. Rudin, ``Monotone normality,'' Top. Appl. 47 (2) (1992), 115127. MR 1193194 (94b:54065)
 [vD]
 E.K. van Douwen, ``The integers and topology,'' in: Handbook of SetTheoretic Topology, K. Kunen and J. Vaughan, eds., NorthHolland (1984) 111167. MR 0776622 (87f:54008)
 [D1]
 A. Dow, ``An introduction to applications of elementary submodels to topology,'' Topology Proceedings 13 (1) (1988), 1772.MR 1031969 (91a:54003)
 [D2]
 A. Dow, ``Compact spaces of countable tightness in the Cohen model,'' in: Set theory and its applications (Toronto, ON, 1987), J. Steprans and S. Watson, eds., Lecture Notes in Mathematics #1401, SpringerVerlag, (1989) 5567.MR 1031765 (91a:54004)
 [DTW]
 A. Dow, F.D.Tall, and W.A.R. Weiss, ``New proofs of the consistency of the normal Moore space conjecture I,'' Topology Appl. 37 (1) (1990), 3351.MR 1075372 (92b:54008a)
 [F]
 V. V. Fedorchuk, ``Bicompacta in which each infinite closed subset is ndimensional,'' Math. USSR Sbornik 25 (1975) 3757.
 [J1]
 I. Juhász, ``A weakening of , with applications to topology'' Comment. Math. Univ. Carolinae 29 (1988), no. 4, 767773. MR 0982796 (90d:54005)
 [J2]
 I. Juhász, ``On the minimum character of points in compact spaces,'' in: Topology. Theory and applications, II (Pécs, 1989), Colloq. Math. Soc. János Bolyai, 55, NorthHolland, Amsterdam, (1993) 365371. MR 1244377 (94k:54004)
 [J3]
 I. Juhász, ``Cardinal functions,'' in: Recent Progress in General Topology, M. Husek and J. van Mill, eds., Elsevier, (1992), 417441.MR 1229134
 [JSz]
 I. Juhász and Z. Szentmiklóssy, ``Convergent free sequences in compact spaces, Proc. Amer. Math. Soc. 116 (4) (1992), 11531160. MR 1137223 (93b:54024)
 [vM]
 J. van Mill, ``On the cardinality of power homogeneous compacta,'' preprint.
 [Ny]
 P. Nyikos, ``Applications of some strong settheoretic axioms to locally compact and hereditarily scwH spaces,'' Fund. Math. 176 (1) (2003) 2545. MR 1971471 (2004k:54008)
 [NyP]
 P. Nyikos and J.E. Porter, ``Hereditarily strongly cwH and separation axioms," in preparation. Preliminary draft: www.math.sc.edu/nyikos/preprints.html
 [R]
 J. Roitman, ``Basic S and L,'' in: Handbook of SetTheoretic Topology, K. Kunen and J. Vaughan, eds., NorthHolland (1984) 295326. MR 0776626 (87a:54043)
 [Sz]
 Z. Szentmiklóssy, ``S spaces and L spaces under Martin's axiom," Coll. Math. Soc. J. Bolyai, 23 (1978), II, 11391145. MR 0588860 (81k:54032)
 [W]
 W.S. Watson, ``Locally compact normal spaces in the constructible universe,'' Canad. J. Math. 34 (5) (1982), 10911096. MR 0675681 (83k:54021)
 [WZ]
 S.W. Williams and H. Zhou, ``Orderlike structure of monotonically normal spaces,'' Comment. Math. Univ. Carolinae 39 (1) (1998), 207217.MR 1623026 (99c:54029)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
03E35,
54A25,
54D15,
54D30,
54F99,
03E50,
54D45
Retrieve articles in all journals
with MSC (2000):
03E35,
54A25,
54D15,
54D30,
54F99,
03E50,
54D45
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, H1117 Hungary
DOI:
http://dx.doi.org/10.1090/S0002993905078615
PII:
S 00029939(05)078615
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.
