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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Two countably compact topological groups: One of size $\aleph_\omega$ and the other of weight $\aleph_\omega$ without non-trivial convergent sequences


Author: Artur Hideyuki Tomita
Journal: Proc. Amer. Math. Soc. 131 (2003), 2617-2622
MSC (2000): Primary 54H11, 54A25, 54A35; Secondary 22A05
Published electronically: March 11, 2003
MathSciNet review: 1974663
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Abstract: E. K. van Douwen asked in 1980 whether the cardinality of a countably compact group must have uncountable cofinality in $\mathrm{ZFC}$. He had shown that this was true under GCH. We answer his question in the negative. V. I. Malykhin and L. B. Shapiro showed in 1985 that under GCH the weight of a pseudocompact group without non-trivial convergent sequences cannot have countable cofinality and showed that there is a forcing model in which there exists a pseudocompact group without non-trivial convergent sequences whose weight is $\omega_1<{\mathfrak c}$. We show that it is consistent that there exists a countably compact group without non-trivial convergent sequences whose weight is $\aleph_\omega$.


References [Enhancements On Off] (What's this?)

  • 1. W. W. Comfort and D. Remus, Imposing pseudocompact topologies on Abelian groups, Fund. Math. 142, n.3, 221-240. MR 94g:22006
  • 2. E. K. van Douwen, The product of two countably compact topological groups, Trans. Amer. Math. Soc. 262 (Dec 1980), 417-427. MR 82b:22002
  • 3. -, The weight of pseudocompact (homogeneous) space whose cardinality has countable cofinality, Proc. Amer. Math. Soc. 80 (1980), 678-682. MR 82a:54009
  • 4. A. Hajnal and I. Juhász, A separable normal topological group need not be Lindelöf, Gen. Top. and its Appl. 6 (1976), 199-205. MR 55:4088
  • 5. K. P. Hart and J. van Mill, A countably compact topological group $ {H} $ such that $ {H} \times {H} $ is not countably compact, Trans. Amer. Math. Soc. 323 (Feb 1991), 811-821. MR 91e:54025
  • 6. P. Koszmider, A. Tomita and S. Watson, Forcing countably compact group topologies on a larger free Abelian group, Topology Proceedings 25 Summer (2000), 563-574.
  • 7. V. I. Malykhin and L. B. Shapiro, Pseudocompact groups without convergent sequences, Math. Notes 37 (1985), no. 1-2, 59-62. MR 87a:22002
  • 8. S. M. Sirota, A product of topological groups and extremal disconnectedness, Mat. Sb. 79(121) (1969), 179-192. MR 39:4315
  • 9. M. G. Tkachenko, Countably compact and pseudocompact topologies on free abelian groups, Izvestia VUZ. Matematika 34 (1990), 68-75. MR 92e:54044
  • 10. A. H. Tomita, On finite powers of countably compact groups, Comment. Math. Univ. Carolinae 37 (1996), no. 3, 617-626. MR 98a:54033
  • 11. -, A group under $\mathrm{MA_{countable}} $ whose square is countably compact but whose cube is not , Top. and its Appl. 91 (1999) 91-104. MR 2000d:54039
  • 12. A. H. Tomita and S. Watson, Ultraproducts, $p$-limits and antichain on the Comfort group order, in preparation.

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Additional Information

Artur Hideyuki Tomita
Affiliation: Department of Mathematics, Universidade de São Paulo, Caixa Postal 66281 CEP 05311-970, São Paulo, Brazil
Email: tomita@ime.usp.br

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06933-8
PII: S 0002-9939(03)06933-8
Keywords: Forcing, countably compact group, cofinality, weight
Received by editor(s): October 15, 2001
Received by editor(s) in revised form: January 15, 2002
Published electronically: March 11, 2003
Additional Notes: This research was partially conducted while the author was visiting the Department of Mathematics of Universidade de Coimbra. This visit was supported by CCINT-USP and the local organizing committee of the Fourth Ibero American Congress of Topology and its Applications
Communicated by: Alan Dow
Article copyright: © Copyright 2003 American Mathematical Society