Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Countably compact groups from a selective ultrafilter

Authors: S. Garcia-Ferreira, A. H. Tomita and S. Watson
Journal: Proc. Amer. Math. Soc. 133 (2005), 937-943
MSC (2000): Primary 54G20, 54D80, 22A99; Secondary 54H11
Published electronically: September 29, 2004
MathSciNet review: 2113947
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the existence of a selective ultrafilter on $\omega$ implies the existence of a countably compact group without non-trivial convergent sequences all of whose powers are countably compact. Hence, by using a selective ultrafilter on $\omega$, it is possible to construct two countably compact groups without non-trivial convergent sequences whose product is not countably compact.

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

  • 1. A. R. Bernstein, A new kind of compactness for topological spaces, Fund. Math. 66 (1970), 185-193. MR 0251697 (40:4924)
  • 2. W. Comfort and S. Negrepontis, The Theory of Ultrafilters, Springer-Verlag, Berlin, 1974. MR 0396267 (53:135)
  • 3. E. K. van Douwen, The product of two countably compact topological groups, Trans. Amer. Math. Soc. 262 (1980), 417 - 427. MR 0586725 (82b:22002)
  • 4. S. Garcia-Ferreira, Quasi $M$-compact spaces, Czechoslovak Math. J. 46 (1996), 161 - 177. MR 1371698 (97b:54033)
  • 5. L. Gillman and M. Jerison, Rings of continuous functions, Lectures Notes in Mathematics No. 27, Springer-Verlag, 1976. MR 0407579 (53:11352)
  • 6. J. Ginsburg and V. Saks, Some applications of ultrafilters in topology, Pacific J. Math. 57 (1975), 403-418. MR 0380736 (52:1633)
  • 7. A. Hajnal and I. Juhász, A separable normal topological group need not be Lindelöf, Gen. Topology Appl. 6 (1976), 199-205. MR 0431086 (55:4088)
  • 8. 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 (1991), 811- 821. MR 0982236 (91e:54025)
  • 9. A. H. Tomita, A group under $MA_{countable}$ whose square is countably compact but whose cube is not, Topology Appl. 91 (1999), 91-104. MR 1664516 (2000d:54039)
  • 10. A. H. Tomita, Countable compactness and finite powers of topological groups without convergent sequences, submitted.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 54G20, 54D80, 22A99, 54H11

Retrieve articles in all journals with MSC (2000): 54G20, 54D80, 22A99, 54H11

Additional Information

S. Garcia-Ferreira
Affiliation: Instituto de Matemáticas (UNAM), Apartado Postal 61-3, Xangari, 58089, Morelia, Michoacán, México

A. H. Tomita
Affiliation: Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, CEP 05315-970, São Paulo, Brasil

S. Watson
Affiliation: Department of Mathematics, York University, 474700 Keele Street, M3P 1P3, Toronto, Ontario, Canada

Keywords: $p$-limit, $p$-compact, selective ultrafilter, countably compact group, topological group
Received by editor(s): March 3, 2003
Received by editor(s) in revised form: November 20, 2003
Published electronically: September 29, 2004
Additional Notes: This research was supported by CONACYT grant no. 40057-F and DGAPA grant no. IN104601
Communicated by: Alan Dow
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society