Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-16T01:17:32.004Z Has data issue: false hasContentIssue false
  • English
  • Français

The Wallace Problem: A Counterexample from MaCountable and p-Compactness

Published online by Cambridge University Press:  20 November 2018

Artur H. Tomita
Artur H. Tomita*
Affiliation:
Department of Mathematics, Universidade de São Paulo, Caixa Postal 66281, CEP 05389-970, São Paulo SP Brasil, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We construct, under MAcountable, a countably compact topological subsemigroup of which is not a group, hence a counterexample for the Wallace problem. We also show that there is no p-compact counterexample for the Wallace problem, answering a question of D. Grant. Finally, we show that—in some sense—our counterexample for the Wallace problem constructed under MAcountable cannot be done in ZFC.

Keywords

22A9954A3503E50Wallace problemMAcountablep-compactnesscountable compactnesstopological semigroups
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 39 , Issue 4 , 01 December 1996 , pp. 486 - 498
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. Comfort, W. W., Problems on topological groups and other homogeneous spaces, Open Problems in Topology, (eds. van Mill, J. and Reed, G. M.), North-Holland, 1990, 311–347.Google Scholar
2. van Douwen, E. K., The product of two countably compact topological groups, Trans. Amer. Math. Soc. 262(1980), 417–427.Google Scholar
3. Ellis, R., Locally compact transformations groups, Duke Math. J. 27(1957), 119–125.Google Scholar
4. Grant, D., Sequentially compact cancellative topological semigroups: some progress on the Wallace Problem, Papers on General Topology and Applications, Seventh Summer Conference at the University of Wisconsin (ed. Andima, S. et al.), 704, Annals of the New York Academy of Sciences, 1993, 150–154.Google Scholar
5. Hart, K. P. and van Mill, J., A countably compact group H such that H x H is not countably compact , Trans. Amer. Math. Soc. 323(1991), 811–821.Google Scholar
6. Kunen, K., Set Theory, North-Holland, 1980.Google Scholar
7. Mukherjea, A. and Tserpes, N. A., A note on countably compact semigroups, J. Austral. Math. Soc. 13(1972), 180–184.Google Scholar
8. Pfister, H., Continuity of the inverse, Proc. Amer. Math. Soc. 95(1985), 312–314.Google Scholar
9. Reznichenko, E. A., Extension of functions defined on products of pseudocompact spaces and continuity of the inverse in pseudocompact groups, preprint.Google Scholar
10. Robbie, D. and Svetlichny, S., An answer to A. D. Wallace s question about countably compact cancellative semigroups, Proc. Amer. Math. Soc, 1996.Google Scholar
11. Tkachenko, M. G., Countably compact and pseudocompact topologies on free abelian groups, Izv. Vyssh. Uchebn. Zaved. Mat. 34(1990), 68–75.Google Scholar
12. Tomita, A., Between countable and sequential compactness in free abelian groups, preprint.Google Scholar
13. Tomita, A., A group under MAcountabie whose square is countably compact but whose cube is not, preprint.Google Scholar
14. Tomita, A., Countable compactness and related properties in groups and semigroups: free abelian groups and the Wallace problem, Ph.D thesis, York University, June, 1995.Google Scholar
15. Vaughan, J., Countably compact and sequentially compact spaces, Handbook of Set-Theoretic Topology, (eds. Kunen, K. and Vaughan, J.), North-Holland, 1984, 569–602.Google Scholar
16. Weiss, W., Versions of Martin's Axiom, Handbook of Set-Theoretic Topology, (eds. Kunen, K. and Vaughan, J.), North-Holland, 1984, 827–886.Google Scholar