Hostname: page-component-7dd5485656-gs9qr Total loading time: 0 Render date: 2025-10-29T21:54:23.042Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on the Normal Moore Space Conjecture

Published online by Cambridge University Press:  20 November 2018

Keith J. Devlin  and
Saharon Shelah
Keith J. Devlin
Affiliation:
University of Lancaster, Lancaster, England;
Saharon Shelah
Affiliation:
The Hebrew University, Jerusalem, Israel
Rights & Permissions [Opens in a new window]

Extract

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.

F. B. Jones (1937) conjectured that every normal Moore space is metrizable. He also denned a particular kind of topological space (now known as Jones' spaces), proved that they were all non-metrizable Moore spaces, but was unable to decide whether or not Jones’ spaces are normal. J. H. Silver (1967) proved that a positive solution to Jones’ conjecture was not possible, and W. Fleissner (1973) obtained an alternative proof by showing that it is not possible to prove the non-normality of Jones’ spaces. These results left open the possibility of resolving the questions from the GCH. In this paper we show that if CH be assumed, then Jones’ spaces are not normal (Devlin, Shelah, independently) and that the GCH does not lead to a positive solution to the Jones conjecture (Shelah). A brief survey of the progress on the problem to date is also included.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 2 , 01 April 1979 , pp. 241 - 251
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Alexandroff, P. and Urysohn, P., Une condition nécessaire et suffisante pour qu'une classe (L) soit une classe (B), Comptes Rendus Hebdomadaires des Séances de l'Acad D. Sci. 177 (1923), 1274–1276.Google Scholar
2. Bing, R. H., Metrization of topological spaces, Can. J. Math. 3 (1951), 175–186.Google Scholar
3. Devlin, K. J., Variations of(), Journal of Symbolic Logic, to appear.Google Scholar
4. Devlin, K. J., Constructibility, in Handbook of Mathematical Logic (Barwise, North Holland 1977).Google Scholar
5. Devlin, K. J. and Johnsbrâten, H., The Souslin problem, Springer Lecture Notes in Mathematics 405 (1974).Google Scholar
6. Devlin, K. J. and Shelah, S.. A weak version of 0 which follows fro 2N0 < 2N1 , Israel J. of Math.. 29 (1978), 239–247.Google Scholar
7. Devlin, K. J. and Shelah, S.. Souslin properties and tree topologies, Proc. of the London Math. Soc. to appear.Google Scholar
8. Fleissner, W. G., When is Jones space normal?, Proc. A.M.S. 50 (1975), 375–378.Google Scholar
9. Jech, T. J., Trees, Journal of Symbolic Logi. 36 (1971), 1–14.Google Scholar
10. Jones, F. B., Concerning normal and completely normal spaces, Bull. A.M.S. 43 (1937), 671–677.Google Scholar
11. Jones, F. B., Remarks on the normal Moore space metrization problem, Proc. of the 1965 Wisconsin Summer Topology Seminar, Annals of Math Studies 60, Princeton (1966).Google Scholar
12. Moore, R. L., Foundations of point set theory, A.M.S. Coll. Publ. 13 (1932).Google Scholar
13. Shelah, S., Whitehead groups may be not free, even assuming CH (I, II), Israel J. of Math, to appear.Google Scholar
14. Tall, F. D., Set theoretic consistency results and topological theorems concerning the normal Moore space conjecture and related problems, Diss. Math.Google Scholar