Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-28T14:16:16.263Z Has data issue: false hasContentIssue false
  • English
  • Français

Compactness in Topological Hjelmslev Planes

Published online by Cambridge University Press:  20 November 2018

J. W. Lorimer
J. W. Lorimer*
Affiliation:
Department of Mathematics, University of Toronto Toronto, Ontario, M5S 1A1
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.

In the theory of ordinary topological affine and projective planes it is known that (1) An affine plane is never compact (2) a locally compact ordered projective plane is compact and archimedean (3) a locally compact connected projective plane is compact and (4) a locally compact projective plane over a coordinate ring with bi-associative multiplication is compact. In this paper we re-examine these results within the theory of topological Hjelmslev Planes and observe that while (1) remains valid (2), (3) and (4) are false. At first glance these negative results seem to suggest we are working in too general a setting. However a closer examination reveals that the absence of compactness in our setting is a natural and expected feature which in no way precludes the possibility of obtaining significant results.

Keywords

51H1051G0551C05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 4 , 01 December 1984 , pp. 423 - 429
Copyright
Copyright © Canadian Mathematical Society 1984

Footnotes

(1)

The author gratefully acknowledges the financial support of the National Sciences and Engineering Research Council of Canada.

References

1. Baker, C., Lane, N. D. and Lorimer, J. W., Order and topology in projective Hjelmslev planes, Journal of Geometry, 19, 1 (1982), 8-42.CrossRefGoogle Scholar
2. Dembowski, P., Finite Geometries, Springer-Verlag, N.Y. (1968).CrossRefGoogle Scholar
3. Engelking, R., Outline of General Topology, John Wiley and Sons Inc. New York (1968).Google Scholar
4. Hjelmslev, J., Einleitung in die allgemeine Kongruenzlehre III Mitteilung, Dansk. Vid. Selsk, Math. Fys. Medd. 19 (1942) no. 12.Google Scholar
5. Kolmogorff, A. N., Zur Begrundung der projektiven Géométrie, Ann. Math. 33 (1932), 175-176.CrossRefGoogle Scholar
6. Lorimer, J. W., Topological Hjelmslev planes, Geom. Dedicata (1978), 185-207.CrossRefGoogle Scholar
7. Lorimer, J. W., Connectedness in Topological Hjelmslev planes, Annali di Mat. para ed appli 118 (1978), 199-216.CrossRefGoogle Scholar
8. Lorimer, J. W., Locally compact Hjelmslev planes and rings, Can. J. Math., XXXIII, 4 (1981), 988-1021.CrossRefGoogle Scholar
9. Lorimer, J. W., Dual numbers and topological Hjelmslev planes, Canadian Math. Bulletin 26, 3 (1983), 297-302.CrossRefGoogle Scholar
10. Pickert, G., Projektive Ebenen (Springer-Verlag, 1975).CrossRefGoogle Scholar
11. Salzmann, Helmut R., Topologische projektive Ebenen, Math. Z, 67 (1957), 436-466.10.1007/BF01258875CrossRefGoogle Scholar
12. Salzmann, Helmut R., Topological Planes, Advances in Math., vol. 2, Fascicle 1 (1967).CrossRefGoogle Scholar
13. Salzmann, Helmut R., Projectivities and the Topology on Lines, Geometry?von Staudfs Point of View, P. Plaumann and K Strambach (eds.), D. Reidel Publishing Co. (1981), 313-337.CrossRefGoogle Scholar
14. Wyler, O., Order and Topology in projective planes, Amer. J. Math 74 (1952), 656-666.CrossRefGoogle Scholar