DOI QR코드

DOI QR Code

TOPOLOGIES AND INCIDENCE STRUCTURE ON Rn-GEOMETRIES

  • Im, Jang-Hwan
  • Published : 2002.01.01

Abstract

An R$^{n}$ -geometry (P$^{n}$ , L) is a generalization of the Euclidean geometry on R$^{n}$ (see Def. 1.1). We can consider some topologies (see Def. 2.2) on the line set L such that the join operation V : P$^{n}$ $\times$ P$^{n}$ \ $\Delta$ longrightarrow L is continuous. It is a notable fact that in the case n = 2 the introduced topologies on L are same and the join operation V : P$^2$ $\times$ P$^2$ \ $\Delta$ longrightarrow L is continuous and open [10, 11]. It is a fundamental topological property of plane geometry, but in the cases n $\geq$ 3, it is no longer true. There are counter examples [2]. Hence, it is a fundamental problem to find suitable topologies on the line set L in an R$^{n}$ -geometry (P$^{n}$ , L) such that these topologies are compatible with the incidence structure of (P$^{n}$ , L). Therefore, we need to study the topologies of the line set L in an R$^{n}$ -geometry (P$^{n}$ , L). In this paper, the relations of such topologies on the line set L are studied.

Keywords

topological geometry;R$^{n}$ -geometry;continuous and open maps

References

  1. H. Salzmann, Topological Planes, Adv. Math. 2 (1967), 1-160 https://doi.org/10.1016/S0001-8708(67)80002-1
  2. D. Betten, Topologische Geometrien auf 3-Mannigfaltigkeiten, Simon Stevin 55 (1981), 221-235
  3. D. Betten, Einige Klassen topologischer 3-Riiume, Resultate der Math. 12 (1987), 37-61 https://doi.org/10.1007/BF03322377
  4. D. Betten and C. Horstmann, Einbettung von topologischen Raumgeometrien auf $R^{3}$ in den reellen affinen Raum, Resultate der Math. 6 (1983), 27-35 https://doi.org/10.1007/BF03323320
  5. H. Busemann, The geometry of geodesics, Academic press, New York, 1965
  6. J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, 1966
  7. R. Engelking, General Topology, Heldermann, Verlag Berlin, 1989
  8. J. Kisyriski, Convergence du Type L, Colloq. Math. 7 (1960), 205-211 https://doi.org/10.4064/cm-7-2-205-211
  9. H. Klein, Models of topological space geometries, J. Geom. 59 (1997), 77-93 https://doi.org/10.1007/BF01229567
  10. J. R. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1975
  11. H. Salzmann, D. Betten, T. Grundhofer, H. Hahl, R. Lowen, and M. Stroppel, Compact Projective Planes, De Gruyter, Berin, New York, 1995
  12. D. Simon, Topologische Geometrien auf dem $R^{3}$, Diplomarbeit, Univ. Kiel, 1985

Cited by

  1. CONDITIONAL FEYNMAN INTEGRAL AND SCHRÖDINGER INTEGRAL EQUATION ON A FUNCTION SPACE vol.79, pp.01, 2009, https://doi.org/10.1017/S0004972708000920