Bull. Austral. Math. Soc. 72(1) pp.17--30, 2005.
Linear geometries on the Moebius strip: a theorem of Skornyakov type
Rainer Löwen |
Burkard Polster |
.
Abstract
We show that the continuity properties of a stable plane are automatically satisfied if we have a linear space with point set a Moebius strip, provided that the lines are closed subsets homeomorphic to the real line or to the circle. In other words, existence of a unique line joining two distinct points implies continuity of join and intersection. For linear spaces with an open disk as point set, the same result was proved by Skornyakov.
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[an error occurred while processing this directive]| (Metadata: XML, RSS, BibTeX) | MathSciNet: MR2162290 | Z'blatt-MATH: 02212182 |
References
- L.V. Ahlfors and L. Sario;
Riemann surfaces,
Princeton Mathematical Series 26 (Princeton University Press, Princeton, N.J., 1960). MR114911 - D. Betten;
Topologische Geometrien auf dem Möbiusband,
Math. Z. 107 (1968), pp. 363--379. MR238331 - M. Brown;
The monotone union of open n-cells is an open n-cell,
Proc. Amer. Math. Soc. 12 (1961), pp. 812--814. MR126835 - H. Groh;
Topologische Laguerreebenen I,
Abh. Math. Sem. Univ. Hamburg 32 (1968), pp. 216--231. MR234343 - H. Groh;
Topologische Laguerreebenen II,
Abh. Math. Sem. Univ. Hamburg 34 (1970), pp. 11--22. MR257857 - T. Grundhöfer and R. Löwen;
Linear topological geometries,
(Chapter 23),
in Handbook of incidence geometry (North Holland, Amsterdam, 1995), pp. 1255--1324. MR1360738 - R. Löwen;
Central collineations and the parallel axiom in stable planes,
Geom. Dedicata 10 (1981), pp. 283--315. MR608147 - R. Löwen;
Ends of surface geometries, revisited,
Geom. Dedicata 58 (1995), pp. 175--183. MR1358231 - M.H.A. Newman;
Elements of the topology of plane sets of points,
(reprinted by Dover Publications, New York, 1992) (Cambridge University Press, New York, 1961). MR0132534, MR1160354 - H. Salzmann;
Topological planes,
Adv. Math. 2 (1967), pp. 1--60. MR220135 - H. Salzmann;
Geometries on surfaces,
Pacific J. Math. 29 (1967), pp. 397--402. MR248855 - H. Salzmann, D. Betten, T. Grundhöfer, H. Hähl, R. Löwen and M.
Stroppel;
Compact projective planes (de Gruyter, Berlin, New York, 1995). MR1384300 - A. Schenkel;
Topologische Minkowski-Ebenen,
Dissertation (Erlangen--Nürnberg, 1980). - L.A. Skornyakov;
Systems of curves on a plane,
(in Russian),
Trudy Moskov. Mat. Obshch. 6 (1957), pp. 135--164. MR88717 - M. Stroppel;
A note on Hilbert and Beltrami systems,
Results Math. 24, pp. 242--247. MR1244287 - R.D. Wölk;
Topologische Möbiusebenen,
Math. Z. 93 (1966), pp. 311--333. MR198332
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