Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 1 10.wxyz/CV72P1 http://www.austms.org.au/Publ/Bulletin/V72P1/ Rainer Löwen Burkard Polster 14 February 2006 2006 2 14 17 30 721-5019-LoPo-2005 10.wxyz/C2005V72P1p17 http://www.austms.org.au/Publ/Bulletin/V72P1/721-5019-LoPo/ \noindent 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. 51H10 MR2162290 02212182 L.V. Ahlfors and L. Sario Princeton Mathematical Series 26 Princeton University Press MR114911 L.V. Ahlfors and L. Sario; \textit{Riemann surfaces}, Princeton Mathematical Series 26 (Princeton University Press, Princeton, N.J., 1960). D. Betten Math. Z. MR238331 D. Betten; Topologische Geometrien auf dem M\"obiusband, \textit{Math. Z.} \textbf{107} (1968), pp.~363--379. M. Brown Proc. Amer. Math. Soc. MR126835 M. Brown; The monotone union of open $n$-cells is an open $n$-cell, \textit{Proc. Amer. Math. Soc.} \textbf{12} (1961), pp.~812--814. H. Groh Abh. Math. Sem. Univ. Hamburg MR234343 H. Groh; Topologische Laguerreebenen I, \textit{Abh. Math. Sem. Univ. Hamburg} \textbf{32} (1968), pp.~216--231. H. Groh Abh. Math. Sem. Univ. Hamburg MR257857 H. Groh; Topologische Laguerreebenen II, \textit{Abh. Math. Sem. Univ. Hamburg} \textbf{34} (1970), pp.~11--22. T. Grundhöfer and R. Löwen (Chapter 23) Handbook of incidence geometry North Holland MR1360738 T. Grundh\"ofer and R. L\"owen; Linear topological geometries, (Chapter 23), in \textit{Handbook of incidence geometry} (North Holland, Amsterdam, 1995), pp.~1255--1324. R. Löwen Geom. Dedicata MR608147 R. L\"owen; Central collineations and the parallel axiom in stable planes, \textit{Geom. Dedicata} \textbf{10} (1981), pp.~283--315. R. Löwen Geom. Dedicata MR1358231 R. L\"owen; Ends of surface geometries, revisited, \textit{Geom. Dedicata} \textbf{58} (1995), pp.~175--183. M.H.A. Newman (reprinted by Dover Publications, New York, 1992) Cambridge University Press MR0132534, MR1160354 M.H.A. Newman; \textit{Elements of the topology of plane sets}, (reprinted by Dover Publications, New York, 1992) (Cambridge University Press, New York, 1961). H. Salzmann Adv. Math. MR220135 H. Salzmann; Topological planes, \textit{Adv. Math.} \textbf{2} (1967), pp.~1--60. H. Salzmann Pacific J. Math. MR248855 H. Salzmann; Geometries on surfaces, \textit{Pacific J. Math.} \textbf{29} (1967), pp.~397--402. H. Salzmann, D. Betten, T. Grundhöfer, H. Hähl, R. Löwen and M. Stroppel de Gruyter MR1384300 H. Salzmann, D. Betten, T. Grundh\"ofer, H. H\"ahl, R. L\"owen and M. Stroppel; \textit{Compact projective planes} (de Gruyter, Berlin, New York, 1995). A. Schenkel Dissertation Erlangen--Nürnberg A. Schenkel; \textit{Topologische Minkowski-Ebenen}, Dissertation (Erlangen--N\"urnberg, 1980). L.A. Skornyakov (in Russian) Trudy Moskov. Mat. Obshch. MR88717 L.A. Skornyakov; Systems of curves on a plane, (in Russian), \textit{Trudy Moskov. Mat. Obshch.} \textbf{6} (1957), pp.~135--164. M. Stroppel Results Math. MR1244287 M. Stroppel; A note on Hilbert and Beltrami systems, \textit{Results Math.} \textbf{24}, pp.~242--247. R.D. Wölk Math. Z. MR198332 R.D. W\"olk; Topologische M\"obiusebenen, \textit{Math. Z.} \textbf{93} (1966), pp.~311--333.