Bull. Austral. Math. Soc. Vol. 72 (2005) [17–30] LINEAR GEOMETRIES ON THE MOEBIUS STRIP: A THEOREM OF SKORNYAKOV TYPE ̈ Rainer Lowen and Burkard Polster

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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.

1. Introduction A flat stable plane (������, ℒ) consists of a point space ������, which is a surface (topological 2manifold), and a system ℒ of lines, which are closed subsets of ������, such that any two points are joined by a unique line and that the operations of join and intersection are continuous. Moreover, it is required that intersection is stable, that is, the set of pairs of distinct intersecting lines is open. For a comprehensive introduction to general stable planes (not only flat ones), see [6]. The lines of a flat stable plane are 1-manifolds ([10]), and if they are connected, then the point surface ������ is either a topological open disk ������, or the compact nonorientable surface ������ of genus 1 (the real projective plane), or an open (that is, boundaryless) Moebius strip ������ , see [8, 11]. The flat stable planes with ������ = ������ and lines homeomorphic to the circle are of particular importance and are called flat projective planes. Conversely, if (������, ℒ) is a linear space (that is, two points are joined by a unique line) and lines are connected 1-manifolds and closed subsets of ������, then it has been proved by Skornyakov for ������ = ������ and by Salzmann for ������ = ������ (see [10, 2.5] and [12, 32.5]), that (������, ℒ) is a flat stable plane, that is, that join and intersection are automatically continuous. Here we prove a similar result for ������ = ������ : Theorem. Let ������ be an open Moebius strip and suppose we are given a system ℒ of closed subsets ������ ⊆ ������ , called lines, such that lines are connected 1-manifolds, and such that any two points are joined by a unique line.
Received 17th January, 2005 Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/05 $A2.00+0.00.

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