Topological Hjelmslev planes

Abstract

An affine or projective Hjelmslev plane (henceforth A.H. or P.H. plane) is a generalization of an ordinary affine or projective plane, where two lines (points) may intersect in (be joined by) more than one point (line). Two points are neighbours if they possess more than one joining line, and two lines are neighbours if each point of either line has a neighbour on the other. These neighbour relations define equivalence relations on the set of points and the set of lines respectively. Their corresponding quotient spaces generate an ordinary plane called the quotient plane of the Hjelmslev plane. A projective (affine) plane is a topological plane if the set of points and the set of lines are Hausdorff spaces, and the operations of join and intersection (and parallelism) are continuous. The purpose of this paper is to investigate analogous topological notions in Hjelmslev planes. Our ideas are naturally highly motivated by the work of Salzmann (cf. [11] and [12]). Every ordinary affine plane has a unique projective extension. This result has a topological analogue, provided the plane is Desarguesian or locally compact and connected (cf. [11], §14 and [12], 7.15). However, for H-planes, the author and N. D. Lane have constructed in [8] a Desarguesian A.H. plane with no Desarguesian projective extension, and in [3] Drake constructed an A.H. plane with no projective extension. Hence, we shall first study topological A.H. planes, in Sections 1 to 6, and then consider topological P.H. planes in Sections 7 and 8. Examples of such planes are constructed in Section 9. A P.H. (A.H.) plane is topological if its point and line sets are topological spaces, the neighbour relations are closed, and join and intersection (and parallelism) are continuous. Our main objectives are, to determine when the quotient plane, endowed with the usual quotient topologies, is a topological plane, and to construct examples of topological H-planes over certain topological rings. In Section 3 we determined relationships between point and line topologies and use these to prove join and intersection are open maps. Section 4 contains the result that the quotient plane is topological if and only if the quotient map is open on the points. If the plane is locally compact and connected or a translation plane, then the quotient plane is shown, in Section 5, to be * The author gratefully acknowledges the support of the National Research Council of Canada.