Criteria for two-dimensional circle planes
The paper is concerned with the following problem. Suppose you are given a circle plane, i.e., a Möbius plane, a Laguerre plane or a Minkowski plane. Suppose the point space of the circle plane can be equipped with a topology such that it is homeomorphic to the point space of the corresponding classical circle plane over the real numbers. Is there a topology on the circle space, such that the circle plane becomes a topological plane, i.e., such that geometric operations are continuous. If the answer is positive, then there is only one such topology on the circle space, namely the topology induced by the Hausdorff metric. The authors present a way to solve this problem by looking at the derived structure. To obtain the derived structure at a point \(s\) one removes all points parallel to \(s\). Circles through \(s\) and parallel classes not containing \(s\) become lines in the derived structure. This yields an affine plane. In the first part of the paper the authors investigate what circles not incident with \(s\) have to become in the derived plane such that the circle plane may become a topological circle plane. These results are then used to show that if for a circle plane as above at every point the derived affine plane is a topological affine plane, then the circle plane can be made a topological circle plane. If a Laguerre plane or a Minkowski plane is given, one only needs that for every point of a parallel class the derived affine plane is a topological affine plane to obtain the desired result. The proof is based on the fact that if a circle plane is given such that the point space is a topological space homeomorphic to the point space of the corresponding classical real circle plane and such that circles are simply closed curves, then the Hausdorff metric induces a topology on the circle set such that the circle plane becomes a topological circle plane. Unfortunately, it is not known whether a similar result holds for 4- dimensional circle planes. To people working with 2-dimensional circle planes the results of the paper might be no surprise but most likely the results will become handy tools to show that certain circle planes are indeed topological circle planes. Unfortunately, people working with 4-dimensional circle planes will have to do without such results.
scientific article; zbMATH DE number 767396