Models of topological space geometries
The study of topological space geometries has been initiated by \textit{D. Betten} [Simon Stevin 55, 221-235 (1981; Zbl 0447.51012)]. Consider a linear space \({\mathcal R} = (\mathbb R^3, \mathfrak L)\) where each line \(L \in \mathfrak L\) is closed in the point space \(\mathbb R^3\) and is homeomorphic to \(\mathbb R\), and where the map of joining \(2\) distinct points by a line is continuous with respect to the Hausdorff topology on \(\mathfrak L\). Very little can be said about such space geometries in general. Therefore, Betten assumed in addition that any triangle is contained in a unique plane and that the geometric operations involving planes are also continuous. The author investigates the consequences of 4 weaker axioms in general topological space geometries. One of these axioms requires that each line is contained in at least one plane. Together with the other axioms, this implies that each incidence-geometric plane is a topological one. The subsets \({\mathcal E}_U\) of all planes intersecting the open pointset \(U\) generate the Hausdorff topology on the space of all planes. Each collineation is continuous, and the collineation group is a Lie group. There is a large class of examples with non-Desarguesian planes.
scientific article; zbMATH DE number 1053891
scientific article