Some classes of topological 3-spaces
A 3-space consists of the point set \(\mathbb R^ 3\) and a system of subsets, called lines, such that any two different points are on a unique line and all lines are closed and homeomorphic to \(\mathbb R\). The 3-space is called topological if the operation of joining different points is continuous with respect to the Hausdorff topology on the set of all closed subsets of \(\mathbb R^ 3\). The paper under review is mainly a collection of examples (and counterexamples) of topological 3-spaces. Most of the given constructions are based on one or more topological plane geometries in the sense of \textit{H. R. Salzmann} [Adv. Math. 2, 1--60 (1967; Zbl 0153.21601)]. These planes are pasted together in a suitable way to yield topological 3-spaces. Included is a direct product construction of two such planes, and some of the constructions make use of a certain Lie group which acts on \(\mathbb R^ 3\) as topological transformation group; cp. also \textit{J. Pfalzgraf} [Result. Math. 12, 172--190 (1987; Zbl 0629.51014)]. The author also discusses 3-dimensional Lie groups which are fibered by their one parameter subgroups. These groups yield examples of topological kinematic spaces.
scientific article; zbMATH DE number 4026231
scientific article