Some classes of topological 3-spaces
type: Item
instance of: scholarly article
title: Some classes of topological 3-spaces
zbMATH Open document ID: 0631.51006
DOI: 10.1007/BF03322377
author: Dieter Betten
published in: Results in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: 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.
reviewed by: Norbert Knarr
Mathematics Subject Classification ID: 51H10
Mathematics Subject Classification ID: 51H20
Mathematics Subject Classification ID: 51J15
zbMATH DE Number: 4026231
zbMATH Keywords: examples
zbMATH Keywords: counterexamples
zbMATH Keywords: topological 3-spaces
zbMATH Keywords: 3-dimensional Lie groups
zbMATH Keywords: topological kinematic spaces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4026231