Some aspects of dimension theory for topological groups
In this paper, the authors discuss dimension theory in the class of all topological groups. In the first section, they refer to the dawn of dimension theory in topology, in particular, introduce the research activities of Luitzen Egbertus Jan Brouwer (1881--1966) with Pavel Sergeyevich Alexandroff (1896--1982) and Pavel Samuilovich Urysohn (1898--1924). In the following sections, the authors concentrate on the dimension theory of topological groups, in particular, non-locally compact topological groups, because many fundamental problems remain unsolved. For example, they pose the following problems in the abstract: Does every connected Polish group contain a homeomorphic copy of \([0,1]\)? Is there a homogeneous metrizable compact space \(X\) such that the homeomorphism group of \(X\) is 2-dimensional? For a topological group \(G\) with a countable network, does it follow that \(\dim G=\mathrm{ind }G=\mathrm{Ind }G\)? Does the inequality \(\dim(G\times H)\leq\dim G+\dim H\) hold for arbitrary topological groups \(G\) and \(H\) which are subgroups of \(\sigma\)-compact topological groups? They introduce several results with respect to the above problems and also pose many related problems.
scientific article; zbMATH DE number 6825024
scientific article