Universal countable-dimensional topological groups
Given a class of topological groups \({\mathcal A}\), a group \(G\in {\mathcal A}\) is called a universal object for \({\mathcal A}\) if \(G\) contains a topologically isomorphic copy of every \(H\in {\mathcal A}\). A universal object for the class of separable metrizable topological groups has first been described by \textit{V. V. Uspenskij} [Funct. Anal. Appl. 20, 160-161 (1986; Zbl 0608.22003)], and later universal objects have been shown to exist for the class of separable metrizable abelian topological groups [\textit{S. A. Shkarin}, Sb. Math. 190, No. 7, 1059-1076 (1999; Zbl 0935.22001)]. In the paper under review, an attempt is made to construct universal objects for classes of separable metrizable abelian topological groups subject to dimensional conditions. A sample result is Theorem 4.1: There exists a countable-dimensional separable metrizable abelian topological group \(G\) containing a copy of every countable-dimensional, separable metrizable abelian topological group \(H\) whose underlying space is an absolute \(G_{\delta\sigma}\). Here, a topological space is called countable-dimensional if it is a countable union of finite-dimensional subspaces. It remains open whether \(G\) can be chosen as an absolute \(G_{\delta\sigma}\) (Question 5.1); this would entail that \(G\) is a universal object for the class of countable-dimensional, absolute \(G_{\delta\sigma}\), separable metrizable abelian topological groups. It also remains open whether there is a universal object for the class of countable-dimensional, separable metrizable abelian topological groups (Question 5.2).
scientific article; zbMATH DE number 1913857