Relative minimality and co-minimality of subgroups in topological groups
A Hausdorff topological group is called minimal if there does not exist any strictly coarser Hausdorff group topology. Recently it was shown by \textit{V. V. Uspenskij} [Topology Appl. 155, No. 14, 1580--1606 (2008; Zbl 1166.22002)] that every Hausdorff group can be embedded in a minimal group. The second named author proved that every Hausdorff group is a group retract of a minimal group [Topology Appl. 155, No. 17--18, 2105--2127 (2008; Zbl 1153.22002)]. In the paper under review the authors investigate minimal group topologies on semidirect products. More precisely, given a continuous biadditive mapping \(w:(E,\sigma)\times (F,\tau)\to (A,\nu)\), they define the generalized Heisenberg group as follows: \(H(w)=(A\times E)\lambda F\) where \[ (a_1,x_1,f_1)\cdot (a_2,x_2,f_2)= (a_1+a_2+f_1(x_2),x_1+x_2,f_1+f_2). \] They prove: \(H(w)\) is a minimal group iff \(A\) is a minimal group and \(w\) is a minimal biadditive mapping which means that for any Hausdorff group topologies \(\sigma_1\subseteq \sigma\) and \(\tau_1\subseteq \tau\) the continuity of \(w:(E,\sigma_1)\times (F,\tau_1)\to (A,\nu)\) implies \(\sigma_1=\sigma\) and \(\tau_1=\tau\). In this context they study relative minimality and co-minimality, where a subset \(X\) of a Hausdorff group \((G,\sigma)\) is called relatively minimal if any coarser Hausdorff group topology \(\sigma_1\) on \(G\) induces the same topology as \(\sigma\) on \(X\). A subgroup \(H\) of \((G,\sigma)\) is called co-minimal if for any coarser Hausdorff group topology \(\sigma_1\) on \(G\) the quotient topologies induced by \(\sigma\) and \(\sigma_1\) on the quotient space \(G/H\) are the same. Many applications and examples are given.
scientific article; zbMATH DE number 5641745
scientific article