Free non-archimedean topological groups. (Q2856933)

For a uniform space \((X,\mathcal{U})\) and a class \(\Omega \) of Hausdorff topological groups, the \(\Omega \)-free group \(F=F_{\Omega }(X,\mathcal{U})\in \Omega \) is defined together with a uniformly continuous mapping \(i\: X\to F\) if for every \(G\in \Omega \) and every uniformly continuous \(\varphi \: X\to G\), there is unique continuous homomorphism \(\Phi \: F\to G\) such that \(\Phi \circ i=\varphi \).NEWLINENEWLINEIt exists if \(\Omega \) is closed under cartesian products and closed subgroups, which is the case of all examples of \(\Omega \) considered in the paper. In particular, the classes of all Hausdorff topological groups, of balanced, Abelian, Boolean, precompact, profinite (inverse limits of finite) groups are studied, respectively.NEWLINENEWLINEThe main point is that all the main results are working with the restriction to non-archimedean uniform spaces, i.e., those which admit a base of uniform covers consisting of partitions. In the same time the classes of topological groups are assumed to be non-archimedean, i.e., they admit a base of the unity consisting of open subgroups. All the above particular classes of groups are studied with this additional property. Some results differ essentially within this context from the recalled known ones.NEWLINENEWLINEThe description of the group topologies of \(F_{\Omega }(X,\mathcal{U})\) gives that metrizability of the non-archimedean \((X,\mathcal{U})\) implies metrizability of the respective \(\Omega \)-free groups in the cases of non-archimedean and Abelian, Boolean, or balanced groups, respectively. The same conclusion holds for the non-archimedean precompact metrizable \((X,\mathcal{U})\) on the non-archimedean precompact or profinite groups, respectively. In most of the nonrestricted cases this is not true for any non-discrete space.NEWLINENEWLINEEvery epimorphism \(f\: G\to H\), \(G,H\in \Omega \), is dense in the restricted (non-archimedean) case, contrary to the non-restricted case. This statement uses automorphizability by the Boolean non-archimedean free group of uniform spaces with particular actions.NEWLINENEWLINESome recent results on surjectively universal groups are obtained and improved describing them as \(\Omega \)-free groups for suitable uniform spaces.