Abstract:
The aim of this paper is to go deeper into the study of local minimality and its connection to some naturally related properties. A Hausdorff topological group (G, tau) is called locally minimal if there exists a neighborhood U of 0 in tau such that U fails to be a neighborhood of zero in any Hausdorff group topology on G which is strictly coarser than tau. Examples of locally minimal groups are all subgroups of Banach-Lie groups, all locally compact groups and all minimal groups. Motivated by the fact that locally compact NSS groups are Lie groups, we study the connection between local minimality and the NSS property, establishing that under certain conditions, locally minimal NSS groups are metrizable. A symmetric subset of an abelian group containing zero is said to be a GIG set if it generates a group topology in an analogous way as convex and symmetric subsets are unit balls for pseudonorms on a vector space. We consider topological groups which have a neighborhood basis at zero consisting of GTG sets. Examples of these locally GIG groups are: locally pseudoconvex spaces, groups uniformly free from small subgroups (UFSS groups) and locally compact abelian groups. The precise relation between these classes of groups is obtained: a topological abelian group is UFSS if and only if it is locally minimal, locally GTG and NSS. We develop a universal construction of GIG sets in arbitrary non-discrete metric abelian groups, that generates a strictly finer non-discrete UFSS ...