Rates of convergence in weakened group topologies for Rn

Metrizable group topologies for Rn that are weaker than the usual topology arise in many contexts, including the study of minimal groups or of Lie groups of transformations. In this paper we study translation-invariant metrics that are defined by choosing a sequence {vi} of elements of Rn and specifying the rate {pi} at which it converges to zero. If {vi} goes to infinity sufficiently fast in the usual topology, then such a metric always exists, and its translation-invariance guarantees that it will make Rn a topological group. Previous papers investigated the effect on the topology of changing the “converging sequence,” and we now determine the consequences of changing the “rate sequence.” The main theorem is that two rate sequences {pi} and {qi} will determine the same topology for Rn if and only if the ratio {pi/qi} is bounded above and has a strictly positive lower bound.