Completeness properties of group topologies for RR

We study the completeness properties of several different group topologies for the additive group of real numbers, and we also compute the corresponding dual groups. We first present two metrizable connected group topologies on RR with topologically isomorphic dual groups, one of which is noncomplete and arcwise connected and the other one is compact (therefore complete), but not arcwise connected. Using a theorem about T  -sequences and adapting a result about weakened analytic groups, we then describe a method for obtaining Hausdorff group topologies RR that are strictly weaker than the usual topology and are complete. They are not Baire, and consequently not metrizable.