Topological Vector Group Topologies Between the Minimal Topology and the Usual Topology on the Real Line

For every positive sequence that tends to zero faster than every fixed exponential, we construct a Hausdorff topological Vector Group topology on the additive group of real numbers. It lies strictly between the minimal Hausdorff topological Vector Group topology and the usual topology. The construction is illustrated by factorial powers, quadratic exponential decay, and prime radicals divided by a quadratic exponential. We also record a finite scalar covering criterion for comparing two such topologies.