Topological groups with many small subgroups

We introduce and study a functorial topology on every group G having as a base the family of all subgroups of G. Making use of this topology, we obtain an equivalent description of the small subgroup generating property introduced by Gould [26]; see also Comfort and Gould [6]. This property implies minimal almost periodicity. Answering questions of Comfort and Gould [6], we show that every abelian group of infinite divisible rank admits a group topology having the small subgroup generating property. For unbounded abelian groups of finite divisible rank, we find a new necessary condition for the existence of a group topology having the small subgroup generating property, and we conjecture that this condition is also sufficient.