Some classes of minimally almost periodic topological groups

Abstract

[EN] A Hausdorff topological group G=(G,T) has the small subgroup generating property (briefly: has the SSGP property, or is an SSGP group) if for each neighborhood U of $1_G$ there is a family $\sH$ of subgroups of $G$ such that $\bigcup\sH\subseteq U$ and $\langle\bigcup\sH\rangle$ is dense in $G$. The class of \rm{SSGP}$ groups is defined and investigated with respect to the properties usually studied by topologists (products, quotients, passage to dense subgroups, and the like), and with respect to the familiar class of minimally almost periodic groups (the m.a.p. groups). Additional classes SSGP(n) for $n<\omega$ (with SSGP(1) = SSGP) are defined and investigated, and the class-theoretic inclusions $$\mathrm{SSGP}(n)\subseteq\mathrm{SSGP}(n+1)\subseteq\mathrm{ m.a.p.}$$ are established and shown proper.In passing the authors also establish the presence of {\rm SSGP}$(1)$ or {\rm SSGP}$(2)$ in many of the early examples in the literature of abelian m.a.p. groups.