Some classes of minimally almost periodic topological groups

Abstract

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 1G there is a family H of subgroups of G such that H ⊆ U and H is dense in G. The class of 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 < ω (with SSGP(1) = SSGP) are defined and investigated, and the class-theoretic inclusionsare established and shown proper. In passing the authors also establish the presence of SSGP(1) or SSGP(2) in many of the early examples in the literature of abelian m.a.p. groups.2010 MSC: Primary 54H11; Secondary 22A05.

SSGP topologies on abelian groups of positive finite divisible rank

“…For every n ∈ N, the property SSGP(n) of topological groups was defined in [7]. We refer the reader to [1,Definition 3.3] for the definition of SSGP(n). It was proved in [1, Remark 3.4, Theorem 3.5] that the following implications hold for an arbitrary topological group:…”

“…1. The small subgroup generating property SSGP Following [4], we define (1) Cyc(A) = {x ∈ G : x ⊆ A} for every A ⊆ G.…”

“…Remark 1.2. The small subgroup generating property was introduced by Gould [7]; see also [8,1]. He says that a topological group G has the small subgroup generating property if for every neighbourhood U of the identity of G, there exists a family H of subgroups of G such that H ⊆ U and H is dense in G. It is easily seen that Gould's definition is equivalent to the one given in 1.1.…”

Contributions to the Bohr topology by W.W. Comfort

“…Minimally almost periodic groups and the small subgroup generating property [13] A topological group G is called minimally almost periodic if the Bohr compactification of G is the trivial group. For more details see [48] It is easy to see that a topological group with SSGP is minimally almost periodic ([48], Theorem 3.1.2).…”

“…For more details see [48] It is easy to see that a topological group with SSGP is minimally almost periodic ([48], Theorem 3.1.2). The joint paper [13] of Wis and F. Gould derives from [48] and extends selected portions of that paper. In particular, they introduced the classes SSGP(n) for 0 ≤ n < ω as follows.…”