Some classes of minimally almost periodic topological groups

W. W. Comfort; Franklin R. Gould

doi:10.4995/agt.2015.3312

Authors

W. W. Comfort Wesleyan University
Franklin R. Gould Central Connecticut State University

DOI:

https://doi.org/10.4995/agt.2015.3312

Keywords:

SSGP group, m.a.p. group, f.p.c. group

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 $1_G$ 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.

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Author Biographies

W. W. Comfort, Wesleyan University

Department of Mathematics and Computer Science,

Franklin R. Gould, Central Connecticut State University

Department of Mathematical Sciences

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