On closed subgroups of precompact groups

2022/03/01 by Hernández, Salvador, Remus, Dieter, Trigos-Arrieta, F. Javier
#FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2203.00334

Abstract

It is a Theorem of W.~ W. Comfort and K.~ A. Ross that if G is a subgroup of a compact Abelian group, and S denotes those continuous homomorphisms from G to the one-dimensional torus, then the topology on G is the initial topology given by S. Assume that H is a subgroup of G. We study how the choice of S affects the topological placement and properties of H in G. Among other results, we have made significant progress toward the solution of the following specific questions: How many totally bounded group topologies does G admit such that H is a closed (dense) subgroup? If C_S denotes the poset of all subgroups of G that are S-closed, ordered by inclusion, does C_S has a greatest (resp. smallest) element? We say that a totally bounded (topological, resp.) group is an SC-group (topologically simple, resp.) if all its subgroups are closed (if G and \e\ are its only possible closed normal subgroups, resp.) In addition, we investigate the following questions. How many SC-(topologically simple totally bounded, resp.) group topologies does an arbitrary Abelian group G admit?

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