On closed subgroups of precompact groups

TL;DR

This paper investigates how the choice of continuous homomorphisms influences the topological properties of subgroups within precompact groups, advancing understanding of subgroup closure and simplicity in totally bounded Abelian groups.

Contribution

It explores the impact of homomorphism sets on subgroup topology, addressing the number of topologies making subgroups closed and the structure of S-closed subgroup posets.

Findings

Characterized the influence of homomorphism sets on subgroup closure.

Determined conditions for the existence of maximal and minimal S-closed subgroup topologies.

Analyzed the number of topologies making a group topologically simple or SC.

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 \textit{SC-group} (\textit{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?