DOI: 10.1515/jgth-2022-0093
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On closed subgroups of precompact groups

Salvador Hern谩ndez,Dieter Remus,F. Javier Trigos-Arrieta

Abelian group
Mathematical analysis
Group (periodic table)
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Abstract It is a theorem of W. W. Comfort and K. A. Ross that if 饾惡 is a subgroup of a compact Abelian group and 饾憜 denotes the continuous homomorphisms from 饾惡 to the one-dimensional torus, then the topology on 饾惡 is the initial topology given by 饾憜. Assume that 饾惢 is a subgroup of 饾惡. We study how the choice of 饾憜 affects the topological placement and properties of 饾惢 in 饾惡. Among other results, we have made significant progress toward the solution of the following specific questions. How many totally bounded group topologies does 饾惡 admit such that 饾惢 is a closed (dense) subgroup? If <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>C</m:mi> <m:mi>S</m:mi> </m:msub> </m:math> C_{S} denotes the poset of all subgroups of 饾惡 that are 饾憜-closed, ordered by inclusion, does <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>C</m:mi> <m:mi>S</m:mi> </m:msub> </m:math> C_{S} have 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 饾惡 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mi>e</m:mi> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> \{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 饾惡 admit?


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