The weights of closed subgroups of a locally compact group
Salvador Hern\'andez, Karl H. Hofmann, and Sidney A. Morris

TL;DR
This paper investigates the structure of closed subgroups in infinite locally compact groups, demonstrating the existence of subgroups with specific weights and exploring implications for compact and abelian groups.
Contribution
It establishes the existence of closed subgroups with prescribed weights and analyzes their implications for the structure of compact and abelian groups.
Findings
Existence of closed subgroups with any weight between countable and the group's weight.
Every infinite compact group contains an infinite closed metric subgroup.
Construction of a family of locally quasiconvex topologies on abelian groups.
Abstract
Let be an infinite locally compact group and a cardinal satisfying for the weight of . It is shown that there is a closed subgroup of with . Sample consequences are: (1) Every infinite compact group contains an infinite closed metric subgroup. (2) For a locally compact group and a cardinal satisfying , where is the local weight of , there are either no infinite compact subgroups at all or there is a compact subgroup of with . (3) For an infinite abelian group there exists a properly ascending family of locally quasiconvex group topologies on , say, , such that . Items (2) and (3) are shown in Section 5.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Homotopy and Cohomology in Algebraic Topology · Advanced Operator Algebra Research
