The weights of closed subgroups of a locally compact group

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 $G$ be an infinite locally compact group and $\aleph$ a cardinal satisfying $\aleph_0\le\aleph\le w(G)$ for the weight $w(G)$ of $G$. It is shown that there is a closed subgroup $N$ of $G$ with $w(N)=\aleph$. Sample consequences are: (1) Every infinite compact group contains an infinite closed metric subgroup. (2) For a locally compact group $G$ and $\aleph$ a cardinal satisfying $\aleph_0\le\aleph\le \lw(G)$, where $\lw(G)$ is the local weight of $G$, there are either no infinite compact subgroups at all or there is a compact subgroup $N$ of $G$ with $w(N)=\aleph$. (3) For an infinite abelian group $G$ there exists a properly ascending family of locally quasiconvex group topologies on $G$, say, $(\tau_\aleph)_{\aleph_0\le \aleph\le \card(G)}$, such that $(G,\tau_\aleph)\hat{\phantom{m}}\cong\hat G$. Items (2) and (3) are shown in Section 5.