The weights of closed subgroups of a locally compact group

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, $(τ_\aleph)_{\aleph_0\le \aleph\le \card(G)}$, such that $(G,{\tau }_{\aleph })\hat{}\cong \hat{G}$. Items (2) and (3) are shown in Section 5.