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