The weights of closed subgroups of a locally compact group

Let $G$ be an infinite locally compact group and $\mathrm{\aleph }$ a cardinal satisfying ${\mathrm{\aleph }}_0\le \mathrm{\aleph }\le w(G)$ for the weight $w(G)$ of $G$. It is shown that there is a closed subgroup $N$ of $G$ with $w(N)=\mathrm{\aleph }$. Sample consequences are: (1) Every infinite compact group contains an infinite closed metric subgroup. (2) For a locally compact group $G$ and $\mathrm{\aleph }$ a cardinal satisfying ${\mathrm{\aleph }}_0\le \mathrm{\aleph }\le \text{lw}(G)$, where $\text{lw}(G)$ 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 $w(N)=\mathrm{\aleph }$. (3) For an infinite abelian group $G$ there exists a properly ascending family of locally quasiconvex group topologies on $G$, say, $({\tau }_{\mathrm{\aleph }}{)}_{{\mathrm{\aleph }}_0\le \mathrm{\aleph }\le \text{card}(G)}$, such that $(G,{\tau }_{\mathrm{\aleph }})\widehat{\phantom{m}}\cong \widehat{G}$. Items (2) and (3) are shown in Section 5.