The weights of closed subgroups of a locally compact group

Abstract

Let G be an infinite locally compact group and let @ be a cardinal satisfying @0 @ w.G/ for the weight w.G/ of G. It is shown that there is a closed subgroup N of G with w.N / D @. Sample consequences are: (1) Every infinite compact group contains an infinite closed metric subgroup. (2) For a locally compact group G and @ a cardinal satisfying @0 @ wloc.G/, where wloc.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 / D @. (3) For an infinite abelian group G there exists a properly ascending family of locally-quasiconvex group topologies on G, say, . @/@0 @ card.G/, such that .G; @/O OG.