Let G be an infinite locally compact group and let be a cardinal satisfying for the weight of G. It is shown that there is a closed subgroup N of G with . 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 , where 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 . (3) For an infinite abelian group G there exists a properly ascending family of locally-quasiconvex group topologies on G, say, , such that .
© 2012 by Walter de Gruyter Berlin Boston