2012/01/18 by Salvador Hernández, Hernández, Salvador, Karl H. Hofmann +3 · 1 citation
Mathematics · #22C05 #22D05 #22D35 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1201.3814
openalex publication_date 2012/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be an infinite locally compact group and ℵ a cardinal satisfying ℵ0≤ℵ≤ w(G) for the weight w(G) of G. It is shown that there is a closed subgroup N of G with w(N)=ℵ. Sample consequences are: (1) Every infinite compact group contains an infinite closed metric subgroup. (2) For a locally compact group G and ℵ a cardinal satisfying ℵ0≤ℵ≤ \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)=ℵ. (3) For an infinite abelian group G there exists a properly ascending family of locally quasiconvex group topologies on G, say, (τ_ℵ)ℵ0≤ ℵ≤ \card(G), such that (G,τ_ℵ)\phantomm≅ G. Items (2) and (3) are shown in Section 5.