Products of topological groups in which all closed subgroups are separable

2016/12/31 by Arkady Leiderman, Arkady G. Leiderman, Leiderman, Arkady G. +3
Mathematics · #46A03 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Primary 54D65 #Rings, Modules, and Algebras #Secondary 22A05 #math.GN #msc:22A05 #msc:46A03 #msc:54D65

paper · pdf · doi:10.48550/arxiv.1701.00084

14 pages

arxiv created 2016/12/31 · openalex publication_date 2016/12/31 · arxiv updated 2017/01/03 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

We prove that if H is a topological group such that all closed subgroups of H are separable, then the product G× H has the same property for every separable compact group G. Let c be the cardinality of the continuum. Assuming 2^ω₁ = c, we show that there exist: (1) pseudocompact topological abelian groups G and H such that all closed subgroups of G and H are separable, but the product G× H contains a closed non-separable σ-compact subgroup; (2) pseudocomplete locally convex vector spaces K and L such that all closed vector subspaces of K and L are separable, but the product K× L contains a closed non-separable σ-compact vector subspace.

Related

• Density character of subgroups of topological groups
• The Separable Quotient Problem for Topological Groups
• Locally Compact Groups with All Dense Subgroups Separable
• Dense-separable groups and its applications in d-independence
• On spaces with σ-closed-discrete dense sets