Products of topological groups in which all closed subgroups are separable Academic Article uri icon

abstract

  • 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 2ω1 = c, we show that there exist: • 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; • 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 … All topological groups and locally convex linear spaces are assumed to be Haus- dorff. The weight of a topological space X, denoted by w(X), is the smallest size of a base for X. A space X is separable if it contains a dense countable subset. If every subspace of …

publication date

  • December 31, 2016