Products of topological groups in which all closed subgroups are separable

TL;DR

The paper investigates conditions under which the product of topological groups preserves the property that all closed subgroups are separable, revealing both positive results and counterexamples under certain set-theoretic assumptions.

Contribution

It proves that the product of a group with all closed subgroups separable and a separable compact group also has this property, and constructs counterexamples under specific set-theoretic conditions.

Findings

Product of certain topological groups retains separability of closed subgroups.

Existence of pseudocompact groups with non-separable closed subgroups in their product.

Counterexamples under set-theoretic assumptions showing failure of the property.

Abstract

We prove that if $H$ is a topological group such that all closed subgroups of $H$ are separable, then the product $G\times H$ has the same property for every separable compact group $G$. Let $c$ be the cardinality of the continuum. Assuming $2^{\omega_1} = c$, 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 $G\times H$ contains a closed non-separable $\sigma$-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 $K\times L$ contains a closed non-separable $\sigma$-compact vector subspace.