Title: Products of topological groups in which all closed subgroups are separable,
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.