Products of topological groups in which all closed subgroups are separable

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: • 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.