Some topological groups with and some without suitable sets

If a discrete subset S of a topological group G with identity 1 generates a dense subgroup of G and S∪{1} is closed in G, then S is called a suitable set for G. We construct in ZFC a Lindelöf topological group L such that t(L)·ψ(L)≤ℵ0 and L does not have a suitable set. We also give a ZFC example of a countably compact topological group H with no suitable set; in addition, the closure of every countable subset of H is compact. It is proved that a non-pseudocompact topological group with a dense strictly σ-discrete subset has a closed suitable set. This implies, in particular, that a free (Abelian) topological group on a metrizable space has a closed suitable set.