Some cases of preservation of the Pontryagin dual by taking dense subgroups

We study the compact-open topology on the character group of dense subgroups of topological abelian groups. Permanence properties concerning open subgroups, quotients and products are considered. We also present some representative examples. We prove that every compact abelian group G with w(G)⩾c has a dense pseudocompact group which does not determine G; this provides (under CH) a negative answer to a question posed by S. Hernández, S. Macario and the third listed author two years ago.