2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61646We address two properties for Abelian topological groups: ``every closed subgroup is dually closed'' and ``every closed subgroup is dually embedded.'' We exhibit a pair of topological groups such that each has both of the properties and the product has neither, which refutes a remark of N. Noble. These examples are the additive group of integers topologized with respect to a convergent sequence as investigated by E.G. Zelenyuk and I.V. Protasov. The proof for the product relies on a theorem on exponential Diophantine equations.6 pagesGeneral TopologyNumber Theory22A05, 43A40 (Primary) 11D61, 11Z05 Secondarytext