Non-productive Duality Properties of Topological Groups

We 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.