  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.
