Non-productive duality properties of topological groups

TL;DR

This paper investigates duality properties of Abelian topological groups, providing counterexamples that challenge previous assumptions about the behavior of these properties under group products.

Contribution

It constructs specific examples of topological groups with certain duality properties that do not persist under taking products, refuting a prior remark by N. Noble.

Findings

Counterexamples of groups with duality properties

Product of such groups can lack these properties

Uses exponential Diophantine equations in proofs

Abstract

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.