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Non-productive duality properties of topological groups

Masasi Higasikawa
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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.

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Masasi Higasikawa. 2001-06-13. Non-productive duality properties of topological groups. https://arxiv.org/abs/math/0106103

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