Non-productive duality properties of topological groups

2001/06/13 by Masasi Higasikawa, Higasikawa, Masasi
Mathematics · #11Z05 Secondary #22A05 #43A40 (Primary) 11D61 #FOS: Mathematics #General Topology (math.GN) #Number Theory (math.NT) #math.GN #math.NT #msc:11D61 #msc:11Z05 #msc:22A05 #msc:43A40

paper · pdf · doi:10.48550/arxiv.math/0106103

6 pages

arxiv created 2001/06/13 · arxiv updated 2009/11/30

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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