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Metrization criteria for compact groups in terms of their dense subgroups

Dikran DikranjanDmitri Shakhmatov — 2013

Fundamenta Mathematicae

According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism Ĝ → D̂ of the dual groups is a topological isomorphism. We introduce four conditions on D that are necessary for it to determine G and we resolve the following question: If one of these conditions holds for every dense (or G δ -dense) subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined by all its...

Productivity of the Zariski topology on groups

Dikran N. DikranjanD. Toller — 2013

Commentationes Mathematicae Universitatis Carolinae

This paper investigates the productivity of the Zariski topology G of a group G . If 𝒢 = { G i i I } is a family of groups, and G = i I G i is their direct product, we prove that G i I G i . This inclusion can be proper in general, and we describe the doubletons 𝒢 = { G 1 , G 2 } of abelian groups, for which the converse inclusion holds as well, i.e., G = G 1 × G 2 . If e 2 G 2 is the identity element of a group G 2 , we also describe the class Δ of groups G 2 such that G 1 × { e 2 } is an elementary algebraic subset of G 1 × G 2 for every group G 1 . We show among others, that Δ is stable...

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