Uncountable products of determined groups need not be determined
The main object of the present paper are determined groups which are defined as follows: If \(H\) is a dense subgroup of the abelian topological group \(G\) then \(H\) determines \(G\) if their character groups endowed with the compact-open topology are topologically isomorphic. An abelian topological group is called determined if it is determined by all its dense subgroups. Every abelian metrizable group is determined. It is shown that any countable sum of determined groups determines the product. Conversely, if the sum of the real line (the integers or a non-trivial locally convex vector space) determines the product, the product must be countable. The main result is the following: If a compact abelian group is determined, then it must be metrizable. This was proved before by \textit{W. W. Comfort, S. U. Raczkowski} and \textit{F. J. Trigos-Arrieta} [Czech. Math. J. 54, No.~2, 509--533 (2004; Zbl 1080.22500)] under the assumption of the continuum hypothesis.
metrizable groups, compact abelian group
scientific article; zbMATH DE number 5355978
scientific article