Concerning the dual group of a dense subgroup (Q2782204)

A topological Abelian group \(G\) is called \textit{determined} if every dense subgroup of \(G\) determines \(G\), meaning that its dual group coincides (topologically) with that of \(G\). This concept was studied previously by \textit{L. Aussenhofer} [Diss. Math. 384, 113 p. (1999; Zbl 0953.22001)] and by \textit{M. J. Chasco} [Arch. Math. 70, 22-28 (1998; Zbl 0899.22001)]. These authors had shown that every metrizable group is determined. The present paper, an expanded version of the second author's Ph.D. thesis, states many more results on determined groups. Thus, assuming the continuum hypothesis, a compact group is determined if and only if its weight is \(\omega\). Whether this holds without the continuum hypothesis is an open question. A large class of examples of determined groups is constructed in the following way: let \(G\) be locally bounded Abelian and determined; on \(G\), replace the given topology by the Bohr topology. The resulting group \(H\) will be determined. If \(G\) is not totally bounded then \(H\) is totally bounded but not metrizable. - A paper containing full proofs is announced.NEWLINENEWLINEFor the entire collection see [Zbl 0983.00046].