Some cases of preservation of the Pontryagin dual by taking dense subgroups (Q639711)

For an abelian topological group \(G\) let \(G^\wedge\) denote the group of continuous characters of \(G\), endowed with the compact-open topology. A dense subgroup \(H\) of an abelian topological group \(G\) is said to \textit{determine} \(G\) if the restriction operator \(G^\wedge\to H^\wedge\) is a topological isomorphism. An abelian topological group is \textit{determined} if each dense subgroup of \(G\) determines \(G\). The authors detect some operations preserving determined groups and present several representative examples of determined and non-determined groups. One of the main results is Theorem 14 saying that each compact abelian group \(G\) of weight \(w(G)\geq \mathfrak c\) contains a dense pseudocompact subgroup which does not determine \(G\).