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Some cases of preservation of the Pontryagin dual by taking dense subgroups. (English) Zbl 1235.22004
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 determine $G$ if the restriction operator ${G}^{\wedge }\to {H}^{\wedge }$ is a topological isomorphism. An abelian topological group is 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 $w\left(G\right)\ge 𝔠$ contains a dense pseudocompact subgroup which does not determine $G$.
##### MSC:
 22A05 Structure of general topological groups