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Zbl 1235.22004
Chasco, M.J.; Dom{\'\i}nguez, X.; Trigos-Arrieta, F.J.
Some cases of preservation of the Pontryagin dual by taking dense subgroups. (English)
[J] Topology Appl. 158, No. 14, 1836-1843 (2011). ISSN 0166-8641

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 {\it determine} $G$ if the restriction operator $G^\wedge\to H^\wedge$ is a topological isomorphism. An abelian topological group is {\it 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 $w(G)\ge \mathfrak c$ contains a dense pseudocompact subgroup which does not determine $G$.
[Taras Banakh (Lviv)]

MSC 2000:: *22A05 Structure of general topological groups

Keywords: topological abelian group; character group; dense subgroup; determined group; determining subgroup; pseudocompact group

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