Throughout this Abstract, G is a topological Abelian group and G is the space of continuous homomorphisms from G into T in the compact-open topology. A dense subgroup D of G determines G if the (necessarily continuous) surjective isomorphism G ։ D given by h → h|D is a homeomorphism, and G is determined if each dense subgroup of G determines G. The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is determined. The authors offer several related results, including these.(1) There are (many) nonmetrizable, noncompact, determined groups.(2) If the dense subgroup Di determines Gi with Gi compact, then ⊕i Di determines Πi Gi. In particular, if each Gi is compact then ⊕i Gi determines Πi Gi.

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1) Every infinite, Abelian compact (Hausdorff) group K admits 2 |K| -many dense, non-Haar-measurable subgroups of cardinality |K|. When K is nonmetrizable, these may be chosen to be pseudocompact.

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If H is a dense subgroup of G, we say that H determines G if their groups of characters are topologically isomorphic when equipped with the compact open topology. If every dense subgroup of G determines G, then we say that G is determined. The importance of this property is justified by the recent generalizations of Pontryagin-van Kampen duality to wider classes of topological Abelian groups. Among other results, we show (a) i∈I R determines the product i∈I R if and only if I is countable, (b) a compact group is determined if and only if its weight is countable. These answer questions of Comfort, Raczkowski and the third listed author. Generalizations of the above results are also given.