Uncountable products of determined groups need not be determined
comunitat-uji-handle:10234/9
comunitat-uji-handle2:10234/7037
comunitat-uji-handle3:10234/8635
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http://dx.doi.org/10.1016/j.jmaa.2008.07.065 |
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Title
Uncountable products of determined groups need not be determinedDate
2008Publisher
ElsevierISSN
0022247XBibliographic citation
Journal of Mathematical Analysis and Applications, 348, 2, p. 834-842Type
info:eu-repo/semantics/articleSubject
Abstract
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 ... [+]
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) ⊕<sub>i ∈ I</sub> R determines the product ∏<sub>i ∈ I</sub> 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. © 2008 Elsevier Inc. [-]
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