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The dual group of a dense subgroup. (English) Zbl 1080.22500
Summary: Throughout this abstract, \(G\) is a topological Abelian group and \(\widehat {G}\) is the space of continuous homomorphisms from  \(G\) into the circle group \(\mathbb T\) in the compact-open topology. A dense subgroup \(D\) of \(G\) is said to determine \(G\) if the (necessarily continuous) surjective isomorphism \(\widehat {G}\twoheadrightarrow \widehat {D}\) given by \(h\mapsto h\big | D\) 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 \(D_i\) determines \(G_i\) with \(G_i\) compact, then \(\oplus _iD_i\) determines \(\Pi _i G_i\). In particular, if each \(G_i\) is compact then \(\oplus _i G_i\) determines \(\Pi _i G_i\).
3. Let \(G\) be a locally bounded group and let \(G^+\) denote  \(G\) with its Bohr topology. Then \(G\) is determined if and only if \({G^+}\) is determined.
4. Let \(\text{non} ({\mathcal N})\) be the least cardinal \(\kappa \) such that some \(X \subseteq {\mathbb T}\) of cardinality  \(\kappa \) has positive outer measure. No compact  \(G\) with \(w(G)\geq \text{non} ({\mathcal N})\) is determined; thus if \(\text{non} ({\mathcal N})=\aleph _1\) (in particular if CH holds), an infinite compact group  \(G\) is determined if and only if \(w(G)=\omega \).
Question. Is there in ZFC a cardinal \(\kappa \) such that a compact group \(G\) is determined if and only if \(w(G)<\kappa \)? Is \(\kappa =\text{non} ({\mathcal N})\)? \(\kappa =\aleph _1\)?

MSC:
22A10 Analysis on general topological groups
22B99 Locally compact abelian groups (LCA groups)
22C05 Compact groups
43A40 Character groups and dual objects
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