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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
##### Keywords:
Bohr compactification; Bohr topology; character; character group
Full Text:
##### References:
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