The Dual Group of a Dense Subgroup

Abstract

Throughout this abstract, G is a topological Abelian group and $$\hat G$$ 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 $$\hat G \to \hat D$$ given by $$h \mapsto h\left| D \right.$$ 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 $$ \oplus _i D_i $$ determines Πi G i. In particular, if each G i is compact then $$ \oplus _i G_i $$ determines Πi 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 non $$\left( {\mathcal{N}} \right)$$ be the least cardinal κ such that some $$X \subseteq {\mathbb{T}}$$ of cardinality κ has positive outer measure. No compact G with $$w\left( G \right) \geqslant non\left( {\mathcal{N}} \right)$$ is determined; thus if $$\left( {\mathcal{N}} \right) = {\mathfrak{N}}_1 $$ (in particular if CH holds), an infinite compact group G is determined if and only if w(G) = ω. Question. Is there in ZFC a cardinal κ such that a compact group G is determined if and only if w(G) < κ? Is $$\kappa = non\left( {\mathcal{N}} \right)?\kappa = {\mathfrak{N}}_1 ?$$