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 Π_i G _i. In particular, if each G _i is compact then $\oplus _i G_i$ determines Π_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 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 $\kappa = non\left( {\mathcal{N}} \right)?\kappa = {\mathfrak{N}}_1 ?$