The Dual Group of a Dense Subgroup |
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Authors: | W W Comfort S U Raczkowski F Javier Trigos-Arrieta |
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Institution: | (1) Department of Mathematics, Wesleyan University, Middletown, CT 06459, USA;(2) USA |
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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 $\kappa = non\left( {\mathcal{N}} \right)?\kappa = {\mathfrak{N}}_1 ?$ |
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Keywords: | Bohr compactification Bohr topology character character group Auß enhofer-Chasco Theorem compact-open topology dense subgroup determined group duality metrizable group reflexive group reflective group |
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