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• W1999321426 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 $$h mapsto hleft| D right.$$ 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 $$wleft( G right) geqslant nonleft( {mathcal{N}} right)$$ 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 $$kappa = nonleft( {mathcal{N}} right)?kappa = {mathfrak{N}}_1 ?$$" @default.
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• W1999321426 date "2004-06-01" @default.
• W1999321426 modified "2023-10-10" @default.
• W1999321426 title "The Dual Group of a Dense Subgroup" @default.
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• W1999321426 doi "https://doi.org/10.1023/b:cmaj.0000042588.07352.99" @default.
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