Concerning the dual group of a dense subgroup

W. W. Comfort, S. U. Raczkowski and F. Javier Trigos-Arrieta

Throughout this Abstract, G is a topological Abelian group and [^G] is the space of continuous homomorphisms from G into T in the compact-open topology. A dense subgroup D of G determines G if the (necessarily continuous) surjective isomorphism [^G]\twoheadrightarrow[^D] given by h --> h|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. Auß enhofer 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 \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 \non(N) be the least cardinal \kappa such that some X subset or equal \TT of cardinality \kappa has positive outer measure. No compact G with w(G) >= \non(N) is determined; thus if \non(N)=\aleph₁ (in particular if CH holds), an infinite compact group G is determined if and only if w(G)=\omega.

Mathematics Subject Classification. 22A10 22B99 22C05 43A40 54H11 (03E35 03E50 54D30 54E35).
Keywords. Bohr compactification, Bohr topology, character, character group, Au{\ss}enhofer-Chasco Theorem, compact-open topology, dense subgroup, determined group, duality, metrizable group, reflexive group, reflective group.

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arXiv

math.GN/0204147

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