Topology Atlas Document # ppae-04


Concerning the dual group of a dense subgroup

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

Proceedings of the Ninth Prague Topological Symposium (2001) pp. 23-35

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 Di determines Gi with Gi compact, then \oplusi Di determines \Pii Gi. In particular, if each Gi is compact then \oplusi Gi determines \Pii Gi.
  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.
Question. Is there in ZFC a cardinal \kappa such that a compact group G is determined if and only if w(G) < \kappa? Is \kappa = \non(N)? \kappa = \aleph1?

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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Copyright © 2002 Charles University and Topology Atlas. Published April 2002.