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Duality properties of bounded torsion topological abelian groups. (English) Zbl 1359.22001
This article mainly discusses the pseudocompactness and the Baire property of the dual group \(G^{\wedge}_{p}\) of a precompact bounded torsion abelian topological group \(G\), where the group \(G^{\wedge}_{p}\) of continuous characters of \(G\) is endowed with the topology of pointwise convergence. In particular, the authors prove that if \(G\) is pseudocompact (Baire), then countably compact (compact) subsets of \(G^{\wedge}_{p}\) are finite. Also, the authors present an example of a precompact Boolean group \(G\) with the Baire property such that the dual group \(G^{\wedge}_{p}\) contains an infinite countably compact subspace without isolated points.

22A05 Structure of general topological groups
54E52 Baire category, Baire spaces
Full Text: DOI
[1] Ardanza-Trevijano, S.; Chasco, M. J.; Domínguez, X.; Tkachenko, M. G., Precompact non-compact reflexive abelian groups, Forum Math., 24, 289-302, (2012) · Zbl 1259.22001
[2] Arhangel’skii, A. V.; Tkachenko, M. G., Topological groups and related structures, Atlantis Ser. Math., vol. I, (2008), Atlantis Press and World Scientific Paris-Amsterdam
[3] Außenhofer, L., Contributions to the duality theory of abelian topological groups and to the theory of nuclear groups, Dissertationes Math., vol. 384, (1999), PWN Warszawa · Zbl 0953.22001
[4] Außenhofer, L.; Gabriyelyan, S., On reflexive group topologies on abelian groups of finite exponent, Arch. Math., 99, 6, 583-588, (2012) · Zbl 1267.22001
[5] Banaszczyk, W., Additive subgroups of topological vector spaces, Lecture Notes in Math., vol. 1466, (1991), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0743.46002
[6] Bruguera, M.; Tkachenko, M., Pontryagin duality in the class of precompact groups and the Baire property, J. Pure Appl. Algebra, 216, 12, 2636-2647, (2012) · Zbl 1278.43003
[7] Chasco, M. J.; Dikranjan, D.; Martín Peinador, E., A survey on reflexivity of abelian topological groups, Topology Appl., 159, 9, 2290-2309, (2012) · Zbl 1247.22001
[8] Chasco, M. J.; Martín Peinador, E.; Tarieladze, V., On MacKey topology for groups, Studia Math., 132, 3, 257-284, (1999) · Zbl 0930.46006
[9] Comfort, W. W.; Ross, K. A., Topologies induced by groups of characters, Fund. Math., 55, 283-291, (1964) · Zbl 0138.02905
[10] Comfort, W. W.; Ross, K. A., Pseudocompactness and uniform continuity in topological groups, Pacific J. Math., 16, 3, 483-496, (1966) · Zbl 0214.28502
[11] Engelking, R., General topology, (1989), Heldermann Verlag Berlin · Zbl 0684.54001
[12] Galindo, J.; Macario, S., Pseudocompact group topologies with no infinite compact subsets, J. Pure Appl. Algebra, 215, 655-663, (2011) · Zbl 1215.54015
[13] Hernández, S.; Macario, S., Dual properties in totally bounded abelian groups, Arch. Math., 80, 271-283, (2003) · Zbl 1025.22003
[14] Juhász, I.; van Mill, J., Countably compact spaces all countable subsets of which are scattered, Comment. Math. Univ. Carolin., 22, 4, 851-855, (1981) · Zbl 0485.54018
[15] McPhail, M. C.; Morris, S. A., Varieties of abelian topological groups and scattered spaces, Bull. Aust. Math. Soc., 78, 3, 487-495, (2008) · Zbl 1168.22002
[16] Mrówka, S.; Rajagopalan, M.; Soundararajan, T., A characterization of compact scattered spaces through chain limits, (TOPO 72 — General Topology and its Applications, Lecture Notes in Math., vol. 378, (1974)), 288-297 · Zbl 0299.54015
[17] Pytkeev, E., Condensations onto compact spaces and complete metric spaces, (1979), Steklov Institute of Mathematics, (in Russian)
[18] Raczkowski, S. U.; Trigos-Arrieta, F. J., Duality of totally bounded abelian groups, Bol. Soc. Mat. Mex. III, 7, 1, 1-12, (2001) · Zbl 1007.22004
[19] Robinson, D. J.F., A course in the theory of groups, (1982), Springer-Verlag Berlin
[20] Tkachenko, M. G., Compactness type properties in topological groups, Czechoslovak Math. J., 38, 324-341, (1988) · Zbl 0664.54006
[21] Tkachenko, M. G., Self-duality in the class of precompact groups, Topology Appl., 156, 12, 2158-2165, (2009) · Zbl 1180.22004
[22] Tkachenko, M. G., Topological groups in which all countable subgroups are closed, Topology Appl., 159, 7, 1806-1814, (2012) · Zbl 1245.22003
[23] Trigos-Arrieta, F. J., Continuity, boundedness, connectedness and the lindelöff property for topological groups, J. Pure Appl. Algebra, 70, 199-210, (1991) · Zbl 0724.22003
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