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Izv. Akad. Nauk SSSR Ser. Mat., 1990, Volume 54, Issue 5, Pages 1090–1107 (Mi izv1064)  

This article is cited in 19 scientific papers (total in 19 papers)

Topologies on abelian groups

E. G. Zelenyuk, I. V. Protasov

National Taras Shevchenko University of Kyiv

Abstract: A filter $\varphi$ on an abelian group $G$ is called a $T$-filter if there exists a Hausdorff group topology under which $\varphi$ converges to zero. $G\{\varphi\}$ will denote the group $G$ with the largest topology among those making $\varphi$ converge to zero. This method of defining a group topology is completely equivalent to the definition of an abstract group by defining relations. We shall obtain characterizations of $T$-filters and of $T$-sequences; among these, we shall pay particular attention to $T$-sequences on the integers. The method of $T$-sequences will be used to construct a series of counterexamples for several open problems in topological algebra. For instance there exists, on every infinite abelian group, a topology distinguishing between sequentiality and the Frechet–Urysohn property (this solves a problem posed by V. I. Malykhin); we also find a topology on the group of integers admitting no nontrivial continuous character, thus solving a problem of Nienhuys. We show also that on every infinite abelian group there exists a free ultrafilter which is not a $T$-ultrafilter.

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English version:
Mathematics of the USSR-Izvestiya, 1991, 37:2, 445–460

Bibliographic databases:

UDC: 512.546
MSC: Primary 20K45, 22A30; Secondary 54A20, 54B99
Received: 03.11.1988

Citation: E. G. Zelenyuk, I. V. Protasov, “Topologies on abelian groups”, Izv. Akad. Nauk SSSR Ser. Mat., 54:5 (1990), 1090–1107; Math. USSR-Izv., 37:2 (1991), 445–460

Citation in format AMSBIB
\by E.~G.~Zelenyuk, I.~V.~Protasov
\paper Topologies on abelian groups
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1990
\vol 54
\issue 5
\pages 1090--1107
\jour Math. USSR-Izv.
\yr 1991
\vol 37
\issue 2
\pages 445--460

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    This publication is cited in the following articles:
    1. E. G. Zelenyuk, “Topological groups in which each nowhere dense subset is closed”, Math. Notes, 64:2 (1998), 177–180  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. J.E. Marcos, “Lacunar ring topologies and maximum ring topologies with a prescribed convergent sequence”, Journal of Pure and Applied Algebra, 162:1 (2001), 53  crossref
    3. Giuseppina Barbieri, Dikran Dikranjan, Chiara Milan, Hans Weber, “Answer to Raczkowski's questions on convergent sequences of integers”, Topology and its Applications, 132:1 (2003), 89  crossref
    4. J.E. Marcos, “The maximum ring topology on the rational number field among those for which the sequence 1/n converges to zero”, Topology and its Applications, 128:2-3 (2003), 157  crossref
    5. Dikran Dikranjan, Mikhail Tkachenko, Ivan Yaschenko, “On transversal group topologies”, Topology and its Applications, 153:5-6 (2005), 786  crossref
    6. Dikranjan D., “Closure operators in topological groups related to von Neumann's kernel”, Topology and Its Applications, 153:11 (2006), 1930–1955  crossref  mathscinet  zmath  isi
    7. M.J. Chasco, E. Martín-Peinador, V. Tarieladze, “A class of angelic sequential non-Fréchet–Urysohn topological groups”, Topology and its Applications, 154:3 (2007), 741  crossref
    8. Dikran Dikranjan, Kenneth Kunen, “Characterizing subgroups of compact abelian groups”, Journal of Pure and Applied Algebra, 208:1 (2007), 285  crossref
    9. S.S. Gabriyelyan, “Characterization of almost maximally almost-periodic groups”, Topology and its Applications, 156:13 (2009), 2214  crossref
    10. S. S. Gabriyelyan, “Reflexive group topologies on Abelian groups”, Journal of Group Theory, 13:6 (2010), 891  crossref
    11. S.S. Gabriyelyan, “On T-sequences and characterized subgroups”, Topology and its Applications, 157:18 (2010), 2834  crossref
    12. S.S. Gabriyelyan, “Groups of quasi-invariance and the Pontryagin duality”, Topology and its Applications, 157:18 (2010), 2786  crossref
    13. S.S. Gabriyelyan, “Characterizable groups: Some results and open questions”, Topology and its Applications, 2012  crossref
    14. Lydia Außenhofer, Daniel de la Barrera Mayoral, “Linear topologies on are not Mackey topologies”, Journal of Pure and Applied Algebra, 2012  crossref
    15. S.S. Gabriyelyan, “Topologies on groups determined by sets of convergent sequences”, Journal of Pure and Applied Algebra, 2012  crossref
    16. D. Dikranjan, S.S. Gabriyelyan, “On characterized subgroups of compact abelian groups”, Topology and its Applications, 2013  crossref
    17. D. Dikranjan, S.S. Gabriyelyan, V. Tarieladze, “Characterizing sequences for precompact group topologies”, Journal of Mathematical Analysis and Applications, 2013  crossref
    18. S.S. Gabriyelyan, “Bounded subgroups as a von Neumann radical of an Abelian group”, Topology and its Applications, 178 (2014), 185  crossref
    19. S.S. Gabriyelyan, “On reflexivity of the group of the null sequences valued in an Abelian topological group”, Journal of Pure and Applied Algebra, 2014  crossref
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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