
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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Mathematics of the USSRIzvestiya, 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. USSRIzv., 37:2 (1991), 445–460
Citation in format AMSBIB
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\jour Izv. Akad. Nauk SSSR Ser. Mat.
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\zmath{https://zbmath.org/?q=an:0728.22003}
\adsnasa{http://adsabs.harvard.edu/cgibin/bib_query?1991IzMat..37..445Z}
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\jour Math. USSRIzv.
\yr 1991
\vol 37
\issue 2
\pages 445460
\crossref{https://doi.org/10.1070/IM1991v037n02ABEH002071}
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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, 128:23 (2003), 157

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S.S. Gabriyelyan, “Topologies on groups determined by sets of convergent sequences”, Journal of Pure and Applied Algebra, 2012

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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

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