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

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

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
\Bibitem{ZelPro90} \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 \mathnet{http://mi.mathnet.ru/izv1064} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1086087} \zmath{https://zbmath.org/?q=an:0728.22003} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1991IzMat..37..445Z} \transl \jour Math. USSR-Izv. \yr 1991 \vol 37 \issue 2 \pages 445--460 \crossref{https://doi.org/10.1070/IM1991v037n02ABEH002071} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
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