This article is cited in 37 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.

UDC: 512.546

MSC: Primary 20K45, 22A30; Secondary 54A20, 54B99

Received: 03.11.1988

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

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