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Topologies on groups determined by sequences. (English) Zbl 0977.54029

Mathematical Studies Monograph Series. 4. Lviv: VNTL Publishers. 111 p. (1999).
In the book under review various themes of the theory of topological groups and the theory of topological rings are treated from a unique viewpoint. The uniqueness is realized by using the following notions which are central in this work. A sequence \(\langle a_n\rangle_{n\in\omega}\) on an Abelian group \(A\) is said to be a \(T\)-sequence provided there exists a group topology \(\tau\) for which \(\langle a_n\rangle_{n\in\omega}\) converges to zero. A group topology on \(A\) is determined by the \(T\)-sequence \(\langle a_n\rangle_{n\in\omega}\) provided \(\tau\) is the strongest group topology in which \(\langle a_n\rangle_{n\in\omega}\) converges to zero.
It is proved that for every infinite Abelian group \(A\) there exists a nontrivial sequence in \(A\) (Theorem 2.1.7) and that a topological Abelian group whose topology is determined by a \(T\)-sequence is complete (Theorem 2.3.11). An example is constructed of a sequential group topology on the group \(\mathbb{Z}\) of integers which is not a Fréchet topology.
A topological space \(X\) is called maximal provided it has no isolated point but X has an isolated point in every stronger topology. It is proved that every maximal topological group contains a countable open Boolean subgroup (Theorem 2.5.13). Under CH an example of a countable Abelian maximally nondiscrete topological group is given (Example 2.5.2). It is proved that on every infinite Abelian group a complete group topology exists for which characters do not separate points (Theorem 2.6.5).
A class of topological groups is called a variety provided it is closed under taking of subgroups, continuous homomorphic images and topological products. It is proved that a variety is minimal if and only if it consists of all totally bounded groups satisfying the identity \(px=0\) for some prime number \(p\) (Theorem 2.6.4).
The authors extend some results from Abelian groups to noncommutative groups and rings. A new proof of the Markov criterion of topologizability of a countable group is given (Theorem 3.2.4) and a new proof of theorem of Arnautov asserting that every countable ring admits a nondiscrete ring topology (Theorem 3.4.8). A topological classification of countable \(k_{\omega}\)-groups is given (Corollary 4.3.10). It is proved that every Abelian group \(A\) furnished with the largest precompact topology has only trivial convergent sequences (Theorem 5.1.4). Every infinite totally bounded group contains a non-closed discrete subset (Corollary 5.2.3).
The authors pose in the book a lot of open questions. The greater part of results exposed in this book were published earlier in articles of the authors, the articles of E.K. van Douwen, V.I. Malykhin et al.

MSC:

54H11 Topological groups (topological aspects)
22A10 Analysis on general topological groups
22-02 Research exposition (monographs, survey articles) pertaining to topological groups
54-02 Research exposition (monographs, survey articles) pertaining to general topology

Keywords:

T-sequence; T-filter