Answer to Raczkowski's questions on convergent sequences of integers
Denote by \(T\) the multiplicative group of complex numbers with modulus 1 equipped with the topology inherited from the complex plane and equipped with its usual Lebesgue measure. It is well known that if \(Z\) denotes the group of integers, the set Hom\((Z,T)\) of homomorphisms from \(Z\) into \(T\) becomes a group topologically isomorphic to \(T\) with operation defined pointwise and with the topology of pointwise convergence. Under this identification it can be proved that given a (dense) subgroup \(H\) of \(T\), the weakest topology \(\tau_H\) on \(Z\) that makes the elements of \(H\) continuous is a (Hausdorff) precompact group topology. That every such topology is given as some \(\tau_H\) follows from a theorem of \textit{W. W. Comfort} and \textit{K. A. Ross} [Fundam. Math. 55, 284-291 (1964; Zbl 0138.02905)]. Thus it is natural to ask how properties of the subgroups of \(T\) influence the properties of the precompact topologies on \(Z\). For example, every \(\tau_T\)-compact subset of \(Z\) must be finite [\textit{H. Leptin}, Abh. Math. Sem. Univ. Hamburg 19, 244-263 (1955; Zbl 0065.01501); \textit{I. Glicksberg}, Can. J. Math. 14, 269-276 (1962; Zbl 0109.02001)], thus in particular there are no non-trivial \(\tau_T\)-convergent sequences [\textit{P. Flor}, Math. Scand. 23, 169-170 (1968; Zbl 0182.36003)]. \textit{W. W. Comfort, T. S. Wu} and the reviewer [Fundam. Math. 143, 119-136 (1993; Zbl 0812.22001); Correction, same journal 152, 97-98] proved that given a sequence \(\sigma:=\langle x_n \rangle\) in \(Z\), the group \(A_\sigma:=\{t \in T: t(x_n) \rightarrow 1\}\) must have measure 0, hence if \(H\) is a non-measurable subgroup of \(T\), then there are no non-trivial \(\tau_H\)-convergent sequences. \textit{S. U. Raczkowski} [Topology Appl. 121, 63-74 (2002; Zbl 1007.22003)] generalized these results as follows: (i) There are \(2^c (c:=2^{\aleph_0})\) non-measurable subgroups of \(T\), hence \(2^c\) (Hausdorff) precompact group topologies on \(Z\) without non-trivial convergent sequences. (ii) There are \(2^c\) precompact group topologies on \(Z\) with non-trivial convergent sequences. (iii) If \(\langle x_n \rangle\) is a sequence in \(Z\) such that \({x_{n+1}}/{x_n} \geq n+1\), then there is a precompact group topology on \(Z\) of weight \(c\) in which \(\langle x_n \rangle\) converges. In the paper under review, the authors deepen our understanding into this line of research. Answering questions of Raczkowski, they prove: (a) (iii) remains true if one requires only \({x_{n+1}}/{x_n} \to \infty\); their proof closely resembles Raczkowski's proof of (iii). (b) If \({x_{n+1}}/{x_n}\) is bounded, then the only precompact group topologies on \(Z\) in which \(\langle x_n \rangle\) converges must be metrizable (hence of weight \(\aleph_0\)). (c) With respect to (i), and using Martin's Axiom, there are \(2^c\)-many subgroups \(H\) of \(T\) of measure zero, such that the topologies \(\tau_H\) have no non-trivial convergent sequences. Their proof is a beautiful illustration of Oxtoby's remark that there is nothing paradoxical about a set being small in one sense and big in another. Reviewer's remark: (1) As the authors point out at the end of the paper, \textit{Hart} and \textit{Kunen} [Limits in function spaces, work in progress] have constructed in ZFC a subgroup \(H\) of \(T\) of measure zero, such that the topology \(\tau_H\) has no non-trivial convergent sequences. (2) Most of the above questions can be raised for arbitrary groups \(G\). In work in progress, Comfort, Raczkowski and the reviewer have proved that every infinite Abelian group \(G\) accepts two families \(\mathcal A\) and \(\mathcal B\) of the maximum number of (Hausdorff) precompact group topologies, namely \(2^{2^{ G }}\), such that (i) in every member of \(\mathcal A\) no non-trivial convergent sequences exist, (ii) all members of \(\mathcal B\) share a common convergent non-trivial sequence.
scientific article; zbMATH DE number 1961619
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