A group topology on the free abelian group of cardinality \(\mathfrak{c}\) that makes its finite powers countably compact
\textit{P. R. Halmos} [Bull. Am. Math. Soc. 50, 877--878 (1944; Zbl 0061.04404)] raised interest in the problem of characterizing those abelian groups which can be made into compact (Hausdorff) topological groups. In the case of infinite free abelian groups, the answer is well-known to be negative; however, there is interest in the problem of endowing them with a topology satisfying at least some weaker covering property. In this respect, under the Continuum Hypothesis, \textit{M. G. Tkachenko} in [Sov. Math. 34, No. 5, 79--86 (1990; Zbl 0714.22001); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1990, No. 5(336), 68--75 (1990)] showed, amongst others, that the free abelian group of size \(\mathfrak c\) (the cardinality of the continuum) admits a countably compact group topology. Various strengthenings of Tkachenko's result have been obtained by A. C. Boero, P. B. Koszmider, R. E. Madariaga-Garcia, A. H. Tomita and S. Watson (see the reference list in the paper under review). In the negative direction, [\textit{A. H. Tomita}, Commentat. Math. Univ. Carol. 39, No. 2, 401--413 (1998; Zbl 0938.54034)] showed that, for every topology on an infinite free abelian group, its \( \omega^{\text{th}}\)-power is not countably compact. The present paper fills the gap between the above results. Assuming the existence of \(\mathfrak c\) selective ultrafilters on \( \omega\), the author topologizes the free abelian group of cardinality \(\mathfrak c\) in such a way that all of its finite powers are countably compact. Remarkably, this furnishes the example of a torsion free topological group whose least power that fails to be countably compact is \( \omega\). Recall that an ultrafilter is selective (also called Ramsey) if it is minimal in the Rudin-Keisler preorder. The existence of even just one selective ultrafilter cannot be proved in ZFC, however, the existence of many selective ultrafilters follows from relatively weak assumptions. See for example [\textit{A. Blass}, in: Handbook of set theory. In 3 volumes. Dordrecht: Springer. 395--489 (2010; Zbl 1198.03058)], [\textit{M. Foreman} (ed.) and \textit{A. Kanamori} (ed.), Handbook of set theory. In 3 volumes. Dordrecht: Springer (2010; Zbl 1197.03001)], [\textit{R. M. Canjar}, Proc. Am. Math. Soc. 110, No. 1, 233--241 (1990; Zbl 0715.03018)]. The proof of the main result in the paper under review is too long to be summarized here in detail. The author carefully defines and constructs certain structures called stacks; roughly, they are special finite sets of sequences. Not every sequence in a finite power is a stack, but a topology is constructed in such a way that there is a correspondence between accumulation points of an arbitrary sequence and ultrafilter convergence of some stack. Once a sufficient number of stacks are constructed, each stack \(\mathcal U\)-converging for an appropriate ultrafilter in the suitable topology, then every sequence has an accumulation point, thus countable compactness follows. Some interesting questions are raised at the end of the paper.
scientific article; zbMATH DE number 6522045