A group topology on the free abelian group of cardinality \(\mathfrak{c}\) that makes its finite powers countably compact
type: Item
instance of: scholarly article
title: A group topology on the free abelian group of cardinality \(\mathfrak{c}\) that makes its finite powers countably compact
zbMATH Open document ID: 1334.54054
author: Artur Hideyuki Tomita
published in: Topology and its Applications
publication date: 2015-12-18T00:00:00Z
review text: \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: Han…
reviewed by: Paolo Lipparini
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 20K20
zbMATH DE Number: 6522045
zbMATH Keywords: countably compact
zbMATH Keywords: topological group
zbMATH Keywords: free abelian group
zbMATH Keywords: selective ultrafilter
zbMATH Keywords: finite product
zbMATH Keywords: integer stack
MaRDI profile type: Publication
full work available at URL: j.topol.2015.05.060
OpenAlex ID: W2232964727
cites work: Sacks forcing and the total failure of Martin's axiom
cites work: A group topology on the free abelian group of cardinality c that makes its square countably compact
cites work: Q3704110
cites work: Abelian torsion groups with a countably compact group topology
cites work: Comment on the real line
cites work: Q4424991
cites work: Countably compact topological group topologies on free Abelian groups from selective ultrafilters
cites work: Q3199667
cites work: The Wallace Problem: A Counterexample from MaCountableandp-Compactness
cites work: Q4955963
cites work: A solution to Comfort's question on the countable compactness of powers of a topological group
cites work: Ultraproducts, \(p\)-limits and antichains on the Comfort group order
DOI: 10.1016/J.TOPOL.2015.05.060
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description: scientific article; zbMATH DE number 6522045