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Countably compact groups without non-trivial convergent sequences. (English) Zbl 1482.22002

It has long been known that the product of two countably compact Hausdorff topological spaces is not pseudocompact (let alone countably compact). However, Comfort and Ross proved that arbitrary products of pseudocompact topological groups are pseudocompact, and this led Comfort to pose the question of whether the product of two countably compact groups is necessarily countably compact. In 1980 van Douwen gave a consistently negative answer by using Martin’s axiom to construct an infinite countably compact Hausdorff group with no nontrivial convergent sequences. In the present paper the authors produce such a topological group using \(\mathsf{ZFC}\) alone. The group is a topological subgroup of \(2^{\mathfrak{c}}\), obtained by starting with \([\omega]^{<\omega}\) (the finite subsets of \(\omega\) under symmetric difference), taking the Bohr topology (i.e., the smallest topology making all homomorphisms to \(\mathbb{Z}_2\) continuous) on \([\omega]^{<\omega}\), and using a carefully-constructed \(\mathfrak{c}\)-indexed family of free ultrafilters on \(\omega\) to “lift” that topology to \([\mathfrak{c}]^{<\omega}\). The group constructed this way is Boolean (i.e., all elements have order 2), and the authors ask\(-\)in a list of interesting open problems\(-\)whether a \(\mathsf{ZFC}\)-example of an infinite countably compact group with no nontrivial convergent sequences has to be a torsion group.

MSC:

22A05 Structure of general topological groups
03C20 Ultraproducts and related constructions
03E05 Other combinatorial set theory
54H11 Topological groups (topological aspects)
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