The Wallace problem and countably compact torsion-free Abelian groups in ZFC

Abstract

We prove in ZFC that every torsion-free Abelian group of cardinality $\mathfrak c$ admits a Hausdorff countably compact group topology without nontrivial convergent sequences. In particular, this applies to the free Abelian group $\mathbb{Z}^{(\mathfrak c)}$, the Baer-Specker group $\mathbb{Z}^ω$ and $\mathbb{Q}^{(\mathfrak c)}$. For the topology constructed on $\mathbb{Z}^{(\mathfrak c)}$, the coordinatewise nonnegative cone is countably compact in the subspace topology. Consequently, there exists in ZFC a commutative Tychonoff countably compact topological semigroup which has two-sided cancellation but is not a group, giving a negative answer to Wallace's question. Combined with earlier results, the main theorem also yields in ZFC a Tychonoff countably compact topological semigroup containing a copy of the bicyclic semigroup and a functionally Hausdorff countably compact paratopological group that is not a topological group.

Disclosure

“Acknowledgments The second author was supported by the São Paulo Research Foundation (FAPESP), grant no. 2025/07302-0. Declaration of generative AI and AI-assisted technologies in the manuscript preparation process OpenAI Codex, powered by the GPT-5.6 Sol model, was used extensively as a generative research and writing tool. In an iterative process directed by the authors, the model carried out most of the exploratory proof devel- opment and generated most”

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2026-08-23T05:22:10.416866Z

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