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 $\mathbb{Z}^\omega$ 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.