Countably compact topological group topologies on free Abelian groups from selective ultrafilters

Roberto E. Madariaga-Garcia, Artur Hideyuki Tomita
2007 Topology and its Applications  
Tkachenko showed in 1990 the existence of a countably compact group topology on the free Abelian group of size c using CH. Koszmider, Tomita and Watson showed in 2000 the existence of a countably compact group topology on the free Abelian group of size 2 c using a forcing model in which CH holds. Wallace's question from 1955, asks whether every both-sided cancellative countably compact semigroup is a topological group. A counterexample to Wallace's question has been called a Wallace semigroup.
more » ... n 1996, Robbie and Svetlichny constructed a Wallace semigroup under CH. In the same year, Tomita constructed a Wallace semigroup from MA countable . In this note, we show that the examples of Tkachenko, Robbie and Svetlichny, and Koszmider, Tomita and Watson can be obtained using a family of selective ultrafilters. As a corollary, the constructions presented here are compatible with the total failure of Martin's Axiom.
doi:10.1016/j.topol.2006.03.028 fatcat:ueqte3swjfcdxge5asrdlugtpy