A group topology on the free abelian group of cardinality c that makes its finite powers countably compact

摘要

We introduce a new approach to the construction of group topologies on free abelian groups which are productively countably compact. We construct an example as in the title (using c incomparable selective ultrafilters) that closes the gap between results of Tkachenko (1990) [9] that showed the free abelian group of cardinality c admits a countably compact group topology using CH and Tomita (1998) [11] that showed a free abelian group cannot be endowed with a group topology that makes its infinite countable power countably compact. This is also connected to Comfort's question 477 in the Open Problems in Topology on countable compactness of powers. In fact, it is the first torsion free example of a topological group whose least power that fails to be countably compact is omega. (C) 2015 Elsevier B.V. All rights reserved.