Provisional solution to a Comfort–van Mill problem

AbstractWe prove under the assumption of Martin's Axiom that an abstract Abelian group G of non-measurable cardinality is the intersection of countably compact subgroups of its Bohr compactification bG. This result is used to show that weakly free countably compact topological groups do not exist, thus answering a question posed by Comfort and van Mill in 1988. In fact, we show under MA that a free (P,CC)-group over a space X exists iff X is empty, where P and CC are the classes of pseudocompact and countably compact topological groups, respectively. On the other hand, we prove the existence of a weakly free (P,CC)-group over an arbitrary space X and show that our construction of such a group is functorial. Similar results remain valid in the Abelian case