A van Douwen-like ZFC theorem for small powers of countably compact groups without non-trivial convergent sequences

Abstract

We show that if κ≤ω and there exists a group topology without non-trivial convergent sequences on an Abelian group H such that Hn is countably compact for each n<κ then there exists a topological group G such that Gn is countably compact for each n<κ and Gκ is not countably compact. If in addition H is torsion, then the result above holds for κ=ω1. Combining with other results in the literature, we show that:a) Assuming c incomparable selective ultrafilters, for each n∈ω, there exists a group topology on the free Abelian group G such that Gn is countably compact and Gn+1 is not countably compact. (It was already know for ω).b) If κ∈ω∪{ω}∪{ω1}, there exists in ZFC a topological group G such that Gγ is countably compact for each cardinal γ<κ and Gκ is not countably compact.