Abstract:It was known that free Abelian groups do not admit a Hausdorff compact group topology. Tkachenko showed in 1990 that, under CH, a free Abelian group of size ${\frak c}$ admits a Hausdorff countably compact group topology. \par We show that no Hausdorff group topology on a free Abelian group makes its $\omega $-th power countably compact. In particular, a free Abelian group does not admit a Hausdorff $p$-compact nor a sequentially compact group topology. Under CH, we show that a free Abelian group does not admit a Hausdorff initially $\omega _1$-compact group topology. We also show that the existence of such a group topology is independent of ${\frak c}= \aleph _2$.
Keywords: free Abelian group, countable compactness, products, initially $\omega _1$-compact, Martin's Axiom
AMS Subject Classification: 54H11, 22B99, 54D30