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. 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 ${\frak C} = \aleph_2$.
@article{119017,
author = {Artur Hideyuki Tomita},
title = {The existence of initially $\omega\_1$-compact group topologies on free Abelian groups is independent of ZFC},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {39},
year = {1998},
pages = {401-413},
zbl = {0938.54034},
mrnumber = {1651991},
language = {en},
url = {http://dml.mathdoc.fr/item/119017}
}
Tomita, Artur Hideyuki. The existence of initially $\omega_1$-compact group topologies on free Abelian groups is independent of ZFC. Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) pp. 401-413. http://gdmltest.u-ga.fr/item/119017/
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