Geodesic
The existence of initially $\omega_1$-compact group topologies on free Abelian groups is independent of ZFC
Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) no. 2, pp. 401-413 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 ${\frak C} = \aleph_2$.
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 ${\frak C} = \aleph_2$.
Classification : 22B99, 54D30, 54H11
Keywords: free Abelian group; countable compactness; products; initially $\omega_1$-compact; Martin's Axiom 
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     author = {Tomita, Artur Hideyuki},
     title = {The existence of initially $\omega_1$-compact group topologies on free {Abelian} groups is independent of {ZFC}},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {401--413},
     year = {1998},
     volume = {39},
     number = {2},
     mrnumber = {1651991},
     zbl = {0938.54034},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1998_39_2_a16/}
}
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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) no. 2, pp. 401-413. http://geodesic.mathdoc.fr/item/CMUC_1998_39_2_a16/