# The existence of initially ${\omega}_{1}$-compact group topologies on free Abelian groups is independent of ZFC

Commentationes Mathematicae Universitatis Carolinae (1998)

- Volume: 39, Issue: 2, page 401-413
- ISSN: 0010-2628

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topTomita, Artur Hideyuki. "The existence of initially $\omega _1$-compact group topologies on free Abelian groups is independent of ZFC." Commentationes Mathematicae Universitatis Carolinae 39.2 (1998): 401-413. <http://eudml.org/doc/248277>.

@article{Tomita1998,

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 $\{\mathfrak \{C\}\}$ 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 $\{\mathfrak \{C\}\} = \aleph _2$.},

author = {Tomita, Artur Hideyuki},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {free Abelian group; countable compactness; products; initially $\omega _1$-compact; Martin’s Axiom; free Abelian group; countable compactness; products; Martin's Axiom},

language = {eng},

number = {2},

pages = {401-413},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {The existence of initially $\omega _1$-compact group topologies on free Abelian groups is independent of ZFC},

url = {http://eudml.org/doc/248277},

volume = {39},

year = {1998},

}

TY - JOUR

AU - Tomita, Artur Hideyuki

TI - The existence of initially $\omega _1$-compact group topologies on free Abelian groups is independent of ZFC

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 1998

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 39

IS - 2

SP - 401

EP - 413

AB - 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 ${\mathfrak {C}}$ 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 ${\mathfrak {C}} = \aleph _2$.

LA - eng

KW - free Abelian group; countable compactness; products; initially $\omega _1$-compact; Martin’s Axiom; free Abelian group; countable compactness; products; Martin's Axiom

UR - http://eudml.org/doc/248277

ER -

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