The existence of initially ω 1 -compact group topologies on free Abelian groups is independent of ZFC

Artur Hideyuki Tomita

Commentationes Mathematicae Universitatis Carolinae (1998)

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

Abstract

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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 ℭ admits a Hausdorff countably compact group topology. We show that no Hausdorff group topology on a free Abelian group makes its ω -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 ω 1 -compact group topology. We also show that the existence of such a group topology is independent of ℭ = ℵ 2 .

How to cite

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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 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 -

References

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