Countably compact group topologies on arbitrarily large free Abelian groups

Matheus K. Bellini; Klaas Pieter Hart; Vinicius O. Rodrigues; Artur H. Tomita

doi:10.1016/j.topol.2023.108538

Countably compact group topologies on arbitrarily large free Abelian groups

Matheus K. Bellini, Klaas Pieter Hart, Vinicius O. Rodrigues^*, Artur H. Tomita

^*Corresponding author for this work

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Abstract

We prove that if there are c incomparable selective ultrafilters then, for every infinite cardinal κ such that κ^ω=κ, there exists a group topology on the free Abelian group of cardinality κ without nontrivial convergent sequences and such that every finite power is countably compact. In particular, there are arbitrarily large countably compact groups. This answers a 1992 question of D. Dikranjan and D. Shakhmatov.

Original language	English
Article number	108538
Number of pages	23
Journal	Topology and its Applications
Volume	333
DOIs	https://doi.org/10.1016/j.topol.2023.108538
Publication status	Published - 2023

Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

Keywords

Countable compactness
Free Abelian group
Selective ultrafilter
Topological group
Wallace's problem

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AU - Hart, Klaas Pieter

AU - Rodrigues, Vinicius O.

AU - Tomita, Artur H.

N1 - Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

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N2 - We prove that if there are c incomparable selective ultrafilters then, for every infinite cardinal κ such that κω=κ, there exists a group topology on the free Abelian group of cardinality κ without nontrivial convergent sequences and such that every finite power is countably compact. In particular, there are arbitrarily large countably compact groups. This answers a 1992 question of D. Dikranjan and D. Shakhmatov.

AB - We prove that if there are c incomparable selective ultrafilters then, for every infinite cardinal κ such that κω=κ, there exists a group topology on the free Abelian group of cardinality κ without nontrivial convergent sequences and such that every finite power is countably compact. In particular, there are arbitrarily large countably compact groups. This answers a 1992 question of D. Dikranjan and D. Shakhmatov.

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KW - Free Abelian group

KW - Selective ultrafilter

KW - Topological group

KW - Wallace's problem

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VL - 333

JO - Topology and its Applications

JF - Topology and its Applications

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ER -

Countably compact group topologies on arbitrarily large free Abelian groups

Abstract

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Keywords

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