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Discrete subsets in topological groups and countable extremally disconnected groups


Authors: Evgenii Reznichenko and Ol’ga Sipacheva
Journal: Proc. Amer. Math. Soc. 149 (2021), 2655-2668
MSC (2020): Primary 54G05, 54H11, 03E35, 22A05
DOI: https://doi.org/10.1090/proc/13992
Published electronically: March 16, 2021
MathSciNet review: 4246814
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Abstract: In 1967 Arhangel’skii posed the problem of the existence in ZFC of a nondiscrete extremally disconnected topological group. The general case is still open, but we solve Arhangel’skii’s problem for the class of countable groups. Namely, we prove that the existence of a countable nondiscrete extremally disconnected group implies the existence of a rapid ultrafilter; hence, such a group cannot be constructed in ZFC. We also prove that any countable topological group in which the filter of neighborhoods of the identity element is not rapid contains a discrete set with precisely one limit point, which gives a negative answer to Protasov’s question on the existence in ZFC of a countable nondiscrete group in which all discrete subsets are closed.


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Additional Information

Evgenii Reznichenko
Affiliation: Department of General Topology and Geometry, Mechanics and Mathematics Faculty, M. V. Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 199991 Russia
MR Author ID: 245922
Email: erezn@inbox.ru

Ol’ga Sipacheva
Affiliation: Department of General Topology and Geometry, Mechanics and Mathematics Faculty, M. V. Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 199991 Russia
MR Author ID: 246154
Email: ovsipa@gmail.com

Received by editor(s): December 23, 2016
Received by editor(s) in revised form: October 16, 2017
Published electronically: March 16, 2021
Additional Notes: This work was supported by the Russian Foundation for Basic Research (project no. 15-01-05369).
Communicated by: Mirna Dz̆amonja
Article copyright: © Copyright 2021 American Mathematical Society