Two observations on extremally disconnected topological groups

Yi Zhou; Jialiang He; Hang Zhang; Shuguo Zhang

doi:10.1016/j.topol.2025.109543

Two observations on extremally disconnected topological groups

Yi Zhou
, Jialiang He
, Hang Zhang
, Shuguo Zhang

Research output: Contribution to journal › Article › peer-review

Abstract

By modifying a method of Malykhin's, we construct two Hausdorff group topologies on the uncountable Boolean group ([ω₁]^<ω,▵) which are both nondiscrete and extremally disconnected. This is accomplished by working under ZFC plus Jensen's Diamond Principle. The first one has the property that all subgroups of the form [ω₁∖α]^<ω are dense and all countable subsets of [ω₁]^<ω are closed and discrete. This answers a question posed by C.A. Martínez-Ranero and U.A. Ramos-García [7, Question 3.4]. The second one has the property that some subgroup (endowed with the subspace topology) fails to be extremally disconnected. This answers a question posed by Arhangel'skii and Tkachenko [2, Open Problems 4.5.1].

Original language	English
Article number	109543
Journal	Topology and its Applications
Volume	373
DOIs	https://doi.org/10.1016/j.topol.2025.109543
State	Published - 1 Nov 2025
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2025 Elsevier B.V.

Keywords

Extremally disconnected
Subgroup
Topological group

Access to Document

10.1016/j.topol.2025.109543

Cite this

@article{38b21b6e6c684edc810dd029a603e1ba,

title = "Two observations on extremally disconnected topological groups",

abstract = "By modifying a method of Malykhin's, we construct two Hausdorff group topologies on the uncountable Boolean group ([ω1]<ω,▵) which are both nondiscrete and extremally disconnected. This is accomplished by working under ZFC plus Jensen's Diamond Principle. The first one has the property that all subgroups of the form [ω1∖α]<ω are dense and all countable subsets of [ω1]<ω are closed and discrete. This answers a question posed by C.A. Mart{\'i}nez-Ranero and U.A. Ramos-Garc{\'i}a [7, Question 3.4]. The second one has the property that some subgroup (endowed with the subspace topology) fails to be extremally disconnected. This answers a question posed by Arhangel'skii and Tkachenko [2, Open Problems 4.5.1].",

keywords = "Extremally disconnected, Subgroup, Topological group",

author = "Yi Zhou and Jialiang He and Hang Zhang and Shuguo Zhang",

note = "Publisher Copyright: {\textcopyright} 2025 Elsevier B.V.",

year = "2025",

month = nov,

day = "1",

doi = "10.1016/j.topol.2025.109543",

language = "אנגלית",

volume = "373",

journal = "Topology and its Applications",

issn = "0166-8641",

publisher = "Elsevier BV",

}

TY - JOUR

T1 - Two observations on extremally disconnected topological groups

AU - Zhou, Yi

AU - He, Jialiang

AU - Zhang, Hang

AU - Zhang, Shuguo

PY - 2025/11/1

Y1 - 2025/11/1

N2 - By modifying a method of Malykhin's, we construct two Hausdorff group topologies on the uncountable Boolean group ([ω1]<ω,▵) which are both nondiscrete and extremally disconnected. This is accomplished by working under ZFC plus Jensen's Diamond Principle. The first one has the property that all subgroups of the form [ω1∖α]<ω are dense and all countable subsets of [ω1]<ω are closed and discrete. This answers a question posed by C.A. Martínez-Ranero and U.A. Ramos-García [7, Question 3.4]. The second one has the property that some subgroup (endowed with the subspace topology) fails to be extremally disconnected. This answers a question posed by Arhangel'skii and Tkachenko [2, Open Problems 4.5.1].

AB - By modifying a method of Malykhin's, we construct two Hausdorff group topologies on the uncountable Boolean group ([ω1]<ω,▵) which are both nondiscrete and extremally disconnected. This is accomplished by working under ZFC plus Jensen's Diamond Principle. The first one has the property that all subgroups of the form [ω1∖α]<ω are dense and all countable subsets of [ω1]<ω are closed and discrete. This answers a question posed by C.A. Martínez-Ranero and U.A. Ramos-García [7, Question 3.4]. The second one has the property that some subgroup (endowed with the subspace topology) fails to be extremally disconnected. This answers a question posed by Arhangel'skii and Tkachenko [2, Open Problems 4.5.1].

KW - Extremally disconnected

KW - Subgroup

KW - Topological group

UR - https://www.scopus.com/pages/publications/105013361716

U2 - 10.1016/j.topol.2025.109543

DO - 10.1016/j.topol.2025.109543

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:105013361716

SN - 0166-8641

VL - 373

JO - Topology and its Applications

JF - Topology and its Applications

M1 - 109543

ER -

Two observations on extremally disconnected topological groups

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this