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axioms Topological Groups Advances, Surveys, and Open Questions Edited by Sidney A. Morris Printed Edition of the Special Issue Published in Axioms www.mdpi.com/journal/axioms Topological Groups Topological Groups Advances, Surveys, and Open Questions Special Issue Editor Sidney A. Morris MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Special Issue Editor Sidney A. Morris La Trobe University and Federation University Australia Australia Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Axioms (ISSN 2075-1680) from 2017 to 2019 (available at: https://www.mdpi.com/journal/axioms/special issues/topological groups). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03897-644-8 (Pbk) ISBN 978-3-03897-645-5 (PDF) c 2019 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Special Issue Editor ...................................... vii Preface to ”Topological Groups” ..................................... ix Arkady G. Leiderman and Sidney A. Morris Separability of Topological Groups: A Survey with Open Problems Reprinted from: Axioms 2019, 8, 3, doi:10.3390/axioms8010003 .................... 1 Taras Banakh Categorically Closed Topological Groups Reprinted from: Axioms 2017, 6, 23, doi:10.3390/axioms6030023 ................... 19 Igor V. Protasov Selective Survey on Spaces of Closed Subgroups of Topological Groups Reprinted from: Axioms 2018, 7, 75, doi:10.3390/axioms7040075 ................... 52 Gianluca Paolini and Saharon Shelah No Uncountable Polish Group Can be a Right-Angled Artin Group Reprinted from: Axioms 2017, 6, 13, doi:10.3390/axioms6020013 ................... 64 George A. Willis Computing the Scale of an Endomorphism of a totally Disconnected Locally Compact Group Reprinted from: Axioms 2017, 6, 27, doi:10.3390/axioms6040027 ................... 68 Stefan Wagner Extending Characters of Fixed Point Algebras Reprinted from: Axioms 2018, 7, 79, doi:10.3390/axioms7040079 ................... 85 Michael Megrelishvili A Note on the Topological Group c0 Reprinted from: Axioms 2018, 7, 77, doi:10.3390/axioms7040077 ................... 89 Ol’ga V. Sipacheva Large Sets in Boolean and Non-Boolean Groups and Topology Reprinted from: Axioms 2017, 6, 28, doi:10.3390/axioms6040028 ................... 95 Dmitri Shakhmatov and V´ıctor Hugo Yanez ˜ Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets Reprinted from: Axioms 2018, 7, 86, doi:10.3390/axioms7040086 ...................113 John R. Martin (L)-Semigroup Sums Reprinted from: Axioms 2019, 8, 1, doi:10.3390/axioms8010001 ....................136 Igor Protasov Varieties of Coarse Spaces Reprinted from: Axioms 2018, 7, 32, doi:10.3390/axioms7020032 ...................141 v About the Special Issue Editor Sidney Morris, Adjunct Professor, La Trobe University and Emeritus Professor, Federation University, Australia. He received BSc (Hons) from University of Queensland in 1969 and Ph.D. from Flinders University in 1970. He held positions of Professor, Department Head, Dean, Vice-President, CAO, and CEO. He has been employed by 13 universities in Australia, UK, USA, and Israel. He was Editor of the Bulletin of Australian Mathematical Society and Journal of Research and Practice in Information Technology, and founding editor of Australian Mathematical Society Lecture Series and Journal of Group Theory. He served on the Council of the Australian Mathematical Society for 25 years and as Vice-President, and received the Lester R. Ford Award from Mathematical Association of America. He has published 160 refereed journal papers and four books for undergraduates, graduates, and researchers, plus an online book, translated into eight languages. In 2016 he edited the book “Topological Groups: Yesterday, Today, Tomorrow”. vii Preface to ”Topological Groups” In 1900 David Hilbert presented an address at the International Congress of Mathematicians in Paris and formulated 23 problems that influenced much of the research in the 20th century. The fifth of these problems asked whether every locally euclidean group admits a Lie group structure. This motivated a great amount of research on locally compact groups, culminating in the 1950s with the work of Gleason, Iwasawa, Montgomery, Yamabe, and Zippin, which gave a positive answer to Hilbert’s question and developed much structure theory of locally compact groups to boot. In the 1940s, the work on the free topological groups of Markov and Graev expanded the study of topological groups in a serious way to non-locally compact groups. In the early 21st century, pro-Lie groups were introduced as a natural and well-behaved extension of the notions of connected locally compact groups and compact groups focusing on their Lie theory. The Special Issue of Axioms called “Topological Groups: Yesterday, Today, Tomorrow” was published as book in 2016 and has had a tremendous reception. It addressed some of the significant research of this 115-year period. A wonderful feature of the book was the inclusion of surveys and a large number of open questions. This second volume on topological groups, called “Topological Groups: Advances, Surveys, and Open Questions”, contains recent articles by some of the best scholars in the world on topological groups including Taras Banakh, Michael Megrelishvili, George A. Willis, Dmitri Shakhmatov, and O’lga V. Sipacheva, as well as a paper by the renowned scholar Saharon Shelah. A feature of the first volume was the surveys, and we continue that tradition in this second volume with three new surveys. These surveys are of interest, not only to the expert but also to those who are less experienced. Particularly exciting to active researchers, especially young researchers, is the inclusion of over three dozen open questions. This volume consists of 11 papers containing many new and interesting results and examples across the spectrum of topological group theory and related topics. The first paper in this book is “Separability of Topological Groups: A Survey with Open Problems” by Arkady Leiderman and Sidney A. Morris. In recent years, Leiderman has been a leader in the study of the separability of topological groups. This paper alone states 20 open questions and puts them in context. The second paper, “Categorically Closed Topological Groups”, is also a survey, this time by Taras Banakh. This paper surveys existing and new results on topological groups that are C-closed for various categories C of topologized semigroups. In particular, it analyzes solutions to a general problem consisting of 45 subproblems. The third paper is “Selective Survey on Spaces of Closed Subgroups of Topological Groups” by Igor V. Protasov. This paper surveys the Chabauty topology and the Vietoris topology on the set of all closed subgroups of a topological group, and in the author’s words “...is my subjective look at this area”. The fourth paper, “No Uncountable Polish Group Can be a Right-Angled Artin Group”, is by Gianluca Paolini and Saharon Shelah. Generalizing results on free groups and free abelian groups, the authors prove that the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. The fifth paper is “Computing the Scale of an Endomorphism of a Totally Disconnected Locally Compact Group” by George A. Willis. While a substantial amount of information about the structure of connected locally compact groups, even almost connected locally compact groups, has been known for over half a century, little of substance was known about totally disconnected locally compact ix groups before the deep contributions of Willis. In this paper, the scale of an endomorphism of a totally disconnected locally compact group is defined and “the information required to compute the scale is reviewed from the perspective of the, as yet incomplete, general theory of totally disconnected, locally compact groups”. The sixth paper is “Extending Characters of Fixed Point Algebras” by Stefan Wagner. Given a dynamical system (A, G, a) with a complete commutative continuous inverse algebra A and a compact group G, it is shown that each character of the corresponding fixed point algebra can be extended to a character of A. The seventh paper is “A Note on the Topological Group c 0” by Michael Megrelishvili. It is shown that Gromov’s compactification of c 0 is not a semigroup compactification. The paper also contextualizes three open questions. The eighth paper is “Large Sets in Boolean and Non-Boolean Groups and Topology” by Ol’ga V. Sipacheva. “Various notions of large sets in groups, including the classical notions of thick, syndetic, and piecewise syndetic sets, and the new notion of vast sets in groups, are studied, with an emphasis on the interplay between such sets in Boolean groups. Natural topologies closely related to vast sets are considered; as a byproduct, interesting relations between vast sets and
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