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Topological Groups: Yesterday, Today, Tomorrow axioms Topological Groups: Yesterday, Today, Tomorrow Edited by Sidney A. Morris Printed Edition of the Special Issue Published in Axioms www.mdpi.com/journal/axioms Sidney A. Morris (Ed.) Topological Groups: Yesterday, Today, Tomorrow This book is a reprint of the Special Issue that appeared in the online, open access journal, Axioms (ISSN 2075-1680) from 2015–2016, available at: http://www.mdpi.com/journal/axioms/special_issues/topological_groups_yesterd ay_today_tomorrow Guest Editor Sidney A. Morris School of Science, IT, and Engineering Federation University Australia Australia Editorial Office MDPI AG St. Alban-Anlage 66 Basel, Switzerland Publisher Shu-Kun Lin Managing Editor Qiang Liu 1. Edition 2016 MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade ISBN 978-3-03842-268-6 (Hbk) ISBN 978-3-03842-269-3 (electronic) Articles in this volume are Open Access and distributed under the Creative Commons Attribution license (CC BY), which allows users to download, copy and build upon published articles even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book taken as a whole is © 2016 MDPI, Basel, Switzerland, distributed under the terms and conditions of the Creative Commons by Attribution (CC BY-NC-ND) license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Table of Contents List of Contributors ............................................................................................................ V About the Guest Editor................................................................................................... VII Sidney A. Morris An Overview of Topological Groups: Yesterday, Today, Tomorrow Reprinted from: Axioms 2016, 5(2), 11 http://www.mdpi.com/2075-1680/5/2/11 .......................................................................... 1 Karl H. Hofmann and Sidney A. Morris Pro-Lie Groups: A Survey with Open Problems Reprinted from: Axioms 2015, 4(3), 294–312 http://www.mdpi.com/2075-1680/4/3/294 ...................................................................... 11 W. W. Comfort and Dieter Remus Non-Abelian Pseudocompact Groups Reprinted from: Axioms 2016, 5(1), 2 http://www.mdpi.com/2075-1680/5/1/2 .......................................................................... 32 Morris W. Hirsch Fixed Points of Local Actions of Lie Groups on Real and Complex 2-Manifolds Reprinted from: Axioms 2015, 4(3), 313–320 http://www.mdpi.com/2075-1680/4/3/313 ...................................................................... 56 Ol’ga Sipacheva Free Boolean Topological Groups Reprinted from: Axioms 2015, 4(4), 492–517 http://www.mdpi.com/2075-1680/4/4/492 ...................................................................... 64 III Mikhail Tkachenko Lindelöf Σ-Spaces and R-Factorizable Paratopological Groups Reprinted from: Axioms 2015, 4(3), 254–267 http://www.mdpi.com/2075-1680/4/3/254 ...................................................................... 95 Saak Gabriyelyan On T-Characterized Subgroups of Compact Abelian Groups Reprinted from: Axioms 2015, 4(2), 194–212 http://www.mdpi.com/2075-1680/4/2/194 .................................................................... 112 Dikran Dikranjan, Anna Giordano Bruno and Daniele Impieri Characterized Subgroups of Topological Abelian Groups Reprinted from: Axioms 2015, 4(4), 459–491 http://www.mdpi.com/2075-1680/4/4/459 .................................................................... 134 Lydia Außenhofer, Dikran Dikranjan and Elena Martín-Peinador Locally Quasi-Convex Compatible Topologies on a Topological Group Reprinted from: Axioms 2015, 4(4), 436–458 http://www.mdpi.com/2075-1680/4/4/436 .................................................................... 171 Viorel Nitica and Andrei Török Open and Dense Topological Transitivity of Extensions by Non-Compact Fiber of Hyperbolic Systems: A Review Reprinted from: Axioms 2015, 4(1), 84–101 http://www.mdpi.com/2075-1680/4/1/84 ...................................................................... 197 IV List of Contributors Lydia Außenhofer Faculty of Computer Science and Mathematics, Universität Passau, Innstr. 33, Passau D-94032, Germany. W. W. Comfort Department of Mathematics, Wesleyan University, Middletown, CT 06457, USA. Dikran Dikranjan Dipartimento di Matematica e Informatica, Università degli Studi di Udine, Via delle Scienze 206, Udine 33100, Italy. Saak Gabriyelyan Department of Mathematics, Ben-Gurion University of the Negev, P.O. 653, Beer-Sheva 8410501, Israel. Anna Giordano Bruno Dipartimento di Matematica e Informatica, Università degli Studi di Udine, Via delle Scienze 206, Udine 33100, Italy. Morris W. Hirsch Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA; Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA. Karl H. Hofmann Department of Mathematics, Tulane University, New Orleans, LA 70118, USA; Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, Darmstadt 64289, Germany. Daniele Impieri Dipartimento di Matematica e Informatica, Università degli Studi di Udine, Via delle Scienze 206, Udine 33100, Italy. Elena Martín-Peinador Instituto de Matemática Interdisciplinar y Departamento de Geometría y Topología, Universidad Complutense de Madrid, Madrid 28040, Spain. Sidney A. Morris Department of Mathematics and Statistics, School of Engineering and Mathematical Sciences, La Trobe University, Bundoora, Victoria 3086, Australia; Faculty of Science and Technology, Federation University Australia, Victoria 3353, Australia. Viorel Nitica Institute of Mathematics of the Romanian Academy, P.O. Box 1–764, RO-70700 Bucharest, Romania; Department of Mathematics, West Chester University, West Chester, PA 19383, USA. Dieter Remus Institut für Mathematik, Universität Paderborn, Warburger Str. 100, Paderborn D-33095, Germany. Ol’ga Sipacheva Department of General Topology and Geometry, Lomonosov Moscow State University, Leninskie Gory 1, Moscow 119991, Russia. V Mikhail Tkachenko Universidad Autónoma Metropolitana, Iztapalapa Campus, Av. San Rafael Atlixco 187, Col. Vicentina, C.P. 09340 Iztapalapa, Mexico City, Mexico. Andrei Török Department of Mathematics, University of Houston, 651 PGH, Houston, TX 77204-3008, USA; Institute of Mathematics of the Romanian Academy, P.O. Box 1–764, RO-70700 Bucharest, Romania. VI About the Guest Editor Sidney Morris, Emeritus Professor of Federation University, Australia and Adjunct Professor of La Trobe University, obtained his BSc(Hons) from the University of Queensland in 1969 and his PhD from Flinders University in 1970. He has been (full) Professor, Department Head, Dean, Vice-President, CAO and CEO. He was employed by nine Australian universities and universities in Israel, the UK, and USA. He was the Editor of the Bulletin of the Australian Mathematical Society and the Journal of Research and Practice in Information Technology, a founding Editor of the Journal of Group Theory, and the Australian Mathematical Lecture Series. He is an Editor of the Gazette of the Australian Mathematical Society and on the Editorial Board of Axioms. He received the Lester R. Ford Award from the Mathematical Association of America. He has published 160 journal papers mostly on topological group theory, and four books, plus an online book which has been translated into eight languages. VII Preface to "An Overview of Topological Groups: Yesterday, Today, Tomorrow" Sidney A. Morris Reprinted from Axioms. Cite as: Morris, S.A. An Overview of Topological Groups: Yesterday, Today, Tomorrow. Axioms 2016, 5, 11. It was in 1969 that I began my graduate studies on topological group theory and I often dived into one of the following five books. My favourite book “Abstract Harmonic Analysis” [1] by Ed Hewitt and Ken Ross contains both a proof of the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups and the structure theory of locally compact abelian groups. Walter Rudin’s book “Fourier Analysis on Groups” [2] includes an elegant proof of the Pontryagin-van Kampen Duality Theorem. Much gentler than these is “Introduction to Topological Groups” [3] by Taqdir Husain which has an introduction to topological group theory, Haar measure, the Peter-Weyl Theorem and Duality Theory. Of course the book “Topological Groups” [4] by Lev Semyonovich Pontryagin himself was a tour de force for its time. P. S. Aleksandrov, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko described this book in glowing terms: “This book belongs to that rare category of mathematical works that can truly be called classical - books which retain their significance for decades and exert a formative influence on the scientific outlook of whole generations of mathematicians”. The final book I mention from my graduate studies days is “Topological Transformation Groups” [5] by Deane Montgomery and Leo Zippin which contains a solution of Hilbert’s fifth problem as well as a structure theory for locally compact non-abelian groups. These five books gave me a good feeling for the most significant research on locally compact group theory in the first 60 years of the twentieth century. My own contribution to understanding the structure of locally compact abelian groups was a small book “Pontryagin Duality and the Structure of Locally Compact Abelian Groups” [6] which was translated into Russian and served to introduce a generation of young Soviet mathematicians to this topic. Far from locally compact groups,
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