Morris S.A. Topology Without Tears. Including a graduate course on Topological Groups

Sidney: Independently published, 2024. - 729 p.A recent addition to the Topology Without Tears book is Appendix 5 which contains the material for a first graduate course on topological groups. In particular, it contains the beautiful Pontryagin-van Kampen Duality Theorem for locally compact abelian groups and a description of the structure of locally compact abelian groups. I first lectured this course at the University College of North Wales in Bangor in 1974. The Inroduction to this Appendix 5 surveys the history of topological group theory and some important material of the last few decades.
In the context of history, all I will say here is that much of the topology described in this book was discovered in the first half of the twentieth century. And one could well say that the centre of gravity for this period of discovery was, Poland. (Borders have moved considerably.) It would be fair to say that World War II permanently changed the centre of gravity!
There are many exercises in this book. Only by working through a good number of exercises will you master this course. I have not provided answers to the exercises, and I have no intention of doing so. It is my opinion that there are enough worked examples and proofs within the text itself, that it is not necessary to provide answers to exercises – indeed it is probably undesirable to do so. Very often I include new concepts in the exercises; the concepts which I consider most important will generally be introduced again in the text. Harder exercises are indicated by an *.
This book has been, or is being, used by professors, graduate students, undergraduate students, high school students, and retirees, some of whom are studying to be, are or were, accountants, actuaries, applied and pure mathematicians, astronomers, biologists, chemists, computer graphics, computer scientists, econometricians, economists, aeronautical, database, electrical, mechanical, software, space, spatial and telecommunications engineers, finance experts, game theorists, neurophysiologists, nutritionists, options traders, philosophers, physicists, psychiatrists, psychoanalysts, psychologists, sculptors, software developers, spatial information scientists, and statisticians in Algeria, Argentina, Australia, Austria, Bangladesh, Bolivia, Belarus, Belgium, Belize, Brazil, Bulgaria, Cambodia, Cameroon, Canada, Chile, Gabon, People’s Republic of China, Colombia, Costa Rica, Croatia, Cyprus, Czech Republic, Denmark, Ecuador, Egypt, Estonia, Ethiopia, Fiji, Finland, France, Gaza, Germany, Ghana, Greece, Greenland, Guatemala, Guyana, Honduras, Hungary, Iceland, India, Indonesia, Iran, Iraq, Israel, Italy, Jamaica, Japan, Jordan, Kenya, Korea, Kuwait, Latvia, Liberia, Lithuania, Luxembourg, Malaysia, Malta, Mauritius, Mexico, New Zealand, Nicaragua, Nigeria, Norway, Pakistan, Panama, Paraguay, Peru, Poland, Portugal, Puerto Rico, Qatar, Romania, Russia, Senegal, Serbia, Sierra Leone, Singapore, Slovenia, South Africa, Spain, Sri Lanka, Sudan, Suriname, Sweden, Switzerland, Syria, Taiwan, Tanzania, Thailand, The Netherlands, Trinidad and Tobago, Tunisia, Turkey, United Kingdom, Ukraine, United Arab Emirates, United States of America, Uruguay, Uzbekistan, Venezuela, and Vietnam.Introduction.
Topological Spaces.
The Euclidean Topology.
Limit Points.
Homeomorphisms.
Continuous Mappings.
Metric Spaces.
Compactness.
Finite Products.
Countable Products.
Tychonoff’s Theorem.
Quotient Spaces.
The Stone-Weierstrass Theorem.
App. 1: Infinite Sets.
App. 2: Topology Personalities.
App. 3: Chaos Theory and Dynamical Systems.
App. 4: Hausdorff Dimension.
App. 5: Topological Groups: A Graduate Course.
App. 6: Filters and Nets.
Bibliography.
Index.True PDF