This elegant book by distinguished mathematician John Milnor, provides a clear and succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem
General Topology is not only a textbook, it is also an invaluable reference work for all mathematicians working the field of analysis. It has long been out of print, but a whole generation of mathematicians, including myself, learned their topology from this book. There are no wasted words in Kelley’s presentation; every sentence is short and to the point, but the student would do well to contemplate each of them, for they are pregnant with subtle implications. The numerous problems that follow each chapter are well chosen to complete the students’ understanding of the topics discussed. THIS VOLUME gives a systematic exposition of the part of general topology which has proven useful in several branches of mathematics and is intended especially as a background for modern analysis. One of the many features of this volume is the wealth and diversity of problem material which includes counter-examples and numerous applications of general topology to different fields. The appendix, which is entirely independent of the rest of the book, includes an axiomatic treatment of set theory. The author has included the most commonly used terminology, and all terms are listed in the index. As a reference, this book offers a unique coverage of topology.
Measure and Category: A Survey of the Analogies between Topological and Measure Spaces (Graduate Texts in Mathematics)
Dugundji is the author of the textbook Topology (Allyn and Bacon, 1966), on general topology. Reviewer M. Edelstein wrote that this was "one of the best among the numerous books on the subject",[4] and it went through numerous reprintings.
Superb one-year course in classical topology. Topological spaces and functions, point-set topology, much more. Examples and problems.
This introduction to topology eases readers into the subject by building a bridge from the familiar concepts of geometry to the formalized study of topology. Focuses on congruence classes defined by transformations in real Euclidean space, continuity, sets, functions, metric spaces, and topological spaces, and more. Exercises and Problems. Includes 101 black-and-white illustrations.
Based on a course given to talented high-school students at Ohio University in 1988, this book is essentially an advanced undergraduate textbook about the mathematics of fractal geometry. It nicely bridges the gap between traditional books on topology/analysis and more specialized treatises on fractal geometry. The book treats such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology. It takes into account developments in the subject matter since 1990. Sections are clear and focused. The book contains plenty of examples, exercises, and good illustrations of fractals, including 16 color plates.
This material is intended to contribute to a wider appreciation of the mathematical words "continuity and linearity". The book's purpose is to illuminate the meanings of these words and their relation to each other.
This text presents the fundamental principles of topology rigorously but not abstractly. It emphasizes the geometric nature of the subject and the applications of topological ideas to geometry and mathematical analysis. The usual topics of point-set topology, including metric spaces, general topological spaces, continuity, topological equivalence, basis, sub-basis, connectedness , compactness, separation properties, metrization, subspaces, product spaces, and quotient spaces are treated in this text. Most of the factual information about topology presented in this text is stated in the theorems and illustrated in the accompanying examples, figures and exercises. This book contains many exercises of varying degrees of difficulty. The notation used in this text is reasonably standard; a list of symbols with definitions appears on the front end-sheets. This text is designed for a one-semester introduction to topology at the undergraduate and beginning graduate levels. It is accessible to junior mathematics majors who have studied multivariable calculus.
Praise for George Francis's A Topological Picturebook: Bravo to Springer for reissuing this unique and beautiful book! It not only reminds the older generation of the pleasures of doing mathematics by hand, but also shows the new generation what ``hands on'' really means. - John Stillwell, University of San Francisco The Topological Picturebook has taught a whole generation of mathematicians to draw, to see, and to think. - Tony Robbin, artist and author of Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought The classic reference for how to present topological information visually, full of amazing hand-drawn pictures of complicated surfaces. - John Sullivan, Technische Universitat Berlin A Topological Picturebook lets students see topology as the original discoverers conceived it: concrete and visual, free of the formalism that burdens conventional textbooks. - Jeffrey Weeks, author of The Shape of Space A Topological Picturebook is a visual feast for anyone concerned with mathematical images. Francis provides exquisite examples to build one's "visualization muscles". At the same time, he explains the underlying principles and design techniques for readers to create their own lucid drawings. - George W. Hart, Stony Brook University In this collection of narrative gems and intriguing hand-drawn pictures, George Francis demonstrates the chicken-and-egg relationship, in mathematics, of image and text. Since the book was first published, the case for pictures in mathematics has been won, and now it is time to reflect on their meaning. A Topological Picturebook remains indispensable. - Marjorie Senechal, Smith College and co-editor of the Mathematical Intelligencer 
This work is a translation into English of the Third Edition of the classic German language work Mengenlehre by Felix Hausdorff published in 1937. From the Preface (1937): ``The present book has as its purpose an exposition of the most important theorems of the theory of sets, along with complete proofs, so that the reader should not find it necessary to go outside this book for supplementary details while, on the other hand, the book should enable him to undertake a more detailed study of the voluminous literature on the subject. The book does not presuppose any mathematical knowledge beyond the differential and integral calculus, but it does require a certain maturity in abstract reasoning; qualified college seniors and first year graduate students should have no difficulty in making the material their own ... The mathematician will ... find in this book some things that will be new to him, at least as regards formal presentation and, in particular, as regards the strengthening of theorems, the simplification of proofs, and the removal of unnecessary hypotheses.''
Learn the basics of point-set topology with the understanding of its real-world application to a variety of other subjects including science, economics, engineering, and other areas of mathematics. Introduces topology as an important and fascinating mathematics discipline to retain the readers interest in the subject. Is written in an accessible way for readers to understand the usefulness and importance of the application of topology to other fields. Introduces topology concepts combined with their real-world application to subjects such DNA, heart stimulation, population modeling, cosmology, and computer graphics. Covers topics including knot theory, degree theory, dynamical systems and chaos, graph theory, metric spaces, connectedness, and compactness. A useful reference for readers wanting an intuitive introduction to topology. |
| Bert Mendelson Introduction to Topology |
Excellent study of sets in topological spaces and topological vector spaces includes systematic development of the properties of multi-valued functions. Topics include families of sets, topological spaces, mappings of one set into another, ordered sets, more. Examples included from different domains.
