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Introduction to Topology: Third Edition (Dover Books on Mathematics)
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From the Back Cover
Highly regarded for its exceptional clarity, imaginative and instructive exercises, and fine writing style, this concise book offers an ideal introduction to the fundamentals of topology. Originally conceived as a text for a one-semester course, it is directed to undergraduate students whose studies of calculus sequence have included definitions and proofs of theorems. The book's principal aim is to provide a simple, thorough survey of elementary topics in the study of collections of objects, or sets, that possess a mathematical structure.The author begins with an informal discussion of set theory in Chapter 1, reserving coverage of countability for Chapter 5, where it appears in the context of compactness. In the second chapter Professor Mendelson discusses metric spaces, paying particular attention to various distance functions which may be defined on Euclidean n-space and which lead to the ordinary topology. Chapter 3 takes up the concept of topological space, presenting it as a generalization of the concept of a metric space. Chapters 4 and 5 are devoted to a discussion of the two most important topological properties: connectedness and compactness. Throughout the text, Dr. Mendelson, a former Professor of Mathematics at Smith College, has included many challenging and stimulating exercises to help students develop a solid grasp of the material presented.Unabridged Dover (1990) republication of the edition published by Allyn and Bacon, Inc., Boston 1975.
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Product details
Series: Dover Books on Mathematics
Paperback: 224 pages
Publisher: Dover Publications; Third edition (July 1, 1990)
Language: English
ISBN-10: 0486663523
ISBN-13: 978-0486663524
Product Dimensions:
5.5 x 0.5 x 8.8 inches
Shipping Weight: 9.1 ounces (View shipping rates and policies)
Average Customer Review:
4.3 out of 5 stars
78 customer reviews
Amazon Best Sellers Rank:
#38,513 in Books (See Top 100 in Books)
This author provides a simple and adequate introduction to the basic concepts of Topology.He begins with sets and their operations, and then moves up the food chain of collectives to functions and vector relationships, and then on to metric spaces, ending with various elementary topics in topology such as topological spaces, connectedness and compactness. All well and good, so far.To the author’s credit, proofs and exercises begin immediately and are coterminous with the text itself.They are well selected too.To his discredit, however, I thought the discussion on functions was minimal to the point of almost being incomplete, if not negligent. However, when I recognized that the book had first been published in 1962, which incidentally was the first year I took Topology, only then did I realize that minimal treatment of functions was par for the course in those days.I mention this in passing only because I remember having a difficult time with the rather subtle idea of “an image of a function,†which I learned the hard way, was to follow us students of elementary Topology all the way into advance Topology a semester later.Of much greater concern to me is the fact that the author went directly from “open and closed sets†directly into “metric spaces†without discussing the intervening structures of “groups†and “rings,†both so fundamental to later developments. How can he be allowed to get away with doing that?How much better off I would have been when I first took Topology had I spent just a bit more time fully understanding all the subtle fine points of functions, groups and rings the first time around.It is true that the “image of a function†is revisited here in the chapter on continuity, which arguably, is perhaps a better place to introduce that idea. However, the omission of groups and rings is very neigh unforgivable.I searched for a rationale for their omission but could find none?Other than these two not so minor details, the rest of the book flows smoothly without hiccups or surprises. Altogether, in the end, I must begrudgingly admit, it is a solid introduction to Topology. Three stars
Overall, great introductory book to topology. The pedagogy was excellent and the development of topics made sense in going from metric spaces (a notion that is general more intuitive) to abstract topological spaces.In particular, it was great for self-study as Mendelson doesn't shy away from fully fleshing-out proofs and repeating relatively similar cases with some additional notes (e.g. when going from metric to topological spaces and proving several ideas there). The book itself can certainly be read by anyone with a set theory background and some intuitive notion of limits/sequences (i.e. a class in pre-calculus), but that doesn't mean it's easy, by any means. I struggled quite a bit with the intuition behind some of the proofs, and have, more than once, rolled around on my bed trying to recall (or prove again) some particular statement that I found quite useful. Sadly, the book doesn't have a section on homotopy equivalence and some other useful notions, but do recall it is an introduction in exactly 200 pages of short text.This book took me at least 20-30 hours to get through, skipping only the very latter section on compactess and doing at least two of the harder problems in each section; but I have very little experience with analysis, something I'm sure would have helped complete this and gain the corresponding intuition much more quickly.Again, great book and would highly recommend it for self-study of topology.
I am a robotic scientist in South Korea. I purchased this book for an introductory course in topology. As a non-specialist in applied mathematics, this book has concise contents and very readable. I am still on my way through this book and recommend this book to who wants to dive into an astonishing world of topology. Highly recommended for non-math-specialists.
This is an entry level book about general topology (or point set topology). The book is written in a clear and well-organized manner, quite easy to follow. It's worth mentioning that aside from the rigorous statement of concepts/theorems, the author also made an effort to explain how and why people get there. This helps me gain the mathematical intuition about the topics, and hence gain better and structured understanding and make the topics really a solid part of my knowledge. The problems are quite gentle and proper for beginners. BTW, I am not majored in mathematics, but I feel quite comfortable going through this book. Quite pleasant reading experience.
Intended for the advanced undergraduate student with a respectable level of mathematical maturity, Mendelson begins with the necessary review of set theory. From there the book delves into metric spaces, topological spaces, connectedness, and compactness. In short, it presents the basics of topology in a clear, linear, very readable fashion.Readers would be well advised to be familiar with the elements of proof, set theory, linear algebra, and abstract algebra in addition to analysis. A knowledge of geometry is also helpful, as one might expect.Weighing the price of this book against the depth and breadth of other texts, this volume offers more to the student who is studying topology on a budget. Unfortunately, as with most books in this category, there is no solution guide provided for the exercises. A selection of hints for the exercises would have been a nice addition but otherwise does not detract from the purpose of the work: to give the beginning topologist an overview of the subject.
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