Introduction to Topology: Third Edition (Dover Books on Mathematics) by Bert Mendelson (PDF)

Description

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.

User’s Reviews

Editorial Reviews: 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.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐couldn’t serve the purpose for which it was procured

⭐The book is what it is. It’s not a casual subject but the textbook could stand a few more examples. However, given the price, its still a good one to use and keep one the shelf.However, the Kindle version is terrible. Not only are mathematical expressions difficult to read on the Kindle version, but pieces of important items, such as parts of theorems, are left out. This continues to be an issue with Kindle editions of technical works and often renders them nearly useless. I only buy them if they are cheap so that I don’t need much convenience to justify the purchase. DO NOT use the Kindle version as your only version.

⭐This book offers a distilled overview of topology and imo is most useful as a supplemental text. For the price, why not?(I’d avoid kindle version)

⭐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.

⭐With all due respect to the late Professor Mendelson, I have struggled with this book. Having said that, I am neither a mathematician nor graduate student of same. I am a professional specialising in theory and am trying to teach myself (with MIT’s help) some belated applied maths useful to theory making. For work I have embarked upon recently I am in need of upgrading my knowledge of things like abstract algebra and topology. This is one of the texts I purchased for self-study.I guess my main gripe is the book is a bit too dry for me. Which led to a lack of clarity on certain topics. I need a bit more flourish of explanation than mere definitions, theorems, examples. Over and over again. But I know, on the other hand Prof M wrote it from his lecture notes, for his students of serious maths. Not for me. So be it. For example, after reading the book through, then pouring over it in selected areas and chapters for weeks, re-visiting things I had not quite grasped, I still cannot tell the difference between a topological space and a topology – other than by theorem definition. Sure, there I can see (X,T) (T being curly T or whatever it is called) is the former, and T itself the latter. But I remain in desperate need of help from an author saying what this means. In clear simple English, preferably.I am not negative about the book as a whole, for I now know something at least about topology. As opposed to before I bought it and read it – when I knew zilch.Recommended for mathematical students; not for inquiring minds of applied people like myself who need more gently, gently.

⭐Just what the title says. I have tried across several different kindles and the online reader as well as the app on ifruit devices. Any time there is a ‘element of’ symbol, the picture insert is so small you can’t read it or zoom it very easily. Probably better to find a PDF and buy it instead.What material I could read was well presented and useful, but bring your magnifying glass if you get the kindle edition. Shame really.

⭐I am shocked that some people recommend this book for self-study and give 5 star.At first glance, it looks excellent: it has very good chapter organization and thin book size.Actually, it is good until the metric space (Chapter 2).Then, from topological spaces (Chapter 3), it is basically unreadable, due to: lack of explanation on terminology, ZERO example, and ZERO figure.I found that there are much better sources out there.This book may be good as a summary, but definitely NOT as a learning source.

⭐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.

⭐This slim tome was my first formal encounter with topology, and I found it reasonably easy to work through on my own. Like many undergrad textbooks, it states that there are no prerequisites other than comfort with proof based maths. However, I would recommend making yourself familiar with basic analytic concepts and being comfortable proving theorems using them before starting the book. While none of this knowledge is strictly necessary for the book, an “analytic way of thinking” will be a helpful springboard into the material, as most of the new concepts covered in the book are presented as generalisations of concepts in Euclidean n-space.It starts off covering the basics of set theory and functions, most of which can be safely skipped by anyone with a semester or two of undergrad under their belt, and merely used as a reference (though it would be a good idea to look at the bit about commutative diagrams for those studying at that level). Chapter 2 covers metric spaces, giving definitions of open sets, neighbourhoods and continuous functions between metric spaces (among other things) in terms of open balls, then linking these concepts through various theorems. Chapter 3 defines topological spaces, then defines various concepts on topological spaces using the by now familiar analogous concepts in metric spaces. Readers may notice that many of the definitions given in Chapter 3 are almost word-for-word copied from theorems in Chapter 2, by design. Chapter 4 introduces the various forms of connectedness, and investigates homotopic paths & the fundamental group, though it doesn’t seem to give a formal definition of the FG anywhere. Chapter 5 introduces compactness of both topological spaces and metric spaces, relating this back to material in chapters 2 and 3.While the exposition is mainly clear and concise, the book is somewhat light on examples in places and occasionally skips over some steps in examples which are not necessarily obvious to a first-time student of topology. The exercises are interesting, useful and have a good difficulty curve between the start and end of a section. The ink is a bit light in some places, which may make it difficult to read, but this can be forgiven due to the inexpensiveness of Dover books in general. While it lacks the depth and scope of weightier tomes such as Munkres, it lives up to its title, and makes a good, cheap first pass at a famously difficult subject.

⭐If you are doing a module in metric spaces or topology you ought to read this, cover to cover (‘cept maybe the first chapter, but this is always the case! Chapter 0 is never interesting) in your first or second year, you should know all the content (like the back of your hand) if you are doing a third year module.It is a brilliant introduction to everything you will need but is just that – an introduction. There’s a superb amount of “hand-holding” in the proofs which I found really useful to boost my confidence, after that I’d start covering proofs and then checking them. This is good!I completely recommend this book, but I do not recommend it is your only topology book (There is another also called “Introduction to topology” with a blue over and an orange torus on the front, from Dover, this is not an introduction it is much more filled out and much faster, if you combine these two, with Munkres’ Topology you’re set)There is one thing I don’t quite like, the treatment of Quotient topologies (or identification topologies) is rather weak and hard to understand, but I cannot write off a brilliant book due to an iffy 5 pages.I have no hesitation in recommending this book. I adore Dover because of the great prices also, I am getting quite the collection!

⭐Excellent

⭐This review is regarding the Kindle version specifically.I have no problem with the content of the book – and would give the book a 4 or 5 for content. However, the layout and formatting in the Kindle edition is absolutely appalling! It almost (almost – but not quite) renders parts of the book unreadable. Another reviewer mentioned the same problem.Summary …Content: 4 or 5Layout / Formatting (in the KINDLE edition): 1 or 2.

⭐I’ve recently finished my MPhys in Theoretical Physics, and going to start a Ph.D in Maths. I bought this to get to grips with topology, as I’ve had no previous exposure and really like the Dover series, I bought this one based on the reviews.The book is structured into manageable chunks, and the topics are very well explained, with lots of questions this book is vital for either studying topology, either self study or as a course supplement.

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