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- Introduction to Topology: Third Edition (Dover Books on Mathematics)
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176 people found this helpful

ByMichael Wischmeyeron April 11, 2003

I was not a mathematics major, and only in recent years have I ventured into abstract mathematics. I was motivated to learn about topology as an aid to understanding a particular 3-D earth modeling application.

I read Introduction to Topology in three stages: as a review of set theory and metric spaces (chapters 1 and 2), then as an introduction to topology (chapter 3), and lastly as a detailed look at two important topological properties, connectedness (chapter 4) and compactness (chapter 5). I had previously read (and reviewed) another book titled Metric Spaces by Victor Bryant, but Mendelson is my first serious look at topology.

My reading of Mendelson - a 200-page text - required about 100 hours, substantially longer than the 40 to 60 hours estimated by an earlier reviewer. No solutions are provided for the section problems, which are generally proofs, not explicit problems.

The first chapter provides a concise overview of set theory and functions that is essential for Mendelson's later chapters on subsequent set-theoretic analysis of metric spaces and topology.

The second chapter is a solid introduction to metric spaces with good discussions on continuity, open balls and neighborhoods, limits from a metric space perspective, open sets and closed sets, subspaces, and equivalence of metric spaces. Chapter 2 concludes with a brief introduction to Hilbert space.

The third chapter introduces topological spaces as a generalization of metric spaces, and many theorems are largely restatements of the metric space theorems derived in chapter 2. I was thankful for this approach.

Mendelson begins chapter 3 by demonstrating that 1) open sets and neighborhoods are preserved in passing from a metric space to its associated topological space and 2) the existence of a one to one correspondence between the collection of all topological spaces and the collection of all neighborhood spaces.

He then reminds us that in a metric space we can say that there are points of a subset A arbitrarily close to a point x if the metric d(x, A) = 0. In characterizing this notion of arbitrary closeness in a topological space, Mendelson introduces the closure of A, the interior of A, and the boundary of A. Other topics included topological functions, continuity, homeomorphism (the equivalence relation), subspaces, and relative topology. The final sections in chapter 3 on products of topological spaces, identification topologies, and categories and functors were more difficult.

In chapter 4 the initial sections (connectedness on the real line, the intermediate value theorem, and fixed point theorems) were largely familiar. But thereafter I became bogged down with the discussions of path-connected topological spaces, especially with the longer proofs involving the concepts of homotopic paths, the fundamental group, and simple connectedness.

Chapter 5, titled Compactness, was even more abstract and difficult, with topics like coverings, finite coverings, subcoverings, compactness, compactness on the real line, products of compact spaces, compact metric spaces, the Lebesgue number, the Bolzano-Weierstrass property, and countability. Perhaps, a reader more familiar with analysis would have less difficulty with the last two chapters.

In summary, Introduction to Topology is quite useful for self-study. Mendelson's short text was intended for a one-semester undergraduate course, and it is thereby ideal for readers that either require a basic introduction to topology, or need a quick review of material previously studied. The last two chapters on connectedness and compactness are substantially more difficult, but are still accessible to the persistent reader.

I read Introduction to Topology in three stages: as a review of set theory and metric spaces (chapters 1 and 2), then as an introduction to topology (chapter 3), and lastly as a detailed look at two important topological properties, connectedness (chapter 4) and compactness (chapter 5). I had previously read (and reviewed) another book titled Metric Spaces by Victor Bryant, but Mendelson is my first serious look at topology.

My reading of Mendelson - a 200-page text - required about 100 hours, substantially longer than the 40 to 60 hours estimated by an earlier reviewer. No solutions are provided for the section problems, which are generally proofs, not explicit problems.

The first chapter provides a concise overview of set theory and functions that is essential for Mendelson's later chapters on subsequent set-theoretic analysis of metric spaces and topology.

The second chapter is a solid introduction to metric spaces with good discussions on continuity, open balls and neighborhoods, limits from a metric space perspective, open sets and closed sets, subspaces, and equivalence of metric spaces. Chapter 2 concludes with a brief introduction to Hilbert space.

The third chapter introduces topological spaces as a generalization of metric spaces, and many theorems are largely restatements of the metric space theorems derived in chapter 2. I was thankful for this approach.

Mendelson begins chapter 3 by demonstrating that 1) open sets and neighborhoods are preserved in passing from a metric space to its associated topological space and 2) the existence of a one to one correspondence between the collection of all topological spaces and the collection of all neighborhood spaces.

He then reminds us that in a metric space we can say that there are points of a subset A arbitrarily close to a point x if the metric d(x, A) = 0. In characterizing this notion of arbitrary closeness in a topological space, Mendelson introduces the closure of A, the interior of A, and the boundary of A. Other topics included topological functions, continuity, homeomorphism (the equivalence relation), subspaces, and relative topology. The final sections in chapter 3 on products of topological spaces, identification topologies, and categories and functors were more difficult.

In chapter 4 the initial sections (connectedness on the real line, the intermediate value theorem, and fixed point theorems) were largely familiar. But thereafter I became bogged down with the discussions of path-connected topological spaces, especially with the longer proofs involving the concepts of homotopic paths, the fundamental group, and simple connectedness.

Chapter 5, titled Compactness, was even more abstract and difficult, with topics like coverings, finite coverings, subcoverings, compactness, compactness on the real line, products of compact spaces, compact metric spaces, the Lebesgue number, the Bolzano-Weierstrass property, and countability. Perhaps, a reader more familiar with analysis would have less difficulty with the last two chapters.

In summary, Introduction to Topology is quite useful for self-study. Mendelson's short text was intended for a one-semester undergraduate course, and it is thereby ideal for readers that either require a basic introduction to topology, or need a quick review of material previously studied. The last two chapters on connectedness and compactness are substantially more difficult, but are still accessible to the persistent reader.

3 people found this helpful

ByIanon June 26, 2015

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.

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.

ByAmazon Customeron December 18, 2015

Overall, great introductory book to topology. The pedagogy was excellent and the development of topics <i>made sense</i> 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, <i>by any means</i>. 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.

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, <i>by any means</i>. 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.

ByChovargoon May 19, 2009

I'd like very much this book. The book is very conceptual and also rigorous. It is self-consistent and this facilate his study. It progress step by step. If you need an Introduction, for self study this is a right book for starting. His writting style is very clear and the edition is also very good. The only defect I found is that there is no solutions for the excersices.

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ByMark Thriceon August 31, 2013

This book fills my desire to read some "real" mathematics, which I have always wanted to do. Though I am a physicist I never really got into the realm of pure mathematics. I picked the right book.

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ByAlex P. Keatonon December 31, 2013

I found this book extremely helpful while working through chapters 2 and 4 of baby Rudin (along with Schramm's analysis book). There are numerous examples and accessible problem sets. In hindsight, I wish I would have worked through this book prior to taking my first course in analysis. This is also a great little book to work through prior to tackling the first half of Munkres. That it is an affordable Dover edition easily makes it a 5 star book.

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ByBaywoofon October 11, 2014

Talk about concise and precise! Wish I had learned my topology through this book 45 years ago: I wouldn't have to relearn everything all over again. First chapter on set theory would have been worth the price of the book. A must for a beginner (the correct way to learn) or oldster (to correct his old learning).

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ByJohn Doeon September 25, 2011

This is a Wonderful book. I have my B.S. in Mathematics and I am in Graduate School for Mathematics. I am signed up for a Topology class next semester and to be ready, I picked up this, Introduction to General Topology by Cain and Topology by Munkres. Out of all of them this one presents Topology in the nicer way. Unlike in Munkres, it reads like a book, and unlike in Cain it doesn't assume steps and connections. This book provided the nice grounding for a more intuitive take on Topology by starting with Metric spaces and then linking everything from there. Cain then allowed for me to practice by giving me some examples and simple exercises. If there is one thing I would suggest to the author, put in some examples and some less general exercises.

Loved it and would suggest this book to anyone.

Loved it and would suggest this book to anyone.

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Bybroke_symlinkon April 26, 2010

While this book is good, its a little overhyped. I did not particularly care for this book's presentation of connectedness and compactness (ie, the last two chapters), but the first three chapters were good. The problems in this book were also pretty good. They were at least interesting and difficult. However, there are no solutions, so it might not be the best book for self study. I personally think introduction to topology by gamelin and greene is better. These books should be used in conjunction with topology by munkres.

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ByFiziksGangstaon January 24, 2012

I had one semester of real analysis prior to picking up this book. I found it's presentation of metric spaces OK, though Mendelson seems happy enough with abstract definitions that he doesn't always try to help with developing some intuition. This book is probably pretty hard to read if you're not familiar with metric spaces.

Mendelson introduces topological spaces very well, though, with lots of examples, counterexamples, and explanations of both. It's dense reading (though not open-dense, ha-ha) but it is worth it. Definitely a good first book on topology.

Mendelson introduces topological spaces very well, though, with lots of examples, counterexamples, and explanations of both. It's dense reading (though not open-dense, ha-ha) but it is worth it. Definitely a good first book on topology.

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Byarpard fazakason March 3, 2012

I enjoyed this book. It is a nice introduction to topology for undergraduates. It presupposes very little mathematical background. It is clear and well-written. Chapter 1 is a good introduction to set theory, Chapter 2 to metric spaces, Chapter 3 to topological spaces, Chapter 4 to connectedness, Chapter 5 to compactness. There are many exercises given. But I have one big problem with this book as a self-study guide: answers to the exercises are not provided. So I give it only 4 stars.

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BySamir Varkeyon October 26, 2010

For all those who want to get into the field of Topology and then do Differential Geometry and then do General Relativity. Take this book as your first step to your final destination!

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