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ByStan Vernooyon July 2, 2000

When I was in a topology course in graduate school, I constantly returned to the Munkres book to get clearer explanations of concepts than any of the graduate-level books could provide. What is noteworthy is that the ease of understanding did NOT come at the price of shallower coverage or lack of mathematical rigor. Although this is an undergraduate text, it covers almost everything you would get in a first-year graduate course in point set topology. If you want to learn that material for the first time without an instructor, then this is the book to use. And, if you are working in another area of mathematics, and come across words like "compact", "metric space", or "connected", and have forgotten what they mean, go straight to Munkres. He always talks to you like a real human being.

8 people found this helpful

ByAmazon Customeron September 23, 2014

This book seems to be generally heralded as a great overview of point-set topology, but honestly I find myself rather unimpressed.

The good: Munkres has very clean, to-the-point prose and his style is entertaining and highly readable. He does a great job of highlighting the important points of the material and showing the reader "the big picture," so to speak.

The bad: Simply put, the proofs are sloppy. Munkres seems to work almost entirely on intuition here, and consequently lots of things that really ought to be done in full rigor (especially towards the beginning, when doing so would not have required a prohibitive amount of extra content) are skimmed over with pictures and hand-waving ("we can construct a homeomorphism" without ever actually demonstrating one is a mainstay, for example). This would be less-frustrating if Munkres did a good job of noting when potentially important subtleties were being overlooked, but one finds, to a nearly maddening extent, that tricky/unobvious parts of many crucial theorems are often simply not mentioned at all.

Additionally, there is a rather absurd disconnect between the difficulty of the exercises and the text itself, and moreover it's not indicated which exercises are "easy" and which require significant amounts of insight, often leaving the reader baffled when a problem requiring an arcane multi-page proof appears directly after a two-line triviality, with no segue.

In all, I'd recommend one avoid this if not for the fact that there really don't seem to be any other standard undergraduate-level topology textbooks available.

The good: Munkres has very clean, to-the-point prose and his style is entertaining and highly readable. He does a great job of highlighting the important points of the material and showing the reader "the big picture," so to speak.

The bad: Simply put, the proofs are sloppy. Munkres seems to work almost entirely on intuition here, and consequently lots of things that really ought to be done in full rigor (especially towards the beginning, when doing so would not have required a prohibitive amount of extra content) are skimmed over with pictures and hand-waving ("we can construct a homeomorphism" without ever actually demonstrating one is a mainstay, for example). This would be less-frustrating if Munkres did a good job of noting when potentially important subtleties were being overlooked, but one finds, to a nearly maddening extent, that tricky/unobvious parts of many crucial theorems are often simply not mentioned at all.

Additionally, there is a rather absurd disconnect between the difficulty of the exercises and the text itself, and moreover it's not indicated which exercises are "easy" and which require significant amounts of insight, often leaving the reader baffled when a problem requiring an arcane multi-page proof appears directly after a two-line triviality, with no segue.

In all, I'd recommend one avoid this if not for the fact that there really don't seem to be any other standard undergraduate-level topology textbooks available.

ByStan Vernooyon July 2, 2000

When I was in a topology course in graduate school, I constantly returned to the Munkres book to get clearer explanations of concepts than any of the graduate-level books could provide. What is noteworthy is that the ease of understanding did NOT come at the price of shallower coverage or lack of mathematical rigor. Although this is an undergraduate text, it covers almost everything you would get in a first-year graduate course in point set topology. If you want to learn that material for the first time without an instructor, then this is the book to use. And, if you are working in another area of mathematics, and come across words like "compact", "metric space", or "connected", and have forgotten what they mean, go straight to Munkres. He always talks to you like a real human being.

ByDavid Elderon June 30, 2003

This is a fantastic book, the type of perfection to which all writers of mathematical texts should aspire. There are plenty of definitions, theorems, and proofs, as well as informative examples and prose exposition. The expository text is what makes this book really stand out. Munkres explains the concepts expressed abstractly in theorems and definitions. That is, he builds motivations for the necessarily abstract concepts in topology. This greatly improves the readability of the book, making it accessibly to general readers in mathematics, science, and engineering.

The book is divided into two sections, the first covering general, i.e. point-set, topology and the second covering algebraic topology. Exercises (without solutions) are provided throughout. The exercises include straight-forward applications of theorems and definitions, proofs, counter-examples, and more challenging problems.

My only complaint with this book is that it does not discuss manifolds and differentiable topology, but other texts fill this gap.

I highly recommend this book to anyone interested in studying topology; it is especially well-suited for self study.

The book is divided into two sections, the first covering general, i.e. point-set, topology and the second covering algebraic topology. Exercises (without solutions) are provided throughout. The exercises include straight-forward applications of theorems and definitions, proofs, counter-examples, and more challenging problems.

My only complaint with this book is that it does not discuss manifolds and differentiable topology, but other texts fill this gap.

I highly recommend this book to anyone interested in studying topology; it is especially well-suited for self study.

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ByMark Arjomandion January 4, 2005

I used to own the first (1975) edition of this title since the late 1990s, but eventually purchased the new edition as well, and donated the old book to our campus library. Despite having very close similarity to the text by Stephen Willard (1970, Dover issue 2004) which points to the fact that both authors must have used the same source articles, Munkres's book stands out as one of the best rigorous introductions for a beginning graduate student. It covers all the standard material for a first course in general topology starting with a full chapter on set theory, and now in the second edition includes a rather extensive treatment of elementary algebraic topology. The style of writing is student-friendly, the topics are nicely motivated, (counter-)examples are given where they were needed, many diagrams provided, the chapter exercises relevant with the correct degree of difficulty, and there are virtually no typos. The 2nd edition fine-tunes the exposition throughout, including a better paragraph formatting of the material and also greatly expands on the treatment of algebraic topology, making up for 14 total chapters as opposed to eight in the first edition. I particularly found useful the discussion of the separation axioms and metrization theorems in the first part, and the classification of surfaces and covering spaces in the second part.

In my opinion, after going through the discussion of algebraic topology in Munkres, the students should be ready to move forward to a (now standard) text such as Hatcher, for further coverage of homotopy, homology and cohomology theories of spaces. Eventhough a few contending general topology texts - such as a recent title published in the Walter Rudin Series - have started to hit the academic markets, Munkres will no doubt remain as the classic, tried-and-trusted source of learning and reference for generations of mathematics students. This is despite the fairly high price tag which could stop some students from buying their own copies, hence encouraging instructors to choose some of the cheaper topology paperbacks readily available through the Dover publications. Also the majority of Munkres's readers would have wished to see more hints and answers provided at the back so as to make the text more helpful for self-study. (I remember suffering from and being lost with my Munkres topology homework exercises in 1998-1999, during my first year of graduate school.) It later became evident to me that those who are newcomers to the topic or are merely testing the waters, should try Fred H. Croom's 1989 topology text, since the latter is a more accessible title similar in the exposition and selection of topics on Munkres (and Willard for that matter), thus nicely serving as a prerequisite for either of the more advanced books.

A couple of ending remarks: A reviewer has correctly mentioned here that Dr. Munkres does not include differential topology in his presentation. This is because of the length consideration, given that he has already written a separate monograph on the topic. In fact it's also necessary to first get a handle on a fair amount of algebraic topology such as the notions of homotopy, fundamental groups, and covering spaces for a full-fledged treatment of the differential aspect. In any case, one high-level reference for a rigorous excursion into this area is the Springer-Verlag GTM title by Morris W. Hirsch which includes introductions to the Morse and cobordism theories. I'd also like to mention that another decent book on general topology, unfortunately out of print for quite some time, is a treatise by "James Dugundji" (Prentice Hall, 1965). The latter would complement Munkres, as for instance Dugundji discusses ultrafilters and some of the more analytical directions of the subject. It's a pity that Dover in particular, has not yet published this gem in the form of one of their paperbacks.

In my opinion, after going through the discussion of algebraic topology in Munkres, the students should be ready to move forward to a (now standard) text such as Hatcher, for further coverage of homotopy, homology and cohomology theories of spaces. Eventhough a few contending general topology texts - such as a recent title published in the Walter Rudin Series - have started to hit the academic markets, Munkres will no doubt remain as the classic, tried-and-trusted source of learning and reference for generations of mathematics students. This is despite the fairly high price tag which could stop some students from buying their own copies, hence encouraging instructors to choose some of the cheaper topology paperbacks readily available through the Dover publications. Also the majority of Munkres's readers would have wished to see more hints and answers provided at the back so as to make the text more helpful for self-study. (I remember suffering from and being lost with my Munkres topology homework exercises in 1998-1999, during my first year of graduate school.) It later became evident to me that those who are newcomers to the topic or are merely testing the waters, should try Fred H. Croom's 1989 topology text, since the latter is a more accessible title similar in the exposition and selection of topics on Munkres (and Willard for that matter), thus nicely serving as a prerequisite for either of the more advanced books.

A couple of ending remarks: A reviewer has correctly mentioned here that Dr. Munkres does not include differential topology in his presentation. This is because of the length consideration, given that he has already written a separate monograph on the topic. In fact it's also necessary to first get a handle on a fair amount of algebraic topology such as the notions of homotopy, fundamental groups, and covering spaces for a full-fledged treatment of the differential aspect. In any case, one high-level reference for a rigorous excursion into this area is the Springer-Verlag GTM title by Morris W. Hirsch which includes introductions to the Morse and cobordism theories. I'd also like to mention that another decent book on general topology, unfortunately out of print for quite some time, is a treatise by "James Dugundji" (Prentice Hall, 1965). The latter would complement Munkres, as for instance Dugundji discusses ultrafilters and some of the more analytical directions of the subject. It's a pity that Dover in particular, has not yet published this gem in the form of one of their paperbacks.

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ByA customeron April 7, 2004

My introduction to Munkres was in an independent study of point set topology in my final semester of undergraduate work. A professor assigned me problems from the book, but my learning was largely self motivated. I found that it was an excellent book for independent study. The text was clear and readable and the exercises helped to cement the concepts that are introduced in the reading.

Later at graduate school, Munkres was also used in a topology class at the beginning graduate level. Highlights were taken from the first section (point set topology), and a large focus of the class was on the algebraic topology in the second section of the book. Sometimes I had difficulty following exactly what the professor was doing at the blackboard, but I could always understand what was going on when I consulted Munkres.

I would stress that this is only to be used as an introduction to algebraic topology, as there is nearly no development of homology groups and other algebraic concepts. However, it gives a very good presentation for the fundamental group. As a whole it would be a very good addition to your mathematical library.

Later at graduate school, Munkres was also used in a topology class at the beginning graduate level. Highlights were taken from the first section (point set topology), and a large focus of the class was on the algebraic topology in the second section of the book. Sometimes I had difficulty following exactly what the professor was doing at the blackboard, but I could always understand what was going on when I consulted Munkres.

I would stress that this is only to be used as an introduction to algebraic topology, as there is nearly no development of homology groups and other algebraic concepts. However, it gives a very good presentation for the fundamental group. As a whole it would be a very good addition to your mathematical library.

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BypiRskwayrdon January 16, 2014

Plenty of people have already mentioned the merits of this book and I wont repeat them. What I wish to mention is that if you choose to order the economy edition (purple cover) then you should know that the quality of the paper is very poor. Everything is readable and its exactly the same (as far as I can tell) as the hardcover but the pages themselves are a dull grey color and not very thick or resilient. The best I can describe it as its the same paper that is used in high school workbooks that you are supposed to write in and then tear out the pages and hand them in. Most people wont care (frankly I don't anymore) so I wont remove any stars from this otherwise excellent text, but just be forewarned if you are that kind of person who is bothered by such things.

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ByRobert Hankon December 11, 2006

I bought this book for my first course in point-set topology. I must say I was quite confused when I began the actual topology portion of the book (chapter 2), but this was due to the difficulty of topology, in general, and not the book itself. Looking back at the introduction to a topology, I can now see it was very good, as was the introduction to the various types of topologies. The rest of the book is almost a reference to the basics of topology with some plan of cohesion behind it. Unfortunately, this lack of a completely cohesive approach is unavoidable, since a course in point-set topology ought to provide a stepping stone one can use for further study in topology and not a mountain one can climb and conquer and thus know the subject completely. This requires that a cursory explanation of many ideas be presented.

Nevertheless, Munkres' book is quite readable and yet still rigorous. His exercises are very instructive in reinforcing the many ideas and provide a wide variety in problem difficulties for a motivated undergraduate in math.

If you are searching for a book that provides a friendly step-by-step aproach to learning point-set topology, I would not recommend this book for it would be too difficult. If you are searching for an introduction to point-set topology that will give you a solid grounding in the basics of point-set topology, but at the same time will give it to you in an easily approached manner, than this book is for you. It is my belief that this book is about as close a math textbook can come to being a "read in front of the fire" sort of book.

Nevertheless, Munkres' book is quite readable and yet still rigorous. His exercises are very instructive in reinforcing the many ideas and provide a wide variety in problem difficulties for a motivated undergraduate in math.

If you are searching for a book that provides a friendly step-by-step aproach to learning point-set topology, I would not recommend this book for it would be too difficult. If you are searching for an introduction to point-set topology that will give you a solid grounding in the basics of point-set topology, but at the same time will give it to you in an easily approached manner, than this book is for you. It is my belief that this book is about as close a math textbook can come to being a "read in front of the fire" sort of book.

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ByRonald Reaganon February 9, 2008

This is the standard into to point-set topology for a reason. It covers general topology very well, with easy-to-follow proofs and exercises that are actually possible to do. The problem is that the algebraic topology portion of this text (around 1/3 of the whole thing), is vastly inferior to Hatcher's "Algebraic Topology" book, which happens to be free. If you're looking for a good way to begin studying topology, then this is the book you're looking for, but if you want to learn about the fundamental group (and various related topics), then Allen Hatcher's book is available for free on his website. It calls to you...[insert spooky noises here]

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ByTodd Eberton August 9, 2002

Although both parts of this book are exceptionally well written,

I've seen even better presentations of general topology in Sutherland's "Introduction to Topological and Metric Spaces", although admittedly Chapters 5 and 8 are not covered there. On the other hand I have found it very difficult to find a better book that covers part 2 of this book, Algebraic Topology. Most textbooks in this area either seem outdated or overly abstract. However, Munkres takes the time to explain concepts like covering spaces and the fundamental group with care and detail, providing a number of concrete examples. Combine this book with his differential topology book, and one can easily self-study his or her way to a mastery of first-year graduate topology.

I've seen even better presentations of general topology in Sutherland's "Introduction to Topological and Metric Spaces", although admittedly Chapters 5 and 8 are not covered there. On the other hand I have found it very difficult to find a better book that covers part 2 of this book, Algebraic Topology. Most textbooks in this area either seem outdated or overly abstract. However, Munkres takes the time to explain concepts like covering spaces and the fundamental group with care and detail, providing a number of concrete examples. Combine this book with his differential topology book, and one can easily self-study his or her way to a mastery of first-year graduate topology.

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ByHassan Sahibzadaon June 13, 2007

This is by far the best undergraduate mathematics textbook I have ever used. I was fortunate enough to have a very good professor for this course and combined with this book was quite possibly the best undergraduate mathematics course I've ever taken. This book is amazing for self study and is truly indispensable. Munkres' explanations are very clear and rigorous at the same time and the problems are very good. Of course, there is no solutions manual which is the only drawback if one is using this text for self study. There really aren't any other options if you want to learn topology for the first time. The book has a nice introduction to set theory then it covers point set topology and algebraic topology. I would recommend this book to anyone who wants to learn topology. I sure had a lot of fun reading through this book!

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Bynow a grad studenton June 18, 2010

I originally used this text in my undergraduate topology class. In the time since then, I have repeatedly returned to this book for reference. While I am specializing in algebraic number theory and algebraic geometry, I find that minor topological considerations still arise fairly often. I also used Munkres as my primary object of study for the topology qualifying exam.

As a textbook, Munkres is clear and precise. He clearly states definitions and theorems, and provides enough examples to get a feel for their usage. The exercises are varied, but none were excessively hard, and they provide a good foundation to understand the flavor of topology. The prose is also very crisp and clear, and it provides motivation without had-holding and there is no needless obfuscation or verbosity. Having looked at many topology texts over the years, this is undoubtedly my favorite as a text. I would venture to say that this is the best introductory topology book yet written.

As a reference, Mukres is still great. It isn't as great a reference as it is a textbook, but it is still wonderful. The book's organization and clarity, which aids its function as a textbook, serves the reference user well. Additionally, it is fairly comprehensive insofar as basic point-set and algebraic topology are concerned. My one problem with Munkres as a reference: it is severely lacking with respect to manifolds and differential topology, even in their most basic form. Still, it is so wonderfully clear with respect to basic point-set and algebraic topology that I can't imagine wanting another book to fill in reference for those basic areas.

Seriously, this is THE book to learn topology, and then it should be kept around as a reference.

As a textbook, Munkres is clear and precise. He clearly states definitions and theorems, and provides enough examples to get a feel for their usage. The exercises are varied, but none were excessively hard, and they provide a good foundation to understand the flavor of topology. The prose is also very crisp and clear, and it provides motivation without had-holding and there is no needless obfuscation or verbosity. Having looked at many topology texts over the years, this is undoubtedly my favorite as a text. I would venture to say that this is the best introductory topology book yet written.

As a reference, Mukres is still great. It isn't as great a reference as it is a textbook, but it is still wonderful. The book's organization and clarity, which aids its function as a textbook, serves the reference user well. Additionally, it is fairly comprehensive insofar as basic point-set and algebraic topology are concerned. My one problem with Munkres as a reference: it is severely lacking with respect to manifolds and differential topology, even in their most basic form. Still, it is so wonderfully clear with respect to basic point-set and algebraic topology that I can't imagine wanting another book to fill in reference for those basic areas.

Seriously, this is THE book to learn topology, and then it should be kept around as a reference.

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