ByAmazon_Customeron 16 June 2017

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.

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.

4 people found this helpful

ByPeteon 3 June 2016

The equations seem to reproduce at fixed pixel dimensions. It's just about usable on old, lower resolution kindles or a PC with a low pixel density screen but modern 300ppi devices are useless, as are all the tablets and phones I've tried. The equations are just too small. Changing the font size doesn't help - it only affects in-text maths, not the standalone equations.

ByPeteon 3 June 2016

The equations seem to reproduce at fixed pixel dimensions. It's just about usable on old, lower resolution kindles or a PC with a low pixel density screen but modern 300ppi devices are useless, as are all the tablets and phones I've tried. The equations are just too small. Changing the font size doesn't help - it only affects in-text maths, not the standalone equations.

ByAmazon_Customeron 16 June 2017

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.

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.

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ByIsomorphEmiton 5 July 2004

This is an excellent book. It is very elegantly written and provides precise understanding of the mathematical concepts required to move from the basic idea of a set to the ideas of topological spaces, via metric spaces.

The six chapters deal with the mathematical concepts clearly and advance the reader's understanding in clear steps. Each chapter divides into sections and the end of each section is a small but comprehensive set of problems which illuminate and extend the section's ideas very successfully.

Given the abstract nature of the material it would be difficult to read successfully without attempting the problems.

The author does not pull any punches with the axiomatic approach required to do this sort of mathematics but neither is the exposition over-formal: the reader benefits greatly from the author's obvious experience of teaching.

The six chapters deal with the mathematical concepts clearly and advance the reader's understanding in clear steps. Each chapter divides into sections and the end of each section is a small but comprehensive set of problems which illuminate and extend the section's ideas very successfully.

Given the abstract nature of the material it would be difficult to read successfully without attempting the problems.

The author does not pull any punches with the axiomatic approach required to do this sort of mathematics but neither is the exposition over-formal: the reader benefits greatly from the author's obvious experience of teaching.

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ByMarco Taboga mtaboga@tiscalinet.iton 4 May 2002

This book is ideal for self-study. If you have not had the luxury of taking a topology course during your undergraduate studies, but you need to know some topology and you have to study it by yourself, this is the book you need. It is very readable and it explains carefully every concept. However, it is just an introductory text and it contains only basic material. You don't have to invest a lot of time to study the material in this book: let's say 40-60 hours of study are enough to grasp everything. I reccomend it especially to those graduate students of applied mathematics, finance, statistics or economics, who need to use some basic result from topology in their work.

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ByOccasional lapse of reasonon 24 January 2008

Nice book, very entertaining.

I am only on the second chapter ( on metric spaces ) but am finding it interesting enough that I am staying up late reading it. It is very well written and clear.

I have minimal mathematical background ( just a little calculus and linear algebra ) and and sadly lacking in knowledge of analysis, but still find this book understandable.

This is well worth the modest cost.

Heck, this is worth ten times the modest cost !

I am only on the second chapter ( on metric spaces ) but am finding it interesting enough that I am staying up late reading it. It is very well written and clear.

I have minimal mathematical background ( just a little calculus and linear algebra ) and and sadly lacking in knowledge of analysis, but still find this book understandable.

This is well worth the modest cost.

Heck, this is worth ten times the modest cost !

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ByAlecon 5 December 2014

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!

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!

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I brought this book a couple of months ago, and have read a few chapters. This book is a great introduction to topology - for someone who isn't necessarily a mathematician by trade (I come from a physics background). The book is well laid out, clear and simple explanations (something I find often lacking in Maths text books), and great value for it's low price. Definitely recommended.

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ByDavid Collinson 18 August 2016

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 5

Layout / Formatting (in the KINDLE edition): 1 or 2.

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 5

Layout / Formatting (in the KINDLE edition): 1 or 2.

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ByLewis Proctoron 1 August 2015

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.

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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ByHarsimrat Kauron 24 December 2012

A useful book to help with learning Topology which is useful for a mathematics student! Have used this book almost everyday since buying it as it is easy to follow and understand. Has examples and explanations which are really useful.

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