# A book in topology

I will have to teach a topology course: it starts in point set topology and ends at fundamental group of $S^1$.

In the past I have used two different books:

• Elementary Topology. Textbook in Problems, by O.Ya.Viro, O.A.Ivanov, V.M.Kharlamov and N.Y.Netsvetaev.
• A First Course in Algebraic Topology by Czes Kosniowski

I like both of these books and my students hate both of them. So I am thinking, maybe I should choose another book this time.

Any suggestions?

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I'm very fond of Munkres - Topology. It covers all the usual point set topology and some dimension theory. Although the second part of the book dealing with Algebraic Topology is not as good as other specialized books in AT such as Hatcher's book (which is free to download on Hatcher's site). –  Asaf Dec 19 '11 at 17:55
Can you provide some more details? What year / level / major are these students? What have they seen and not seen yet? –  Qiaochu Yuan Dec 19 '11 at 20:29

A fairly streamlined book, although initially gentle, is Essential Topology by Crossley. It goes up to homotopy and homology. See also Celebrating Swansea University Authors to view Crossley talking about his book.

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Essential Topology looks good, but not suitable for me. Crossley does not think that fundamental group could be the highest point in the course. Nevertheless, this is the best answer I have got so far. Thank you very much. –  ε-δ Jan 9 '12 at 0:24

I'd recommend a combination. Topology by Munkres for the point set stuff, and Algebraic Topology by Hatcher for the algebraic topology. You get all the advantages of two more specialized textbooks, and since Hatcher's text is free, your students won't need to buy two textbooks.

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Additionally, further courses in algebraic topology can continue using Hatcher. It's nice to get used to his writing style early. –  Zack Wolske Dec 19 '11 at 20:19
and for Alg.Top., you can couple Hatcher's textbook with Munkres' Elements of Algebraic Topology (since Hatcher takes a geometric approach and Munkres takes an algebraic one). –  Chris Gerig Dec 19 '11 at 21:17

I am bound to recommend my book

Topology and Groupoids, (2006) Ronald Brown,

available from amazon.com . An e-version is also available from www.kagi.com for £5.

See my web page http://www.bangor.ac.uk/r.brown/topgpds.html for links to reviews.

It takes a geometric approach, and at the same time a categorical view, that is, there is an emphasis on constructing continuous functions. The approach to the fundamental group via groupoids goes a long way beyond a first course, but then the results go beyond other books, for example on the fundamental group(oid) of an orbit spaces, and a gluing theorem on homotopy equivalences.

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I should say that I chose the groupoid view in the first 1968 edition as it seemed to me more intuitive and more powerful. For example, to describe journeys between towns, you look at all journeys, without a special emphasis on return journeys. –  Ronnie Brown Dec 19 '11 at 18:45

Introduction to Topology: Pure and Applied, by Colin Adams and Robert Franzosa. Immediately after proving that there is no retraction from the disk onto its circle boundary, they use degree theory to analyze sudden cardiac death. There is a chapter on knots, a chapter on dynamical systems, sections on Nash equilibrium and digital topology, a chapter on cosmology.

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It's a great book to introduce applied topology, although it stops just short of using groups. –  J W Dec 19 '11 at 20:19

I'm fond of Wilson Sutherland's book Introduction to Metric and Topological Spaces. It covers topics such as completeness and compactness extremely well. In particular, the motivation of compactness is the best I've seen. (It doesn't do any algebraic topology, though.) I just taught a class using it, and it was generally well liked.

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+1: Sutherland is where I learned point-set –  Yemon Choi Dec 19 '11 at 19:46
Thank you, the book seems to be very good. But, what would you choose for $\pi_1(S^1)$ from it? –  ε-δ Dec 20 '11 at 21:35
I don't know. I don't have a favourite book for the fundamental group. –  Tom Leinster Dec 21 '11 at 19:10
I like a book with lots of examples of applications of major theorems. So as part of a course in analysis I used as a source R.P. Boas, A primer of real functions, for lots of fun applications of the Baire category theorem; and I see these as the main point of the theorem. It is difficult to find a book at this level which also does in a basic and example oriented way the Hausdorff metric on compact subsets of $\mathcal R ^n$; there is an argument that graduands in maths should have heard of this background to fractals and chaos theory. Students do find this fun. –  Ronnie Brown Dec 30 '11 at 18:04

The notes from when I learned topology were eventually published as a UTX book called "A taste of topology" by Volker Runde. It starts with metric spaces but ends at the same place your intended course.

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A point-set topology book that students seem to love is Topology without Tears by Sidney A. Morris. And it doesn't cost anything.

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Do you know, what is all of this business about having to get a password in order to print the book? I can see that the second batch of files[^1] is encrypted, but I can't see what stops me from printing the first batch. (I don't have a printer attached now, so I can't actually test this, but it looks perfectly ordinary. Even with an American printer, it looks like I could print it with no more trouble than funny margins.) [^1]: I only looked at the first file in each batch, trusting that the translations work the same way. –  Toby Bartels Dec 31 '11 at 6:17
I actually don't know. I only know that in a course I was a TA for, all student used this book as their reference for point set topology, instead of the assigned text. I never tried printing it. –  Michael Greinecker Dec 31 '11 at 6:40

Willard's General Topology is my favourite book on point-set topology (together with Bourbaki, but the latter is not suited as course text for several reasons). It also defines the fundamental group, but doesn't really do anything with it.

More geometric is Lee's Introduction to Topological Manifolds, it is also very student friendly.

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Willard's book is great, but probably too advanced for the students in question. –  Michael Greinecker Dec 19 '11 at 20:23
I took the course from Willard and found it fine. The textbook is very efficient and encyclopaedic. Very much a point-set-topology-is-a-subject-in-its-own-right kind of outlook. It's not designed for a very general audience. But for students that have had a strong set theory or analysis course(s) beforehand, it's a great book. –  Ryan Budney Dec 19 '11 at 23:57

From several points of view i.e. group theory and computability and visualization I suggest 3 books:

1.Topology and Groupoids

Prof Ronnie Brown

Chapter 1-4 are one of the best approaches to the topology I have ever seen. The students learn the concepts fast, their theoretical language to explicate honed, and their visualization skills improved. From chapter 5 and on it provides one of the most modern theoretical works in Topology and group theory and their inter-relationships. The exercises are superbly chosen and the examples are wonderful in pushing the theory forwards. Both the language and presentation are modern and allows for much room for visualization computational development.

2.Topology

Klaus Janich

This book is excellent for visualization and at the same precise theoretical treatment of the subject.

3.Counter-examples in Topology

Author?? (book is not with me right now)

Lots of weird spaces, really great to flex muscles for the topological bodybuilders.

I do not recommend Munkres I work with both his books on manifolds and topology and the students did not grasp much of the theory. The presentation is old and tired.

Dara

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Jänich is gorgeous - there is no way your students won't like it! –  Peter Arndt Dec 20 '11 at 0:11
Jänich is great for revisiting topology, but I don't think it is a good book for learning the material the first time. It is far too chaotic and chatty, and one needs a lot of background to appreciate the connections he draws to other areas of mathematics. It also doesn't have enough theorems and proofs to immerse oneself in the new concepts. Giving that topology is very terminology-intensive, this is a real problem. –  Michael Greinecker Dec 20 '11 at 5:39
Counterexamples in Topology, a real classic, is by Lynn Steen and Arthur Seebach. –  Toby Bartels Dec 31 '11 at 6:19

I'm assuming that the students are not familiar with point-set topology and it's the first course in topology for them. I'd recommend a combination of Munkres and Intuitive topology by V. V. Prasolov. There will be a great deal of precision and intuition all together.

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A book that I find very readable is "Topology" by John G. Hocking and Gail S. Young. I have little teaching experience, but I remember being a student and based on that I believe that a few years ago I would have also liked this book.

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