A book in topology - MathOverflow

A book in topology

2011-12-19T17:44:15Z

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?

Answer by Ronnie Brown for A book in topology

2011-12-19T18:17:23Z

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.

Answer by Michael Biro for A book in topology

2011-12-19T18:21:34Z

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.

Answer by Simon Rose for A book in topology

2011-12-19T18:33:55Z

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.

Answer by Joseph O'Rourke for A book in topology

2011-12-19T18:53:07Z

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.

Answer by Tom Leinster for A book in topology

2011-12-19T19:40:28Z

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.

Answer by Michael Greinecker for A book in topology

2011-12-19T19:57:04Z

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

Answer by wildildildlife for A book in topology

2011-12-19T20:22:25Z

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.

Answer by J W for A book in topology

2011-12-19T20:25:39Z

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.

Answer by Ehsan M. Kermani for A book in topology

2011-12-19T21:06:20Z

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.

Answer by unknown (google) for A book in topology

2011-12-19T21:38:10Z

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

Answer by Daniele for A book in topology

2011-12-29T16:51:52Z

I am an undergraduate student. I think that when you begin to study a new subject it is better to start from books not too broad. For a basic course in topology, I recommend these books (based on my experience as student)

J. Dugundji, Topology;
C. Kosniowski, A first course in algebraic topology;
L.C. Kinsey, Topology of surfaces.