The German mathematician Klaus Janich has a wonderful response to this question in his book on topology, which is intentionally very. Topology. Klaus Janich. This is an intellectually stimulating, informal presentation of those parts of point set topology that are of importance to the nonspecialist. Topology by Klaus Janich: Forward. Content. Sample. Back cover. Review.
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A fairly streamlined book, although initially gentle, is Essential Topology by Crossley.
It goes up to tipology and homology. I’d recommend a combination. Topology by Klaaus 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. See my web page http: It takes a geometric approach, and at the same time a categorical view, that is, there is an emphasis janoch constructing continuous functions.
The approach to the fundamental group via groupoids goes a long way beyond a first klau, 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. 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. 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.
It starts with metric spaces but ends at the same place your intended course. A point-set topology book that students seem to love is Topology without Tears by Sidney A. And it doesn’t cost anything. 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 Janifh to Topological Manifoldsit is also very student friendly. From several points of view i. Chapter 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 janidh and the examples are wonderful in pushing the theory forwards.
Both the language and presentation are modern tkpology allows for much topologgy for visualization computational topo,ogy. This book is excellent for visualization and at the same precise theoretical treatment of the subject. I do not recommend Munkres I work with both his books on manifolds and topology and the topollgy did not grasp much of the theory.
The presentation is old and tired. 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. There will be a great deal of precision and intuition all together. A book that I find very readable is “Topology” by John G. Hocking and Gail S. I have little teaching experience, but I remember being a student and based on that I believe that a few years ago I would klais also liked this book.
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In the past I have used two different books: Textbook in Problems, by O. 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 janoch good as klqus specialized books in AT such as Hatcher’s book which is free to download on Hatcher’s site.
Can you provide some more details? What have they seen and not seen yet? 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. Additionally, further courses in algebraic topology can continue using Hatcher.
It’s nice to get used to his writing style early. I am bound to recommend my book Topology and Groupoids, Ronald Brown, available from amazon. An e-version is also available from www. I should say that I chose the groupoid view in the first 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. Thank you, the book seems to be very good.
I don’t have a favourite book for the fundamental group. 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. Boas, A primer of real toopologyfor lots of fun applications of the Baire category theorem; and I see these as the main point of the theorem.
Students do find this fun.
It’s a great book to introduce applied topology, although it stops just short of using groups. Do you know, what is all of this business about having to get a password in order to print the book? 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. I only looked at the first file in each batch, trusting that the translations work the same way. 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. Willard’s book is great, but probably too advanced for the students in question. 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. I agree that Willard’s is the very best. It was helpful to me as a college sophomore taking this course because he really parses the issues cleanly: It also addresses a lot of matters like uncountable ordinals that students will likely not have seen, but which are useful in understanding the role of paracompactness, for example.
The exercises are extensive and very helpful. It makes every other book look disorganized and scattershot.
textbook recommendation – A book in topology – MathOverflow
Topology Klaus Janich This book is excellent for visualization and at the same precise theoretical treatment of the subject. Counter-examples in Topology Author?? 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.
Undergraduate Texts in Mathematics: Topology by Klaus Jänich (1994, Hardcover)
Giving that topology is very terminology-intensive, this is a real problem. I know it’s been a while and I entirely don’t expect an answer, but do you still hold this view? Reviews have said that their book is somewhat outdated, but I can’t be sure if that’s the case.
It seems to cover a large range of topics, which is nice. For a basic course in topology, I recommend these books based on my experience as student J. Kosniowski, A first course in algebraic topology; L.
Kinsey, Topology of surfaces. It is better to klays the question before giving an answer: