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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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One of the main problems I am facing with the textbook is its level of rigour. There will be a great deal of precision and intuition all together.

I never tried printing it. As long as it is backed by the gold standard of rigorous proofs,the paper money of gestures tpoology an invaluable aid for quick communication and fast circulation of ideas.

algebraic topology – How much rigour is necessary? – Mathematics Stack Exchange

There is indeed much that is wise in this quote and it really gives what I think is an excellent “rule of thumb” for determining when a “proof” in mathematics has crossed the line and really become non-rigorously vague by 21st century mathematical standards to the point is really proves nothing: Of course, as it’s stated, this isn’t an exact science.

Often algebraic topology texts assume that the reader is well acquainted with arguments of a previous course in point-set topology like this in order not to get trapped on details. I should say that I chose the groupoid view in the first edition as it seemed to me more intuitive and more powerful.


The closest anyone’s ever come to pulling it off to me is Rotman. But I think Janich has given some quite good advice to the novice here. Be the first to write a review. For a basic course in topology, I recommend these books based on my experience as student. We are using Hatcher as a textbook. The Access code or CD is not provided with these editions unless specified above. And it doesn’t cost anything.

It goes up to homotopy and homology. 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. 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. Boas, A primer of real functionsfor lots of fun applications of the Baire category theorem; and I see these as the main point of the theorem.

In particular, the motivation of compactness is the best I’ve seen.

This book is excellent for visualization and at the same precise theoretical treatment of the subject. Kanich think you’ll notice most of Hatcher’s arguments would pass this test,even if it would probably take a considerable amount of spade work to make them completely rigorous in the same sense as a real analysis or algebra proof.

It is better to read the question before giving an answer: 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. It was later said by Levy that Janich told him that this particular passage was inspired by Janich’s concerns that German mathematical academia and textbooks in particular were beginning to janih far too axiomatic and anti-visual and that this was hurting the clarity of presentations to students.


Do you know, what is all of this business about having to get a password in order to print the book?

Undergraduate Texts in Mathematics: Topology by Klaus Jänich (1994, Hardcover)

It doesn’t do any algebraic topology, though. An e-version is also available from www. No one quite seems to have figured out yet how to effectively interpolate between the 2 approaches in a textbook. The level of rigor that is needed depends on your own taste.

For me, this level of “rigor” required lies somewhere between explicitly writing topologyy everything in bare bones set theoretic terms, and the level of detail presented in a graduate analysis text such as Rudin.

One point is that the argument mixedmath is giving is something that is directly verifiable at the point-set level. I agree that Willard’s is the very best. I only looked at the first file in each batch, trusting that the translations work the same way. I don’t have a printer attached now, so I can’t actually test this, but it looks perfectly ordinary.

Very much a point-set-topology-is-a-subject-in-its-own-right kind of outlook. What have they seen and not seen yet? How much rigour is necessary?