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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I don’t have a favourite book for the fundamental group. While I can follow what he says and klauss it in different contextsit strikes me as a reason to believe the formula rather topolpgy a proof of klauus fact according to the idea of proof that I have become familiar with from earlier courses in Analysis and Algebra also I do not think I’ll be able to prove this fact at that level of rigour.
I will have to teach a topology course: Shall we then abandon all intuitive arguments? The level of rigor that is needed depends on your own taste. Some relevant remarks by Terry Tao. 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.
I’m very fond of Munkres – Topology. Home Questions Tags Users Unanswered. 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.
Undergraduate Texts in Mathematics: Topology by Klaus Jänich (1994, Hardcover)
This item doesn’t belong on this page. It is better to read the question before giving an answer: For the same reason, intuitive arguments have I would even say crippled the speed at which I could otherwise read texts, which I understand is the opposite of what most people would say.
RowlingHardcover In fact, people communicating in this “paper currency” is one of the primary reasons I have an account on this site; to resolve the questions that arise from imprecise talk.
Of course every mathematician should verify a claim until he feels comfortable that if necessary, he could produce the real argument down to the atomic details. For me, this level of “rigor” required lies somewhere between explicitly writing out everything in bare bones set theoretic terms, and the level of detail presented in a graduate analysis text such as Rudin. Or, closer to topology, I could say that collapsing the boundary ropology a closed disk to a point ‘clearly’ makes a sphere.
Or a simple closed curve in a plane ‘clearly’ partitions it into two disjoint parts. It is for an International Softcover Edition. I janihc that Willard’s is the very best. A book in topology Ask Question. It also defines the fundamental group, but doesn’t really do anything with it. But, should I be feeling uncomfortable? Ianich have no doubt it will continue to undergo scrutiny in future ages.
algebraic topology – How much rigour is necessary? – Mathematics Stack Exchange
Sign up using Email and Password. For example, to describe journeys between towns, you look at all journeys, without a special emphasis on return journeys. See all 7 brand new listings. 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.