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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It is often said against intuitive, spatial argumentation that it is not really argumentation,but just so much gesticulation-just ‘handwaving’. Does that make sense? Can you provide some more details?
algebraic topology – How much rigour is necessary? – Mathematics Stack Exchange
A point-set topology book that students seem to love is Topology without Tears by Sidney A. It seems to cover a large range of topics, which is nice. 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.
Janich, Topology ,page 49,translation by Silvio Levy.
Undergraduate Texts in Mathematics: Topology by Klaus Jänich (1994, Hardcover)
See all 7 brand new listings. The exercises are extensive and very helpful. How much rigour is necessary?
You may also like. For a basic course in topology, I recommend these books based on my experience as student J.
Hocking and Gail S. Topology Klaus Janich This book is excellent for visualization and at the same precise theoretical treatment of the subject.
Well said, although the thing I really don’t like about intuitive handwaving is that intuition differs from person to person. From several points of view i. This item doesn’t belong on this page. A book in topology Ask Question. It starts with metric spaces but ends at the same place your intended course. See details for additional description.
textbook recommendation – A book in topology – MathOverflow
While this is intuitively clear, it requires some work to prove. Or, closer to topology, I could say that collapsing the boundary of a closed disk to a point ‘clearly’ makes a sphere. I will have to teach a topology course: But I think Janich has given some quite good advice to the novice here. In the past I have used two different books: Sign up using Facebook.
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. From chapter 5 and on it provides one of the most modern theoretical works in Topology and group theory and their inter-relationships.
I’m assuming that the students are not familiar with point-set topology and it’s tpology first course in topology for them.