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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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Shall we then abandon all intuitive arguments? Kkaus example, to describe journeys between towns, you look at all journeys, without a special emphasis on return journeys.

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

For a basic course in topology, I recommend these books based on my experience as student J. Well said, although the thing I really don’t like about intuitive handwaving is that intuition differs from person to person. It was helpful to me as a college sophomore taking this course because he really parses the issues cleanly: It also doesn’t have enough theorems and proofs to immerse oneself in the new concepts.

Jeff 3, 1 12 It takes a geometric approach, and at the same time a categorical view, that is, there is an emphasis on topollgy continuous functions.

While I can follow what he says and reproduce it in different contextsit strikes me as a reason to believe the formula rather than a proof of that 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. It is often said against intuitive, spatial argumentation that it is not really argumentation,but just so much gesticulation-just ‘handwaving’.


Very much a point-set-topology-is-a-subject-in-its-own-right kind of outlook.

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This item doesn’t belong on this page. Of course your right.

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

Nevertheless, this is the best answer I have got so far. You may also like. Essential Topology looks good, but not suitable for me. No one quite seems to have figured out yet how to effectively interpolate topopogy the 2 approaches in a textbook. If the intuitive argument cannot be rewritten in a completely axiomatic and pedantic manner where there is a completely logical progression from premises to a conclusion, then we should seriously reconsider whether or not our intuitive argument ilaus a valuable one.

textbook recommendation – A book in topology – MathOverflow

So I am thinking, maybe I should choose another book this time. A point-set topology book that students seem to love is Topology without Tears by Sidney A. Sign up or log in Sign up using Google. From chapter 5 and on it provides one of the most modern theoretical works in Topology and group theory and their inter-relationships. The students learn the concepts fast, their theoretical language to explicate honed, and their visualization skills improved.


Textbook in Problems, by O.

I’m very fond of Munkres – Topology. 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. Immediately after proving that there is no retraction from the disk onto its circle boundary, they use degree theory to analyze sudden cardiac death.

Post as a guest Name. It is far too chaotic and chatty, and one needs a lot topolofy background to appreciate the connections he draws to other areas of mathematics. Counter-examples in Topology Author??

But, is this the right level of rigour? The closest anyone’s ever come to pulling it off to me is Rotman. Janich, Topology ,page 49,translation by Silvio Levy 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 become far too axiomatic and anti-visual and that this was hurting the clarity of presentations to students.

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. Or a simple closed curve in a plane ‘clearly’ partitions it into two disjoint parts.