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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In particular, the motivation of compactness is the best I’ve seen. About this product Synopsis Contents: Hocking and Gail S.
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.
The exercises are superbly chosen and the examples are wonderful in pushing hanich theory forwards. I never tried printing it. Email Required, but never shown. Of course every mathematician should verify a claim until he feels comfortable that if necessary, topilogy could produce the real argument down to the atomic details. I just taught a class using it, and it was generally well liked. I should say that I chose the groupoid view in the first edition as it seemed to me more intuitive and more powerful.
This item doesn’t belong on this page. It is better to read the question before giving an answer: If your students hate that book, they will grow up. As for the uanich question of how much rigor is necessary, I think a proof should contain enough rigor to provide a high-level sketch of how one would create a mechanically checkable proof, given enough time and energy. And it doesn’t cost anything.
Nevertheless, this is the best answer I have got so klzus. There is a chapter on knots, a chapter on dynamical systems, sections on Nash equilibrium and digital topology, a chapter on cosmology. From chapter 5 and on it provides one of the most modern theoretical works in Topology and group theory and their inter-relationships.
textbook recommendation – A book in topology – MathOverflow
In fact, people communicating in this “paper currency” is one of the primary reasons I topollogy an account on this site; to resolve the questions that arise from imprecise talk. The down side of this approach is that it completely disconnects the subject from it’s geometric roots and it becomes simply another branch of algebra whose roots are utterly mysterious. A wise choise because Kosniowski’s “A first course in algebraic topology” is an user-friendly book to learn basic definitions and theorems about general topology, homotopy theory and fundamental group.
I have no doubt it will continue to undergo scrutiny in future ages.
Is this level of rigour acceptable? Looking back, I was never uncomfortable with the kind of justifications we used to give in high school calculus and this current discomfiture stems from the fact that I have taken a few courses in Analysis in between.
Or, closer to topology, I could say that collapsing the boundary of a closed disk to a point ‘clearly’ makes a sphere. RowlingHardcover No one quite seems to have figured out yet how to effectively interpolate between the 2 approaches in a textbook.
From several points of view i. Skip topologyy main content. As long as it is backed by the gold standard of rigorous proofs,the paper money of gestures is an invaluable aid for quick communication and fast circulation of ideas. Kinsey, Topology of surfaces. How much rigour is necessary?
Undergraduate Texts in Mathematics: Topology by Klaus Jänich (1994, Hardcover)
Additional Details Number of Volumes. It was helpful to me as a college sophomore taking this course because he really parses the issues cleanly: Post as a guest Name. The details of the questions you asked can be diligently unwound. For example, to describe journeys between janixh, you look at all journeys, without a special emphasis on return journeys.
It is often said against intuitive, spatial argumentation that it is not really argumentation,but just so much gesticulation-just ‘handwaving’. Janich, Topology ,page 49,translation by Silvio Levy. But since algebraic topology is so closely related to classical geometry, completely abstract reasoning would probably strip away much understanding of the sources of most of the central concepts,which I believe was Hatcher’s reason for writing the text in this manner.
Undergraduate Texts in Mathematics: Topology by Klaus Jänich (, Hardcover) | eBay
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. But, is this the right level of rigour? Show More Show Less. Also, Lagrange’s theorem is also immediate in your sense because quotienting a finite group by a subgroup is yopology the collapsing of its cosets to a point, so clearly the number of elements in the quotient is exactly the number of cosets.