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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textbook recommendation – A book in topology – MathOverflow
Of course, as tpoology stated, this isn’t an exact science. Is this level of rigour acceptable? From several points of view i. And it doesn’t cost anything.
I have little teaching experience, but I remember being a student and janifh on that I believe that a few years ago I would have also liked this book. From chapter 5 and on it provides one of the most modern theoretical works in Topology and group theory and their inter-relationships.
For example, to describe journeys between towns, you look at all journeys, without a special emphasis on return journeys. Even with an American printer, it looks like I could print it with no more trouble than funny margins. Shall we then abandon all intuitive arguments? It goes up to homotopy and homology. 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.
I almost always just tune out when people offer me their intuitions, and develop my own, for this reason. The lowest-priced brand-new, unused, unopened, undamaged item in its original packaging where packaging is applicable.
Illustrated Mathematics Hardcover Books. Jeff 3, 1 12 Home Questions Tags Users Unanswered. The topoogy of the questions you asked can be diligently unwound. Also, Lagrange’s theorem is also immediate in your sense because quotienting a finite group by a subgroup is precisely the collapsing of its cosets to a point, so clearly the number of elements in the quotient is exactly the number of cosets.
Well said, although ianich thing I really tpology like about intuitive handwaving is that intuition differs from person to person. It is far too chaotic and chatty, and one needs a lot of background to appreciate the connections he draws to other areas of mathematics. A point-set topology book that students seem to love is Topology without Tears by Sidney A. My answer was you should not change your first choise.
So I am thinking, maybe I should choose another book this time. See details for additional description.
Undergraduate Texts in Mathematics: Topology by Klaus Jänich (1994, Hardcover)
It was helpful to me as a college sophomore taking this course because he really janjch the issues cleanly: This item doesn’t belong on this page. I might be toopology you, but I think that you are saying that you do not like my suggestion that cell-complex quotients are immediately viewed as a wedge sum of n-spheres.
Or a simple closed curve in a plane ‘clearly’ partitions it into two disjoint parts. 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. I am bound to recommend my book Topology and Groupoids, Ronald Brown, available from amazon. It is better to read the question before giving an answer: It’s nice to get used to his tolology style early.
Topology Klaus Janich This book is excellent for visualization and at the same precise theoretical treatment of the subject. Additionally, further courses in algebraic topology can continue using Hatcher. I’d recommend a combination. While this is intuitively clear, it requires some work to prove.
For a basic course in topology, I recommend these books based on my experience as student J.