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Topology (Undergraduate Texts in Mathematics) 1st ed. 1984. 2nd Corr. printing 1994 Edition

by Klaus Jänich (Author), S. Levy (Translator)
5 out of 5 stars 6 customer reviews
ISBN-13: 978-0387908922
ISBN-10: 0387908927
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Product Details

  • Series: Undergraduate Texts in Mathematics
  • Hardcover: 193 pages
  • Publisher: Springer; 1st ed. 1984. 2nd Corr. printing 1994 edition (January 1, 1995)
  • Language: English
  • ISBN-10: 0387908927
  • ISBN-13: 978-0387908922
  • Product Dimensions: 6.1 x 0.6 x 9.2 inches
  • Shipping Weight: 15.2 ounces (View shipping rates and policies)
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (6 customer reviews)
  • Amazon Best Sellers Rank: #874,572 in Books (See Top 100 in Books)

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27 of 27 people found the following review helpful
great as motivation but not a textbook
By Malcolm on May 3, 2008
While I agree with the other reviewers here that Jaenich's "Topology" is very well written, goes to great lengths to explain the "hows and whys" of topology, and includes many, many figures (about 1 per page on average), it is probably more popular with people who already know topology than with beginning students, even though it is an introductory text intended for undergraduates. This is due to both a frequent lack of precision or formality in proofs and definitions coupled with a tendency to discuss much more advanced material with which a student at this level wouldn't be familiar. I believe that experienced mathematicians, who perhaps learned point-set topology from books such as those of Munkres, Kelley, or Bredon (or even an analysis book such as Royden), appreciate how this book focuses on motivating the concepts, explaining how the various objects are used elsewhere in mathematics - for that purpose this is one of the finest books I have seen. However, too much material is mentioned that is certainly over the heads of most students new to topology, such as the Pontrjagin-Thom construction, the spectrum of commutative Banach algebras, or Lie groups, often in a very cursory manner that would serve only to confuse beginners. Concepts are often used before they are defined, or are not defined precisely, which is liable to frustrate these students as well. Many topics are given such short attention it makes you wonder why the author even bothered - such as a page devoted to Frechet spaces followed by a section consisting of a single paragraph on locally convex topological vector spaces. Much of the material is not covered very deeply - only a definition and maybe a theorem, which half the time isn't even proved but just cited.Read more ›
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24 of 24 people found the following review helpful
Students: BUY THIS BOOK!!!
By Dr. Lee D. Carlson HALL OF FAMEVINE VOICE on August 8, 2002
It is not too often that a book about topology is written with the goal of actually explaining in detail what is going on behind the formalism. The author does a brilliant job of teaching the reader the essential concepts of point set topology, and the book is very fun to read. The reader will walk away with an appreciation of the idea that topology is just not abstract formalism, but has an underlying intuition that is rich in imagery. The author has a knack for allowing readers to "see into the future" of what kind of mathematics is waiting for them and how topology is indispensable in its study.
At the end of chapter three, which deals with the quotient topology, the author writes the following paragraph: "If is often said against intuitive, spatial argumentation that it is not really argumentation but just so much gesticulation - just 'handwaving'. Shall we then abandon all intuitive arguments? Certainly not. 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. Long live handwaving!". This has to rank as one of the best paragraphs that has every appeared in a mathematics book, for it nicely summarizes the need for developing a feel for the concepts behind mathematics before moving on to the rigorous proofs. Physicists in particular, who must assimilate mathematics very quickly in order to apply it to real problems must have a pictorial, "playful" understanding of the mathematical constructions.
Thus the language that the author employs is informal, and a listing of the best discussions in the book would really entail a listing of every one in the book. There is not one part of the book that is not helpful or interesting, and the author delves into many different areas that involve the use of topology.
If you are a beginning student in mathematics, BUY AND STUDY THIS BOOK...BUY AND STUDY THIS BOOK. You will take away so much for the price paid.
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18 of 19 people found the following review helpful
Full of motivations
By A Customer on August 28, 2003
This book is fun to read. In a weekly homework meeting for an Algebraic Geometry class, I complained to one grad student "Geometry textbooks should have many pictures", and he asked "Define 'many'?" I said "One on each page". Now this topology book is certainly close to that. (It has more than 180 illustrations.) Though its written style is a bit informal, 'handwaving' arguments can serve as outlines of rigorous proofs.
Since it does not have any problem sections, I can see why Munkres' book is more popular in college. It still gives some inspiring questions from time to time. Besides the basic pot-set topology, it also covers some algebraic topology and differential topology. The author does not hesitate to use examples from those advanced areas without formal definitions, and this was a bit annoying when I read it the first time. In this sense, the book is not really selfcontained. However, when finally a notion is formally defined, I can see it from various aspects in those examples. This really helps me understand topology better, and makes me want to explore them. After reading the existence thm of covering spaces in chapter 9, I realized that mathematics is really an art.
The index in the back of the book is in the format of short definitions, which can be used as a quick reference.
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