A Taste of Topology (Universitext) [Paperback]

Volker Runde (Author)

Book Description

Publication Date: July 6, 2005 | ISBN-10: 038725790X | ISBN-13: 978-0387257907

This should be a revelation for mathematics undergraduates. Having evolved from Runde’s notes for an introductory topology course at the University of Alberta, this essential text provides a concise introduction to set-theoretic topology, as well as some algebraic topology. It is accessible to undergraduates from the second year on, and even beginning graduate students can benefit from some sections. The well-chosen selection of examples is accessible to students who have a background in calculus and elementary algebra, but not necessarily in real or complex analysis. In places, Runde’s text treats its material differently to other books on the subject, providing a fresh perspective.

Editorial Reviews

Review

From the reviews: "This is an introduction to set-theoretic topology … . Each section of each chapter ends with exercises and each chapter has a section with remarks of a historical nature and suggestions for further reading. I consider this ‘Taste of Topology’ an interesting universitext that will make the appetite of the students grow during the years." (Corina Mohorianu, Zentralblatt MATH, Vol. 1079, 2006) "The intention of ‘A Taste of Topology’ is to provide a first introduction to topology for students … in calculus and basic algebra. … Each chapter contains well-chosen examples and exercises and concludes with historical remarks. The presentation is based on nets instead of filters throughout, certainly to the advantage of the beginner in the field. Several novel approaches to standard material … round off the picture of a welcome and timely addition to the introductory literature on topology." (M. Kunzinger, Monatshefte für Mathematik, Vol. 150 (3), 2007) "The book is an introductory text for a study of topology. In general, the book is oriented to second-year undergraduates. It presents the basic language used in various fields of modern mathematics. The book covers classical topics of topology. … The text also presents several non-typical approaches to various topics. … The book is also a source of exercise on basic topological notions." (EMS Newsletter, September, 2006)   "…Runde does something very interesting and, I think, very useful, in the book under review: he explicitly plays down what accordingly might to the beginner appear synthetic and artificial, and plays up material which resonates with other, already familiar, mathematical notions from, for example, analysis; and he goes well beyond what is ordinarily found in a first topology course. ..The highlight of this presentation is the proof of the equivalence of having a normed space be finite dimensional, of having its closed unit ball be compact, and of having each closed and bounded subset be compact. Appearing at the end of the book in the form of an appendix this elegant characterization is a nice encore to what came before… A Taste of Topology is also rich in exercises of varying degrees of difficulty and contains excellent historical material, primarily contained in the sections labeled "Remarks" at the ends of all chapters...Manifestly Runde possesses the gifts of understatement and a light touch which contributes substantially to the book’s readability… I will now amplify this by noting that I intend to use Runde’s A Taste of Topology as the main textbook…I highly recommend the book." [Michael Berg; posted to MAA Reviews 12/1/2005]

From the Back Cover

If mathematics is a language, then taking a topology course at the undergraduate level is cramming vocabulary and memorizing irregular verbs: a necessary, but not always exciting exercise one has to go through before one can read great works of literature in the original language. The present book grew out of notes for an introductory topology course at the University of Alberta. It provides a concise introduction to set-theoretic topology (and to a tiny little bit of algebraic topology). It is accessible to undergraduates from the second year on, but even beginning graduate students can benefit from some parts. Great care has been devoted to the selection of examples that are not self-serving, but already accessible for students who have a background in calculus and elementary algebra, but not necessarily in real or complex analysis. In some points, the book treats its material differently than other texts on the subject: * Baire's theorem is derived from Bourbaki's Mittag-Leffler theorem; * Nets are used extensively, in particular for an intuitive proof of  Tychonoff's theorem; * A short and elegant, but little known proof for the Stone-Weierstrass theorem is given.

Product Details

• Paperback: 192 pages
• Publisher: Springer (July 6, 2005)
• Language: English
• ISBN-10: 038725790X
• ISBN-13: 978-0387257907
• Product Dimensions: 9 x 5.9 x 0.5 inches
• Shipping Weight: 9.6 ounces (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #1,468,199 in Books (See Top 100 in Books)

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32 of 32 people found the following review helpful:

5.0 out of 5 stars Flawless exposition, great examples, short enough to read cover to cover, January 21, 2006

By 

Stephen A. Orzel (Fenton, MI) - See all my reviews
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This review is from: A Taste of Topology (Universitext) (Paperback)

This skinny little math book from the Springer Universitext series achieves excellence on many levels. First of all, anyone familiar with the old quip "A topologist is someone who cannot tell the difference between a coffee mug and a donut" will instantly smile when they see the cover. The exposition is downright beautiful, and the organization of the material could not be more perfect. The remarkable thing is that the examples not only demonstrate the concepts, but also play a large role in the development. The choice of fonts and notation is well thought-out and, although minor, contributes greatly to the excellence of the book. One of the best features of this book is its length. With less than 200 pages, one can reasonably set a goal to read it cover to cover. The well-chosen examples not only aid in understanding, but also serve to introduce the reader to concepts from other areas of mathematics. On that note, not only those seeking an introduction to topology, but also anyone new to advanced mathematics, and in addition seasoned mathematicians who are thinking about writing books themselves, will benefit greatly from reading this book.

The author divides the material into five chapters-- 1. Set Theory, 2. Metric Spaces, 3. Topological Spaces, 4. Function Spaces, and 5. Basic Algebraic Topology. There are a number of good exmples from chapters 2 and 3 that serve to compare and contrast properties of metric spaces and topological spaces, as can be expected in any topology text, however the examples used here are interesting in their own right in other areas of math. The author uses the Zariski topology on the prime ideals of a commutative ring in many places. The reader will also meet various function spaces and see how pointwise vs. uniform convergence manifest themselves through suitably chosen topologies.

A number of unique features worth noting are the proof of the Baire category theorem, which is derived from the so called Mittag-Leffler theorem (this is probably the only introductory text which proves this), and Tychonoff's theorem is proved using nets by expressing compactness as every net has a convergent subnet. Also of interest are proofs of the Stone-Weierstrass theorem and the Arzela-Ascoli theorem. On top of all this, there is still some room left at the end to introduce some basic homotopy theory. The fundamental group is defined and covering spaces are introduced. The author proves that homotopy-equivalent spaces have isomorphic fundamental groups, shows that paths and path homotopies can be lifted, and uses this to establish that the fundamental group of the circle is isomorphic to the integers. This is used to prove the Brouwer fixed-point theorem.