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Introduction to Topology: Third Edition (Dover Books on Mathematics) [Englisch] [Taschenbuch]

Bert Mendelson , Mathematics
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  • Taschenbuch: 224 Seiten
  • Verlag: Dover Pubn Inc; Auflage: 0003 (1. Juli 1990)
  • Sprache: Englisch
  • ISBN-10: 0486663523
  • ISBN-13: 978-0486663524
  • Größe und/oder Gewicht: 21,4 x 13,8 x 1,1 cm
  • Durchschnittliche Kundenbewertung: 5.0 von 5 Sternen  Alle Rezensionen anzeigen (1 Kundenrezension)
  • Amazon Bestseller-Rang: Nr. 26.701 in Fremdsprachige Bücher (Siehe Top 100 in Fremdsprachige Bücher)

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1 von 2 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen A great book, especially for the price 16. Mai 2000
I know that some people don't like Dover, but I think Dover is great, and Mendelson's Introduction of Topology is an example of why.
Although the book is very short (around 150 pages), it covers the basics of topology very thoroughly and should prepare the reader for the considerably more abstruse Spanier's Algebraic Topology or other texts of such ilk.
If you are a recreational topologist, or are simply tryinging to figure out which way is up in your first topology course, this is for you.
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Die hilfreichsten Kundenrezensionen auf (beta) 4.6 von 5 Sternen  33 Rezensionen
151 von 154 Kunden fanden die folgende Rezension hilfreich
4.0 von 5 Sternen Good Introduction to Metric Spaces and Topology 11. April 2003
Von Michael Wischmeyer - Veröffentlicht auf
I was not a mathematics major, and only in recent years have I ventured into abstract mathematics. I was motivated to learn about topology as an aid to understanding a particular 3-D earth modeling application.

I read Introduction to Topology in three stages: as a review of set theory and metric spaces (chapters 1 and 2), then as an introduction to topology (chapter 3), and lastly as a detailed look at two important topological properties, connectedness (chapter 4) and compactness (chapter 5). I had previously read (and reviewed) another book titled Metric Spaces by Victor Bryant, but Mendelson is my first serious look at topology.

My reading of Mendelson - a 200-page text - required about 100 hours, substantially longer than the 40 to 60 hours estimated by an earlier reviewer. No solutions are provided for the section problems, which are generally proofs, not explicit problems.

The first chapter provides a concise overview of set theory and functions that is essential for Mendelson's later chapters on subsequent set-theoretic analysis of metric spaces and topology.

The second chapter is a solid introduction to metric spaces with good discussions on continuity, open balls and neighborhoods, limits from a metric space perspective, open sets and closed sets, subspaces, and equivalence of metric spaces. Chapter 2 concludes with a brief introduction to Hilbert space.

The third chapter introduces topological spaces as a generalization of metric spaces, and many theorems are largely restatements of the metric space theorems derived in chapter 2. I was thankful for this approach.

Mendelson begins chapter 3 by demonstrating that 1) open sets and neighborhoods are preserved in passing from a metric space to its associated topological space and 2) the existence of a one to one correspondence between the collection of all topological spaces and the collection of all neighborhood spaces.

He then reminds us that in a metric space we can say that there are points of a subset A arbitrarily close to a point x if the metric d(x, A) = 0. In characterizing this notion of arbitrary closeness in a topological space, Mendelson introduces the closure of A, the interior of A, and the boundary of A. Other topics included topological functions, continuity, homeomorphism (the equivalence relation), subspaces, and relative topology. The final sections in chapter 3 on products of topological spaces, identification topologies, and categories and functors were more difficult.

In chapter 4 the initial sections (connectedness on the real line, the intermediate value theorem, and fixed point theorems) were largely familiar. But thereafter I became bogged down with the discussions of path-connected topological spaces, especially with the longer proofs involving the concepts of homotopic paths, the fundamental group, and simple connectedness.

Chapter 5, titled Compactness, was even more abstract and difficult, with topics like coverings, finite coverings, subcoverings, compactness, compactness on the real line, products of compact spaces, compact metric spaces, the Lebesgue number, the Bolzano-Weierstrass property, and countability. Perhaps, a reader more familiar with analysis would have less difficulty with the last two chapters.

In summary, Introduction to Topology is quite useful for self-study. Mendelson's short text was intended for a one-semester undergraduate course, and it is thereby ideal for readers that either require a basic introduction to topology, or need a quick review of material previously studied. The last two chapters on connectedness and compactness are substantially more difficult, but are still accessible to the persistent reader.
60 von 64 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Ideal for self-study 4. Mai 2002
Von Marco Taboga - Veröffentlicht auf
This book is ideal for self-study. If you have not had the luxury of taking a topology course during your undergraduate studies, but you need to know some topology and you have to study it by yourself, this is the book you need. It is very readable and it explains carefully every concept. However, it is just an introductory text and it contains only basic material. You don't have to invest a lot of time to study the material in this book: let's say 40-60 hours of study are enough to grasp everything. I reccomend it especially to those graduate students of applied mathematics, finance, statistics or economics, who need to use some basic result from topology in their work.
27 von 28 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Great book on Topology 17. Januar 2007
Von Brian Lauer - Veröffentlicht auf
I bought this book for my own enlightenment after already having a course in Topology here at Penn State University. What I find most interesting about this book is that the author explains the philosophy on the ideas and what we are really trying to say with these definitions and theorems. The book I used in my course didn't explain much at all so it would have been much more difficult to teach yourself from this book. Topology is somewhat abstract so if you're looking to study Topology this is a great book to start. A word of advice, read over a theorem and proof and try to reproduce it on paper from your mind. Help yourself from the book a bit along the way if necessary. You will learn much more this way as opposed to following along the proofs in the book as you read. You might also be interested in Counterexamples in Topology, a book with thousands of counterexamples.
13 von 13 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen very mindful of the student 1. Juli 2009
Von amr - Veröffentlicht auf
I highly recommend this book. The problems are excellent. They really hit home and force you to truly understand the content. They get to the crux of the issues (some problems specifically test to make sure you didn't misinterpret a definition for example) and they're also interesting.

The book is carefully written in a simple style. It's a bit hard to explain... For lack of a better explanation, an analogy would be to how Mac computers are simple to use but not lacking in function. One specific example that I can pinpoint is that the author avoids using symbols excessively.

It is not a "layman" book at all however. Some problems take a lot of thinking. Some of them take me a few hours of scribbling in my notebooks and some of them take a few days of mulling over on top of that. But I'm not a math student or math practitioner (only a hobby at this point) so mathematicians-to-be should have an easier time than I.
9 von 9 Kunden fanden die folgende Rezension hilfreich
4.0 von 5 Sternen A solid introduction 7. November 2010
Von Serious Inquirer - Veröffentlicht auf
Format:Taschenbuch|Verifizierter Kauf
Intended for the advanced undergraduate student with a respectable level of mathematical maturity, Mendelson begins with the necessary review of set theory. From there the book delves into metric spaces, topological spaces, connectedness, and compactness. In short, it presents the basics of topology in a clear, linear, very readable fashion.

Readers would be well advised to be familiar with the elements of proof, set theory, linear algebra, and abstract algebra in addition to analysis. A knowledge of geometry is also helpful, as one might expect.

Weighing the price of this book against the depth and breadth of other texts, this volume offers more to the student who is studying topology on a budget. Unfortunately, as with most books in this category, there is no solution guide provided for the exercises. A selection of hints for the exercises would have been a nice addition but otherwise does not detract from the purpose of the work: to give the beginning topologist an overview of the subject.
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