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

Bert Mendelson (Author), Mathematics (Author)

Book Description

Publication Date: July 1, 1990 | ISBN-10: 0486663523 | ISBN-13: 978-0486663524 | Edition: 3

Highly regarded for its exceptional clarity, imaginative and instructive exercises, and fine writing style, this concise book offers an ideal introduction to the fundamentals of topology. It provides a simple, thorough survey of elementary topics, starting with set theory and advancing to metric and topological spaces, connectedness, and compactness. 1975 edition.

Product Details

• Paperback: 224 pages
• Publisher: Dover Publications; 3 edition (July 1, 1990)
• Language: English
• ISBN-10: 0486663523
• ISBN-13: 978-0486663524
• Product Dimensions: 8.4 x 5.4 x 0.4 inches
• Shipping Weight: 8 ounces (View shipping rates and policies)
• Average Customer Review: 4.8 out of 5 stars  See all reviews (17 customer reviews)
• Amazon Best Sellers Rank: #3,541 in Books (See Top 100 in Books)
  • #30 in Books > Professional & Technical > Professional Science > Mathematics

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Customer Reviews

Most Helpful Customer Reviews

112 of 115 people found the following review helpful:

4.0 out of 5 stars Good Introduction to Metric Spaces and Topology, April 11, 2003

By 

Michael Wischmeyer (Houston, Texas) - See all my reviews
(VINE VOICE)    (REAL NAME)   

This review is from: Introduction to Topology: Third Edition (Dover Books on Mathematics) (Paperback)

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.

39 of 43 people found the following review helpful:

5.0 out of 5 stars Ideal for self-study, May 4, 2002

By 

Marco Taboga [e-mail] (Torino) - See all my reviews

This review is from: Introduction to Topology: Third Edition (Dover Books on Mathematics) (Paperback)

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.

10 of 10 people found the following review helpful:

5.0 out of 5 stars Great book on Topology, January 17, 2007

By 

Brian Lauer - See all my reviews
(REAL NAME)   

This review is from: Introduction to Topology: Third Edition (Dover Books on Mathematics) (Paperback)

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.