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.
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Customer Reviews
Most Helpful Customer Reviews
116 of 119 people found the following review helpful
Format: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.
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.
44 of 48 people found the following review helpful
Format: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.
11 of 11 people found the following review helpful
Format: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.
Most Recent Customer Reviews
Prompt delivery, great book
I ordered this book to get a little overview of topology before taking the introduction to topology course that is offered by my university. I like it so far!!! Read more
Published 2 days ago by Ben
Very fun to work through
This is a very readable and very fun book to work through.
Readers possessing mathematical maturity should be able to
finish it fairly quickly. HIghly recommended.
Readers possessing mathematical maturity should be able to
finish it fairly quickly. HIghly recommended.
Published 8 days ago by Manolis Tsakiris
Great Book
I used this for a topology class in college. It was a great book. I learn better on my own so naturally I like textbooks that are useful without additional input from a lecture or... Read more
Published 1 month ago by jc22
good review, but no answers for the exercises!
I enjoyed this book. It is a nice introduction to topology for undergraduates. It presupposes very little mathematical background. It is clear and well-written. Read more
Published 2 months ago by arpard fazakas
Great for self study
I bought this book to fill in some of the gaps in Pugh's book Real Mathematical Analysis. Chapter 2 on metric spaces was a good repetition from Pugh's presentation but chapter 3 on... Read more
Published 3 months ago by Hertzberg
Great for the price
I had one semester of real analysis prior to picking up this book. I found it's presentation of metric spaces OK, though Mendelson seems happy enough with abstract definitions... Read more
Published 3 months ago by FiziksGangsta
Topology
I am enthusiast about the organization of this book. The chapter/paragraph subdivision is clear and simple and the contents are outstanding.
Published 4 months ago by Francesco Erriques
Wonderful
This is a Wonderful book. I have my B.S. in Mathematics and I am in Graduate School for Mathematics. Read more
Published 7 months ago by John Doe
Introduction to Topology: Third Edition
Introduction to Topology: Third EditionI bought the book to review the subject. I had long ago studied set theory and analysis. I think this book is flat out wonderful. Read more
Published 7 months ago by Engineer by Training and by Preference
A solid introduction
Intended for the advanced undergraduate student with a respectable level of mathematical maturity, Mendelson begins with the necessary review of set theory. Read more
Published 18 months ago by Serious Inquirer
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Citations (learn more)
17
books
cite this book:
See all 17 books citing this book
- The Thread of Discourse (Janua linguarum) by Joseph Evans Grimes on page 38, and Back Matter
- Dictionary of Mathematical Games, Puzzles, and Amusements by Harry E. Eiss on page 251
- Lebesgue Integration and Measure by Alan J. Weir in Back Matter
- Computational Geometry in C (Cambridge Tracts in Theoretical Computer Science) by Joseph O'Rourke in Back Matter
- Information, Coding and Mathematics (The Springer International Series in Engineering and Computer Science) by Mario Blaum on page 277
See all 17 books citing this book
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