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Counterexamples in Topology (Dover Books on Mathematics) [Paperback]

Lynn Arthur Steen , J. Arthur Seebach Jr.
4.2 out of 5 stars  See all reviews (13 customer reviews)

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Book Description

September 22, 1995 048668735X 978-0486687353
Over 140 examples, preceded by a succinct exposition of general topology and basic terminology. Each example treated as a whole. Over 25 Venn diagrams and charts summarize properties of the examples, while discussions of general methods of construction and change give readers insight into constructing counterexamples. Includes problems and exercises, correlated with examples. Bibliography. 1978 edition.

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Counterexamples in Topology (Dover Books on Mathematics) + Counterexamples in Analysis (Dover Books on Mathematics) + Elements of the Theory of Functions and Functional Analysis (Dover Books on Mathematics)
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Product Details

  • Series: Dover Books on Mathematics
  • Paperback: 272 pages
  • Publisher: Dover Publications (September 22, 1995)
  • Language: English
  • ISBN-10: 048668735X
  • ISBN-13: 978-0486687353
  • Product Dimensions: 8.4 x 5.4 x 0.5 inches
  • Shipping Weight: 9.1 ounces (View shipping rates and policies)
  • Average Customer Review: 4.2 out of 5 stars  See all reviews (13 customer reviews)
  • Amazon Best Sellers Rank: #103,946 in Books (See Top 100 in Books)

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Customer Reviews

4.2 out of 5 stars
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Most Helpful Customer Reviews
43 of 43 people found the following review helpful
5.0 out of 5 stars Essential if you want to be good in point set topology February 25, 2004
A distinct characteristic of point set topology is that it builds on counterexamples. If you thumb through any PST text, many theorems are in the form "If the space T is A,B,C, then the space is X,Y,Z". The point of point set topology (pun unintended) is too determine what A,B,C are, and to weaken the hypothesis. "Can we take condition B out? Maybe hypothesis C can be weaken considerably?" How can we answer these questions? You're right, by counterexamples. Students who want to master point set topology should know the various counterexamples, no matter how contrived or unnatural they seem. While textbooks usually present a counterexample to show why Theorem Three Point Five Oh will not work on a weaker assumption -- most students (and teachers) tend to skip these parts. A collection of counterexamples presented in this book (excellent organisation, by the way) is an essential supplement of a topology course; it enables one to 'see' between the points, so to speak.
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42 of 42 people found the following review helpful
4.0 out of 5 stars a good book to combine with a regular textbook December 3, 2002
This book has examples in it that are "missing", so to speak, from many regular topology books. It aims to shore up some of these shortcomings, with examples that the student can see and understand. There are charts and graphs, as well as a detailed explanation. Some "problems" often found in regular topology books are solved. Very few proofs, if any, are given. This is not a book meant to be studied without a regular textbook on topology, only to be used as an overall review of problems and short basic premises of topology. Use this in addition to your regular fare, but keep it close at hand when doing homework or preparing for an exam.
There are fundamentals on Cantor's Theorem, the countability or uncountability of sets, compactness, closed and bounded functions, open sets, continuity, connectedness, etc. All these are basic to topology, and this book does address them, but in a brief way. It then shows a basic overview of topology that helps greatly to understand the different fields of topology.
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30 of 30 people found the following review helpful
5.0 out of 5 stars Great Book December 11, 2001
As a graduate I encountered a book called "counter examples in analysis" which I found very useful. I always dreamed of such a book in topology, this book exceeds my dreams. It is great. It does not cover all the examples that I have used over the decades but it does cover some that I have never seen. The style is quite readable for a professional topologist. The book goes into a lot of interesting details (and some while not interesting to me would be another person). In short for me it is an essential book. The question is to whom else would this be interesting to. It is clearly of little use to a first year student and less to more advanced student. It's brand of topology is not the current cutting edge. So the audience for this book is limited to a small group and for these people it is top notch.
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20 of 20 people found the following review helpful
5.0 out of 5 stars concepts become clear November 23, 1999
This is an excellent book to really start understanding all the general topology learned in an introductory (undergrad or grad level) class. The first section of the book is basically a terminology review. The second part of the book is the real meat here and contains all the counter-examples. These spaces tend to clarify all the concepts, their differences and relative strengths and weaknesses. Of course the nice introduction to meterization theory in the appendix also adds value to the book. In short no student of topology should be without this book.
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15 of 15 people found the following review helpful
5.0 out of 5 stars a veritable mine of information.... May 27, 2004
To paraphrase Chandrasekhar's review of Watson's Bessel functions text, this is "a veritable mine of information... indispensable to those who have occasion to use point-set topology." I don't think this book is intended to be a text (& I think the authors say so), in which case it would be terrible because it doesn't explain the concepts very much. It's mostly a catalogue of every kind of set you can come up with, every kind of topology you can put on it, and what properties it has such as what T_i axioms the space satisfies, whether it's compact, para compact, etc etc. Most of the time such things are proven, but be prepared to think hard sometimes about the proofs or fill in details. I'm the kind of student where I have trouble understanding things which are highly 'counter-intuitive' so I had trouble proving things, even when I knew definitions, when I did topology for the first time last term. Once I saw this book though I got used to all the weird things in topology (like the ordered square, R in the lower-limit topology, Sorgenfrey plane, etc etc). This book is incredibly useful as a reference.
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14 of 14 people found the following review helpful
By A Customer
This is not your typical topology text. In the first part of the book, the authors give a crash course on basic point-set topology. Rather than proving theorems, the emphasis is on defining and explaining concepts, especially as the various concepts relate to each other. The explanations are not always sufficient in themselves for the student's understanding, but that wasn't the book's mission. In the second part, the book provides "Counterexamples": quite a few topologies, both the predictable and the quirky kinds. The topologies given vary considerably in level of difficulty. Thirdly, several pages at the end provide charts showing which properties the book's listed topologies have. The reader can use these charts to find a suitable topology for many applications or disproofs. Counterexamples in Topology is both useful and enjoyable, particularly for people who benefit from charts and outlines. It is not a book to be plowed through, page by page, like a textbook. Rather, it is a compendium of several interesting cases, any of which can be studied independently of the rest.
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Most Recent Customer Reviews
5.0 out of 5 stars Good book to have around
I purchased this (on my own, not as a recommended or required text) for an introductory undergraduate topology class I took around a year ago. Read more
Published 16 months ago by Josh Maddux
4.0 out of 5 stars A Really Cheap yet Great Resource for Topology
I really like the compactness of this book; that is, this book is the finite subcover for any arbitrary cover on the subject of Topology. Read more
Published on November 24, 2011 by Verrückter
4.0 out of 5 stars Excellent book but not without mistakes
This is no more and no less than a catalogue (which may or may not be "complete" as far as it goes) of instances of point-set topology. Read more
Published on April 17, 2011 by Matt Westwood
4.0 out of 5 stars Resource for deep knowledge of Point-Set Topology
This book is a must have for anyone who wants to contribute to remaining questions in point-set topology. Read more
Published on February 4, 2009 by Biblio
1.0 out of 5 stars Counterexamples in Topology
I have found this book to be confusing to use and therefore of little to no value. If I had seen in a bookstore and not Online I would not have purchased it. Read more
Published on March 27, 2007 by Topology Student
4.0 out of 5 stars A book on Topolgy with a map
This book gave me an inspiration: it isn't the best written or the best organized, just one of the best topology books I've ever found! Read more
Published on December 10, 1999 by Roger Bagula
5.0 out of 5 stars A very cool catalogue of topologies.
On its own, this is not a good book to learn topology from. When combined with a standard textbook in topology (such as Baum's book) it makes an invaluable guide for the student. Read more
Published on June 11, 1998 by Michael C Smith
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