3 comments about Teach Yourself Calculus.

1. Hundreds of Calculus texts claim to teach this rich field of mathematics easily or well. Most texts send students running out of class. As an educator, my goal is to explain all of the fundamental techniques of Calculus, with their proofs, in a manner and pace that a tenth grade inner-city student can comprehend. I believe this is quite feasible using this book as a comprehensive guide if the teacher loves math, uses the text, and loves the students. The self-tests are cogent and progress gradually in complexity - keeping pace with the growing power of the chapters. This is the first book I reach for when a Geometry or Algebra student asks me "Do you think I could ever take Calculus?" You bet! Hugh Neill's masterful "Teach Yourself Calculus". I recommend it without reservation. N. Montero, MD



2. I was looking for a little book of calculus I could carry around with me, my textbook from school is just too big. I knew that such a book wouldn't cover every single topic in calculus, but if it covered the major stuff and had good presentation, I would be pleased, since the rest could be filled in using online resources, etc.
   I saw this book at the bookstore, and thought it looked pretty decent, it was only $13, so I bought it. I'm glad I did, the introduction to e and natural logarithms is the best I've seen. The author is able to explain concepts in the most clear and concise manner, such as is rarely seen.
   This book is the best little book of calculus I've seen.



3. I have not yet finished the book; I do think it is clearly written.
   
   I would have awarded four stars but due to some incorrect answers that cause one to waste time trying to find where the reader (me) made a mistake, a penality of one star is assessed.
   
   There is nothing more frustrating.

5 comments about Pre-Algebra Demystified.

1. "Demystified" is more like "profoundly confusing." It's just another book rushing through problems with little or no explanation. It really angers me that there are so many books like this that promise enlightenment but provide headache. Maybe it'll work for you, but if you're like me and REALLY need this stuff taught SLOWLY step-by-step (meaning no steps skipped), then try another book. Upon reading this book, I felt as though it assumed I had experience in the subject it was supposed to introduce me to. Try Painless Algebra by Lynette Long-while not flawless, it is the best I have used thus far and has brought me, a math loser, into the game.



2. Although the DeMystified series promises to demystify math, it helped me in some areas;however, with this particular title I found the book seriously weak on practice problems. I bought it in order to brush up on my basic math skills, but in my experience with it, it wasn't enough to throw out five or ten practice problems per section and assume that I'd know the concepts well. Some, like me, need to be able to practice quite a few problems to get the idea or just to be sure that I understand the math concept correctly. I found this to be the books greatest weakness.
   
   One would do better to just study from their class texts since these works often do give you the practice and detailed explanation you may need. If one is looking to cram (shame on you), then this book may work. However, then again, it may not since there are so few practice problems.



3. HAHA! Ok, I'll explain why I am laughing. I have been working on this book and everything has been good until chapter 11 page 211. I remembered when I first bought this book, I read this comment and thought "ok the guy (S.Brown) just doesn't know how to answer the problem" I came on here looking for this comment to see if it was the same comment I had. That is why I am laughing. The author really does just give up after chapter 10! Although there are a few errors in this book, it's a pretty ok book. Considering I didn't even know my multiplication table before I started this book and now I can pass math tests easily. Yah! Except the graphing. Luckily I have three other math books that can help with this.



4. I bought this book (and a bunch of others) because I wanted to prepare myself for a college math assessment test and score high enough that I could jump straight into statistics (I know, pretty ambitious!).
   
   My current situation?
   
   1) I haven't studied algebra (or any math at all for that matter!) in about 20 years
   
   2) I didn't learn math in English
   
   So, after opening the first algebra book and realizing I didn't remember how to do even more basic operations, like dividing large numbers by large numbers, I decided I needed to get something that went even further back.
   
   I bought about 10 books, thinking that it was a good investment, as skipping the basic math classes at about $300, plus the cost of books, for each, would save me quite a bit of money.
   
   After reading about 40 pages of the first basic math book, I was very disappointed. It was unclear, complicated and just didn't make much sense to me. I started to wonder if things had changed THAT much since I used to study. I put the book aside and pulled this one from the pile. Well, it has 12 chapters and I have completed 8 so far. The book is great! I read some bad reviews and, who knows, maybe chapters 9, 10, 11, and 12 are rubbish... But I doubt it!
   
   The approach is logical, practical and easy to follow. There are quite a few exercises and tests at the end of each chapter, as well as a complete test at the end of the book. I usually complete a couple of chapters, then take all the chapter-end tests from the previous chapters again before proceeding. This way, the information stays fresh in my mind.
   
   I liked this book so much that I bought about 4 or 5 more books from the DeMystified collection, including Algebra and Math Word Problems (the latter by the same author). Ok, so maybe I buy books compulsively at times, but I am very excited about math, probably for the first time in my life.
   
   If you need to take a refresher course, or if you need support for classes you are taking, I truly recommend this book!
   
   Good luck!



5. Thank you, Mr. Bluman. You have cracked into my brain, and now I can do math, without a calculator, without feeling like my head will explode! Mr. Bluman is a gifted, no-nonsense teacher who can teach anyone math with this straightforward, brilliantly crafted self-teaching manual. He even tackles math phobia -it is real- and dispatches it effectively with tips to get yourself going every day. This book should be in every junior high and high school library. He gives you the tools and then with practice, you can solve the problems. The format of explanations step-by-step, repetition, practice, and practice tests and final tests in every chapter is an excellent approach to truly "getting it". Thanks a million!

3 comments about MP: Basic Mathematical Skills with Geometry.

1. The book and the publisher's MathZone show a nice attempt at integrating the power of the Internet with a traditional maths text. The material is for high school readers. You can of course treat the book just as a conventional text, and refrain from accessing MathZone. In this respect, the book is well polished, being in its 6th edition, and very logically internally consistent. As befits Euclidean geometry.
   
   Now if you do want to use MathZone, what to do at the website? Perhaps the most fruitful approach, if you are disciplined enough, is to take those tests offered there. In addition to doing the exercises in the book, of course. The tests are a valuable metric of how well you comprehend the material. The authors and publisher have put a lot of time into MathZone. Go for it!



2. I have the third edition which I used in college back in the mid- 1990's. Of all the math books that I have used, This text is my favorite. The authors of this text really make math fun and easy to learn. I highly recommend this book to all who just haven't understood math and hate it. This book will make math fun.



3. I am still waiting for my product. I emailed the seller asking when I should expect it to arrive. The delivery dates were between July 22 to approximately August 13. The seller emailed on August 13 to say she was out of town and asked if I still wanted the textbook. I replied to the email stating that "Yes, I still need the book". I emailed her two more times since then, no response from her and still no book. I will wait until August 25 to see if the book arrives. If not, I will refute the charges.

3 comments about Sacred Mathematics: Japanese Temple Geometry.

1. For anyone who truly loves mathematics, this book is a must have.
   Simply put, the book tells the story of sangaku, geometry problems which were painted in color on wooden tablets and displayed at Buddhist temples and Shinto shrines throughout Japan. Most of the sangaku were composed by people from all walks of life-priests, farmers, children women, samurai, etc.-between 1600 and 1900. Approximately 900 of the old tablets have survived and even today one is occasionally found at an abandoned temple/shrine. Tony Rothman has assisted Mr. Fukagawa Hidetoshi, a retired Japanese high school teacher, who is one of the world's foremost experts in sangaku, in producing a beautiful book. Various chapters discuss Japan and temple geometry, the Chinese foundation of mathematics, Japanese mathematics and mathematicians of the Edo period. In addition, the book contains over 200 sangaku problems ranging from very elementary to extremely difficult. The book also contains extensive excerpts from the diary of Yamaguchi Kanzan, a Japanese mathematician, who treked through Japan during the 1800s collecting sangaku problems. Finally, there are chapters on East and West, Japanese attempts to handle differentiation and integration, and inversion. The book contains numerous diagrams which accompany the problems and there are 16 color plates. In summary, this book captures a beautiful form of vanished mathematics which was artistic/religious in nature. Mr. Fukagawa Hidetoshi and Mr. Rothman are to be congratulated for producing a superb book which tells the story of this vanished mathematical/religious art form. Buy your copy today. This book contains enough history, mathematics, art, and religion to keep one's intellect perplexed for years.



2. I am always interested in what Tony Rothman has to say. He is the real deal, teaches physics at Princeton, Harvard, etc., who comes up with revolutionary insights you just can't find anywhere else. SACRED MATHEMATICS is a revelation and a tremendous challenge, another brilliant one in this writer's repertoire.
   
   I began my Rothman studies after reading INSTANT PHYSICS, which pretty much brought me up to speed in what had always intrigued yet baffled me. Then I was amazed with his majestic DOUBT AND CERTAINTY followed by the jaw-dropping, myth-busting EVERYTHING'S RELATIVE. I couldn't get enough so I started backtracking and discovered the Pulitzer Prize nominated A PHYSICIST ON MADISON AVENUE and SCIENCE A LA MODE, where he maybe first established his continual theme of treating science with the skeptical irreverence it often deserves. In between, I discovered articles in SCIENTIFIC AMERICAN, DISCOVER, ISAAC ASIMOV'S SCIENCE FICTION MAGAZINE and THE NEW REPUBLIC, not to mention some weighty scientific papers and reports. Finally, I found his science fiction novel, THE WORLD IS ROUND, with which the movie industry might finally have the tools to do justice.
   
   Tony Rothman is a great and gifted writer and SACRED MATHEMATICS is a beautifully illustrated book of art, religion, history and geometry. I see from his web site that a novel about The Great Seige of Malta is next. I anxiously anticipate that and hope that both APOCHRYPHA and the plays there mentioned will soon be published.
   
   I strongly recommend SACRED MATHEMATICS and, in fact, everything written by Tony Rothman to anyone, who in a world too often full of nonsense and lies, cherishes instead reality and truth. Rothman's voice is beautiful and unique.



3. The last (for the moment) title of Fukagawa&Rothman is really excellent. Not only the printing is superb, but the mathematical content is also outstanding. Strongly recomended to every lover of geometry...

5 comments about Computational Geometry: Algorithms and Applications.

1. Pro:
   (1) Each chapter begins with a practical example. For example, the chapter computing intersections of lines starts with a discussion of a map-making application that goes into enough detail to see how the algorithms they present would be useful. This is a considerable step up from the common practice in algorithms literature of motivation by way of vaguely mentioning some related field (i.e. "These string matching algorithms are useful in computational biology"). This book does a much better job of motivating the material it presents, but if you're primarily interested in the abstract problem, these sections can be skipped.

   (2) Each chapter is relatively self-contained. Feel free to skip ahead to subjects that interest you.

   (3) Surprisingly readable. Unlike most technical material, one can read an entire chapter in a single sitting without missing much. Generally, each chapter will develop a single algorithm for a single kind of problem.

   (4) It's very up to date. This second edition is less than two years old, it includes some new results in the field.

   Con:
   (1) Algorithms are only given in pseudocode. The emphasis is on describing algorithms and data structures clearly and completely. If you're looking for a "cookbook" with code to copy and paste into an application, perhaps O'Rourke's "Computational Geometry in C" would be a better choice.

   (2) There are many important advanced results that are not discussed in the main text. An obvious example is the first chapter, which describes a well-known convex hull algorithm that takes O(n log n) time but algorithms that are faster for most inputs are mentioned only in the "Notes and Comments" at the end of the chapter. Someone interested in lots of gory details would be well-served to combine this book with Boissonnat and Yvinec's more detailed and mathematical "Algorithmic Geometry".




2. This is one of the really few computational geometry books available. It fills a niche and does it decently. However it could be better:
   
   1. The chapter layout is not very good. There are many "revisiting this" and "we saw in chapter so-and-so".
   
   2. The mathematical proofs are often written in a single paragraph full of "English" interspersed with mathematical notation, instead of the tried and true way of numbered equations and one-per explanations. This makes for disconcerting reading.
   
   3. The book in general could have done with more math and code, and less "English", not to mention more and better diagrams -- they tend to be sparsely detailed (ie. a picture is worth only a hundred words). The arrangement of diagrams also needs to be better: some are in the margins, some are in the middle, again not easy and intuitive to follow.
   
   Hopefully a future edition will address this issues.



3. The authors amass an impressive array of algorithms related to finding geometrical properties. Where these algorithms are performed on a computer. The book itself does not advocate any particular programming language. The algorithms are given in pseudocode, and you are expected to manually convert these to code in your choice of language. Given the calibre of the discussion in the text, which suggests that the readers are quite experienced, then this manual step should be easy to most.
   
   There are numerous contexts in which the text might prove useful. Ranging from graphics to GIS to robotics. Thus, there is an entire chapter on the planning of robotic motion. The robot can in general translate and rotate.
   
   Each chapter comes with an exercise set. Which helps make the book suitable as a graduate or even undergraduate text.



4. This book is extremely well written, easy to understand, and actually is the standard text for Computational Geometry classes, as far as I know. The only thing I didn't like about it was that there seemed to be a few errors in some of the pseudocode. But, it's to be expected when publishing a textbook, and I think it'll probably be cleared up in future editions.
   
   Overall, great book. I'd recommend it to anyone taking graphics or a computational geometry class.



5. The authors did a great job of introducing the reader to all the important aspects of the field of computational geometry while keeping it simple and understandable.

3 comments about The Works of Archimedes.

1. Again I feel I must post a review to counter misleading
   information in an earlier review. The author of the
   previous review seems to think these works were _not_
   available to scholars during the Renaisance. In fact,
   the famous "Archimedes Palimpsest" that resurfaced in
   the 1990s is only a small part of the works of Archimedes
   found in this book. Moreover, this book is a reprint of
   the translation published in 1897 by Thomas L. Heath.
   Heath _did_ have access to the Palimpsest (or maybe to
   a translation into German or to a copy--of this I am
   unsure) and did include a translation in this book in
   1897. It is true that after the Palimpsest resurfaced
   in the 1990s and began to be examined by modern methods,
   some lacunae were filled in. But that's not even most
   of the Palimpsest, let alone most of the contents of
   this book. Moreover, the newly discovered material is
   not in this book (but Heath's translation of the Palimpsest
   is). The previous reviewer is _extremely_ confused about
   the history.
   
   Now to the contents of the book. The famous Palimpsest
   actually is my favorite part. Prepare to be dazzled.
   Many 20th-century calculus texts, saying that integral
   calculus was anticipated by Archimedes in the 3rd century
   BC, are so phrased that they may give their readers
   the impression that Archimedes worked with something similar
   to Riemann sums, or similar nonsense. The truth is far more
   interesting. Archimedes used infinitesimals explicitly.
   His proofs were amazingly efficient. If you think that a
   brilliant proof by an ancient mathematician having only
   relatively primitive methods at his disposal must be longer
   and more complicated than a proof by modern methods, think
   again. Modern methods are indeed more efficient, but not
   because one writes _shorter_ proofs; rather it is because
   (at least in the present case) one writes _fewer_ proofs.
   Archimedes introduced the concept of center of gravity.
   In the Palimpsest, he finds not only areas and volumes,
   but centers of gravity (that of a solid hemisphere of
   uniform volume is 5/8 of the way from the "north pole" to
   the center of the sphere, Archimdes shows in one of his
   startlingly efficient proofs--just one example).
   
   It was not only by the use of infinitesimals that Archimedes
   solved problems that would now be treated by integral calculus.
   For example, one of the methods (just one of them) by which
   Archimedes found the area between a parabola and one of its
   secant lines involved dividing that area into an infinite
   sequence of triangles, the sum of the areas of which is a
   geometric series. Many other examples are in these pages.



2. I enjoyed the previous review, but do not wholly agree. It seemed to me the method of centers of gravity was the one by which Archimedes discovered, rather than proved, his results. His proofs do seem to me to involve limiting arguments which are at least reminiscent of riemann sums. Indeed even the method of centers of gravity involved slicing up solids in a way that to me suggests again riemann sums. Perhaps i have not read as carefully as the previous reviewer. I agree however that the works are startlingly wonderful and inspiring.
   
   The key to Archimedes' geometry solutions was the principle of parallel slices, that two figures all of whose slices parallel to a given reference line or plane have equal areas, or lengths, themselves have equal volume, or area. This is of course the fundamental theorem of calculus for equating areas, and the cavalieri principle, for equating volumes. Note it does not suffice to calculate them, merely to equate two such areas. thus Archimedes had to bootstrap up from one known area or volume to another.
   
   Thus starting from an actual decomposition of a cube into three pyramids, one sees that a right pyramid has volume 1/3 of cube. Then by parallel slices one sees the same for any pyramid or cone. then by taking complements one computes the volume of a sphere, by showing that horizontal slices of a cone and a sphere add up to the slice of a cylinder. Knowing cylinder and cone volume thus gives a sphere's volume.
   
   Finally one of the hard problems we give students is finding the volume of a bicylinder, the intersection of two transverse cylinders. After seeing Archimedes' solution of the volume of a sphere, by the principle of parallel slices, equating the volume of a sphere, slice by slice, with that of the complement of a (double) cone in a cylinder, one easily intuits his (still lost) solution of the volume of a bicylinder, as that of the complement of a square based (double) pyramid in a block! (of course reading further one sees it was rediscovered by Zeuthen 100 years ago, but so what, it is fun to do it oneself.)



3. EVERYTHING that Archimedes is supposed to have "discovered" already existed in Africa, thousands of years before "WHITE" Greeks existed. The Ancient Egyptians "THE MASTER BUILDERS" had already discovered "ALL" of the Arts & Sciences. The Greeks & Romans were students of the Ancient Black Egyptians, before they destroyed the Egyptian Civilization by raping the women, killing the Priests, forbidding the speaking of the language & burning the Library of Alexandria. Ask yourself this question, if the Greeks were such Great Mathematicians why did they go all the way to Africa to set up this Library, and where are their Pyramids? Huh?
   
   Africa & Africans were the fountainhead of knowledge, at a time when the Whites had recently emerged from the Caves of & Hillsides of Europe, where they were walking on all fours and eating their meat raw, not having the knowledge of fire. Go back and read the ancient historical accounts by Herodotus, where he describes not only the Scientific Wonders of the Ancient Egyptians, but also describes their race as being of "Burnt Skin & Woolly Hair, & that they describe themselves as "THE" Most Ancient of Peoples.
   
   WHY ARE THERE NO ANCIENT RUINS IN WHITE CIVILIZATIONS BUILT BY WHITE PEOPLES? (Stonehenge and other monuments in Europe were built by Blacks who peopled what is called Europe millions of years before the first Whites arrived. Google "Grimaldi Negro", the first inhabitants of Europe. Also see "The Making of the White Man" by Paul Guthrie & "Black Spark, White Fire".
   
   THIS IS THE SAME TYPE OF RACIST LOGIC THAT POSITS THAT CHRISTOPHER COLUMBUS DISCOVERED AMERICA, WHEN EVERYBODY KNOWS THAT BOTH INDIANS & BLACKS WERE HERE FIRST, BUILDING PYRAMID CIVILIZATIONS.
   
   For further edification read: "The African Origin of Civilization" by Cheik Anta Diop (Renowned Senegalese Physicist & Linguist), "Stolen Legacy" by George M. James (Greek Scholar) & "Black Athena" by Martin Bernal (which shows that Early Greece was peopled by two successive waves of African colonization who laid the foundation of both Minoan & Greek Civilization. Take a close look at the Minoans, they are of African stock, as were the early Greeks prior to the invasions of the Barbaric White Dorians, who brought no Civilizing influence to Greece.
   
   Racist White historical analysis cannot replace cold hard facts such as the Pyramid Civilizations appearing only in Black Civilizations such as Egypt, Mexico etc. The Pyramid culture in the Americas begins with the Thick Lipped, Broad Nosed, Wooly Haired Olmec Civilization, "THE MOTHER CIVILIZATION" of the Americas.
   
   FURTHERMORE, WHOSE TO SAY THAT ARCHIMEDES WAS WHITE, AS GREEK CIVILIZATION AT THAT TIME, HAD BLACKS AS WELL AS WHITES.
   
   Truth crushed to Earth will Rise Again!!!

5 comments about Introduction to Smooth Manifolds.

1. I really like this book. Physically, it looks much like Lang's algebra book, but I assure you that it contains none of the snide remarks. Though, it does have a picture of the author in a berra which is odd. I'm sure I mis-spelled that, but it's the french hat that people like to use to make fun of artist types.
   
   In any case, this book is long and contains a lot of problems for you to do. Unfortunately I do not do them, but that is a different story. I'm nowhere near finishing all the stuff this book has to tell me, but whenever I need to find something I don't know this book tends to have it. The index is great. It might be the best of any book I've used. The greatness of this book is a little surprising juxtaposed with Lee's book on Riemannian geometry which is not exceptional.
   
   Since this book is so large, and it says it's a graduate math book right on the cover, I like to take it out with me when I go out on the town. I find it's a great ice breaker with the ladies. I only wish it was the nice burnt orange of the newer springer books.
   
   All in all, this is a great book, and really puts Spivak to shame.



2. that lends itself to self-studying then this is not a good book, but excellent. All complaints reported in other reviews are actually answered in the preface: the book is about the mathematical machinery ordinated under the title smooth manifold theory. It is not a book on Riemannian geometry that's why there is no extensive treatment of metrics or any treatment of connections. Each topic comes up whenever the prerequisite tools have been built and enough motivation can be given, that's why it is a pleasure to read this book. If you like encyclopedic expositions there are plenty of them out there. The appendix contains a compilation of virtually all the results you need from calculus, linear algebra, and topology in order to study smooth manifolds. Perhaps the first time I actually read the appendix of a book (Arnol'd 's books form an exception) It is obvious that the author belongs to that group of people who like to excel in whatever they do, as all his books not only teach you the subject of their titles but also how to write a book if it happens to reach that point in your mathematical career. They are in some sense both books and meta-books on mathematics.
   This review is not intended to comment on other reviews, but let us be honest and agree on the fact that an author never faces the danger of being too clear: as to the length and the pace of the book, I wish this book were only one volume of a series from the same author starting with topology and culminating with the interplay of differential geometry and PDE. There is a drawback however, reasonably not anticipated. Most math books are not written to be actually read (aphoristic but true). This one makes an exception and thus the binding proves insufficient quickly. A hardcover version would be convenient. Suggestion for clever math students: learn the stuff from Lee and then pretend you are reading Lang's "introduction"...



3. I should say first that I was already familiar with manifold theory before picking up this book. I had already wrestled with some of the definitions, theorems, and whatnot, so I can't necesarily say I was a complete beginner before reading this book. Also, I'm not sure if I can say how great this book would be if you have no idea what a manifold (or tangent space, etc.) is. However, that stuff aside, this is an amazing text. I'm studying this book on my own, and it's great. The concepts are woven throughout the text instead of being lumped into chapters devoted to them (though some people might prefer the latter). Also, they're used to reinforce and build on each other.
   
   As an example, Spivak doesn't treat Lie groups until the second to last chapter. Lee introduces them in the second chapter, uses them as examples throughout the text, builds up the theory of Lie groups as the book goes on, uses Lie groups (and their actions on other manifolds) in developing certain other areas (it really streamlines the development) and ends with a nice big chapter on them. Of course, this is just one example.
   
   Lee developes manifold theory so that it would appeal to a physicist, geometer, algebraist, topologist, etc. Everything gets talked about! This means, however, that he can't treat any one subject in too much detail. For instance, he leaves curvature and other parts of Riemannian geometry to his other Riemannian Geometry text, but it's definitely worth the trade off. This book trashes Spivak. Buy it today!



4. My field lies somewhere at the intersection of algebra, geometry and physics. This is a very handy reference, meaning a few pages accessible and contains most of the basic notions of differentiable manifold and some useful beyond-elementary topics for me. A good book to look into if you can't remember something in details.



5. This book is an antidote to the more common style of math text. So many math books feel like they were written by mathematicians, which is to say their authors prize being terse over being understandable. Far too many textbooks out there have me jumping on the internet to find explanations and examples of key concepts that absolutely must be understood completely for the sake of even being able to read the next ten pages. Lee spares his readers the trouble by taking things slowly and including all the steps they need to keep up with the discussion.
   
   This was the textbook for my first class on smooth manifolds, and it is worth noting that it was the only class I took that semester in which I learned more from the textbook than from the professor's notes.
   
   If you are a grad student, this book will make your life considerably easier.

No comments about Origami Tessellations: Awe-Inspiring Geometric Designs.

1 comments about SuperVisions: Impossible Optical Illusions (Supervisions).

1. This book will keep my kids busy for hours. I needed the rest and they needed the fun.

5 comments about Counterexamples in Topology.

1. 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.



2. 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.



3. 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.



4. 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.



5. 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. I also purchased Schaum's Outline of General Topology which is very good.