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Rating: 
Summary: Insightful and fun introduction
Review: I'm surprised that several previous reviewers have given this book low ratings. This book is far superior to the standard introductions. As someone who has studied topology for several years now, I have found that the greatest failing of many introductory texts is the inability to give a real 'feel' for the subject. By 'feel' I mean not only familiarity with the necessary tools and ways of thought needed to progress to higher levels of understanding but also experience with the kinds of problems that plague(excite?) topologists on a daily basis. Several texts proceed in the logical progression from point set topology to algebraic topology. Munkres is among the best of this style. But the logical order is not always pedagogically best, especially in topology. To start one's topology career by spending one or more semesters on point set topology is utterly ridiculous, given that such point set subtleties are to a large degree not used to study the beginnings of geometric or algebraic topology. This is how these texts fail to give students the 'feel' for topology; the student has no idea what it is that most topologists do, and in fact will not get a good idea until much later. Armstrong tries (and succeeds for the most part) in grounding concepts in real applications, the way the tools are actually used by research mathematicians. Perhaps this is part of why it may be confusing to the novice; introducing topological groups and group actions on spaces right after the section on quotient spaces may appear a bit much, but those concepts are a big part of *why* quotient spaces are so important! Incidentally, the material on quotient spaces is the most complete I've ever seen in an introductory book; Armstrong covers cones and also gluing/attaching maps. The book is certainly fun. Imagine learning about space-filling curves right after the section on continuous functions. Armstrong keeps things spiced up throughout the book. He also goes at some length into triangulations, simplicial approximation, and simplicial homology. Then he *applies* this stuff to get results like Borsuk-Ulam, Lefschetz fixed-pt thm, and of course dimension invariance. Throw in less standard material like Seifert surfaces, and you have quite an interesting mix. The exercises can be quite varied and hard, but are designed to give the reader a realistic view of the difficulties of the subject. The reader will get considerable insight from them, and loads of fun too. I say this, because as someone who already knows the stuff, I find more than a few of the problems enjoyable even now. Having wrote all that, I should add that I did *not* learn out of this book! But I wish greatly that I had! I would have known sooner whether topology was the right subject for me to pursue and had some 'lead time' to absorb some very fundamental concepts early on. If you pass over this book, be warned that you are shorting yourself in the long run.
Rating:
Summary: Insightful and fun introduction
Review: I'm surprised that several previous reviewers have given this book low ratings. This book is far superior to the standard introductions. As someone who has studied topology for several years now, I have found that the greatest failing of many introductory texts is the inability to give a real 'feel' for the subject. By 'feel' I mean not only familiarity with the necessary tools and ways of thought needed to progress to higher levels of understanding but also experience with the kinds of problems that plague(excite?) topologists on a daily basis. Several texts proceed in the logical progression from point set topology to algebraic topology. Munkres is among the best of this style. But the logical order is not always pedagogically best, especially in topology. To start one's topology career by spending one or more semesters on point set topology is utterly ridiculous, given that such point set subtleties are to a large degree not used to study the beginnings of geometric or algebraic topology. This is how these texts fail to give students the 'feel' for topology; the student has no idea what it is that most topologists do, and in fact will not get a good idea until much later. Armstrong tries (and succeeds for the most part) in grounding concepts in real applications, the way the tools are actually used by research mathematicians. Perhaps this is part of why it may be confusing to the novice; introducing topological groups and group actions on spaces right after the section on quotient spaces may appear a bit much, but those concepts are a big part of *why* quotient spaces are so important! Incidentally, the material on quotient spaces is the most complete I've ever seen in an introductory book; Armstrong covers cones and also gluing/attaching maps. The book is certainly fun. Imagine learning about space-filling curves right after the section on continuous functions. Armstrong keeps things spiced up throughout the book. He also goes at some length into triangulations, simplicial approximation, and simplicial homology. Then he *applies* this stuff to get results like Borsuk-Ulam, Lefschetz fixed-pt thm, and of course dimension invariance. Throw in less standard material like Seifert surfaces, and you have quite an interesting mix. The exercises can be quite varied and hard, but are designed to give the reader a realistic view of the difficulties of the subject. The reader will get considerable insight from them, and loads of fun too. I say this, because as someone who already knows the stuff, I find more than a few of the problems enjoyable even now. Having wrote all that, I should add that I did *not* learn out of this book! But I wish greatly that I had! I would have known sooner whether topology was the right subject for me to pursue and had some 'lead time' to absorb some very fundamental concepts early on. If you pass over this book, be warned that you are shorting yourself in the long run.
Rating: 
Summary: NO NO
Review: The fact that the author does not explicitly define things is a bad enough reason to stay away from this book. If you only want a light treatment of point-set topology, go for Munkres, otherwise, Hatcher.
Rating:
Summary: A Well Written, Excellent Introductory Book
Review: This book was used in an undergraduate intro to topology class I took. It is short, the text is incredibly concise to the point of being incomprehensible, the exercises are all fairly standard but range from difficult to impossible since the text gives no examples or applications of the theorems, and it tries to cover far too much material in too few pages. Look for something by Munkres if you want to learn topology.
Rating: 
Summary: Run. Munkres, Massey are better
Review: This is the book we used in my first undergraduate course in topology. I remember it as being one of the worst textbooks I ever came across in my undergraduate math studies. The explanations were too short and many definitions were buried into the text. I remember constantly having to flip through the pages of this book to find something I was looking for. Stay away from this book!
Rating:
Summary: Run. Munkres, Massey are better
Review: this text is required for my course on introductory topology. Not only does it omit entire branches of the subject, for example only giving cursory treatments of vector field topology, metric topolgy, and combinatorial techniques; but, the presentation is mostly unexplained. The author has concentrated on following a very dry quick example-theorem-proof technique instead of theorem-proof interspersed with discussion and example technique. I have yet to find a superb text, though for algebraic things i like massey
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