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In this broad introduction to topology, the author searches for topological invariants of spaces, together with techniques for their calculating. Students with knowledge of real analysis, elementary group theory, and linear algebra will quickly become familiar with a wide variety of techniques and applications involving point-set, geometric, and algebraic topology. Over 139 illustrations and more than 350 problems of various difficulties help students gain a thorough understanding of the subject.
- Sales Rank: #1203919 in Books
- Brand: Brand: Springer
- Published on: 2013-10-04
- Released on: 2013-10-04
- Original language: English
- Number of items: 1
- Dimensions: 9.25" h x .62" w x 6.10" l, .84 pounds
- Binding: Paperback
- 251 pages
- Used Book in Good Condition
Review
"The book is very good, its material sensibly chosen...It has good and plentiful illustrations...for the author, topology is above all a geometric subject..." -- MATHEMATICAL GAZETTE
Most helpful customer reviews
21 of 24 people found the following review helpful.
A very welcome, intuitive approach to topology
By Alexander C. Zorach
Many of the standard introductions to Topology (Munkres comes to mind) focus more on the logical flow of the material, and less on the motivation for the material. This book focuses on the motivation, but after the first few chapters, the logical development is sound too.
The Armstrong book starts out with some fairly advanced concepts, outlining some interesting topological results before giving the modern definition of topological spaces in terms of open sets. Typically, authors give the open set definition of a Topology at the outset, before explaining what topology really is, and without explaining why that definition is used or how it was developed. Armstrong instead shows the historical motivation of the subject, and actually leads the reader through this development, starting with the less elegant but more intuitive definition of spaces in terms of neighborhoods. The equivalent open set definition is then taken in chapter two. However, once things get going, this book does not move slowly at all--quotient spaces and the fundamental group are presented early and covered in depth, and it is not long before the reader encounters genuinely advanced material, in rigorous form.
It's true that this book doesn't cover the same amount of raw material that a book like the Munkres does, and it's true that the book does not follow the most concise logical order, but it offers history, motivation, and initial exposure to more interesting results. Perhaps more importantly, it develops the reader's intuition. In many ways, this book is a complement to the Munkres, and an enthusiastic self-learner would benefit greatly from using both books simultaneously.
At the same time, this book does get into some more advanced topics. It has a particularly clear exposition of simplicial homology. My last word of praise about this book is that although it gives lots of motivation, it is still very concise. I think it's hard to go wrong with this book.
11 of 12 people found the following review helpful.
Valuable, and generally pleasant, introduction to topology
By maisonsuspendue
I can see how this book has left very few people happy. To generalize broadly, one often finds two types of reviewers of math books on Amazon: those who find the text too difficult, and denounce it; and those who implicitly denounce the first group by means of vigorous support of the book in question. Typically this latter group goes on to write reviews of books *supported* by the first group in which they denounce the excessive "hand-holding," the pandering to the reader's "intuition," and the general attempts to make the material accessible.
This book, however, manages to both require a non-trivial amount of effort and sophistication from the reader (thus alienating the first group), all while also appealing to intuition and giving large numbers of examples (thus alienating the second).
The following example should make the author's approach clear. On several occasions, Armstrong gives a non-standard definition of an idea. This is usually a definition that is more intuitive (to the beginner), but which is harder to use to complete proofs. This non-standard definition is followed by the standard definition, and the equivalence of both formulations is established. This is the case with connectedness, for instance. First, connectedness is defined by appealing to the idea that a space "should be one piece," leading to the formulation that whenever a connected topological space is decomposed into two subsets, the intersection of the closure of one of these sets with the other set is always nonempty. Soon thereafter, the standard formulation (the formulation which one almost always uses to actually write proofs) is introduced and established as equivalent, namely that a connected topological space is one in which the only sets that are both open and closed are the entire space itself and the empty set.
It is true that this approach makes for a bad reference book. It is also certainly not the most elegant and streamlined presentation. But the book is clearly not meant to be a reference or to be a showcase of exceptional concision and elegance. It is meant to be a book to learn from. Adding to this, the chapters are all full of examples, many of them quite interesting.
I will concede that the writing and layout can be irritating at times. In particular, as has been pointed out many times before, the author does not isolate and highlight all definitions and corollaries. So this adds to the difficulty of using the book as a reference, and it even makes it somewhat unpleasant to read as a learning text at times. But to say that the author does not define things is simply wrong. (In the first chapter the author sketches an overview of the material contained in the text, and it consequently does not contain many formal definitions or proofs. By and large, however, all subsequent chapters are independent of this chapter. So if you are truly scandalized by someone attempting to give a loose overview of the subject, you are entirely free to skip this chapter and refer to it as necessary (which will be infrequently).)
All and all, I thought this was a good first topology text. You are always given good examples to chew on while you are sorting out the technicalities. The problems are also generally good. While many are fairly straightforward, I have found that they are almost all at least thought provoking, and some develop new material entirely. And there are more than a handful of difficult ones.
Finally, it should be emphasized that one can realistically be introduced to the rudiments of wide range of topics in a single semester: general topology, identification spaces, topological groups, the fundamental group, triangulations (including Seifert-Van Kampen), and simplicial homology. (To be clear, the book contains more than that, but I am only outlining what could be done in about 13 weeks.) Moreover, unlike some texts which are only meant to give the flavor of a subject to undergraduates, I have found that the foundations set by this books were substantial enough to build on.
11 of 13 people found the following review helpful.
Needlessly Obtuse
By Edmonton Euler
I'm quick to praise textbooks. I've written fan mail to authors of texts that I've truly loved. I keep those books in a special place on my shelf, and treasure them the way one might a favorite movie, or a beloved single malt. Armstrong's Topology is not among those treasured books.
There are four main complaints I have with this text:
-There are not enough diagrams to go with the provided examples. I understand there are space constraints, but this book is practically a pamphlet, and there is room for twice as many pictures. Topology is visual, and Armstrong does the reader a disservice by skimping on the graphics.
-The majority of the book is a wall of text, and it is impossible to reference prior results in less than 30 seconds. An attentive reader should be able to turn right to any previous result immediately, especially for such a tiny textbook.
-There is almost no expository material. Whereas Munkres takes care to introduce every single concept that will be discussed before jumping into the actual topological content, Armstrong just throws terms -- and worse, notation -- out there. An "Undergraduate Text in Mathematics" should be more thorough than this.
-Armstrong makes excessive appeals to obviousness. This sort of technical writing is the fastest way to leave a reader feeling helpless and insulted. (It's also a good way to make readers who 'get it' feel in on the joke, which I think partially explains the polarizing effect this book has.)
There are some things I like:
-The pictures that are included are top-notch.
-The introductory section which uses Euler's Polyhedron Formula to motivate future results is exceptional. I remember how sunny I felt about the rest of the book after reading it...
Honestly, though, Munkres' book is more extensive, and friendlier. Read that if you have any choice in the matter.
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