Customer Reviews

Essential Topology (Springer Undergraduate Mathematics Series)

5 star:		(3)
4 star:		(1)
3 star:		(2)
2 star:		(0)
1 star:		(0)

 

 

I have a major in math, many years ago. I have moved into economics, but miss the elegance of math, hence I decided to revisit some old topics, and started with topology. As a student we used lecture notes and no real textbook, so my choice now was this textbook. It is a pure pleasure to read. I wish we had used it as a text book when I studied.

The topics are well motivated. Crossley does a good job in explaining why we should care about these particular lemmas and theorems. The proofs are usually elegant. I find the estetic pleasures a good math book should provide.

I have never seen such a beatiful explanation on continuity and its relations to series and sets. Now I understand why, when mathematics is lousily explained,everything seemms to be so hard. I recommend strongly this book for someone for self study on topology. Hope the author can write on other topics of mathematics.

This is actually not that bad of a book. It is reasonably well-motivated and has tons of examples (although some are pretty tedious). Where it fails is its large number of errors. There are tons of minor errors scattered throughout, making the book more difficult to read. There are also some pretty major mistakes. The two page proof of theorem 10.11 is blatantly erroneous, and is the standout example. I should note, however, that I have the first printing of the book, so it is easily possible that many of these issues have been resolved in the second printing. I also feel that it would make more sense to have chapters seven and eight switched, so that the chapter on homotopy groups would follow the chapter on homotopy and the chapter on simplicial homology would follow the chapter covering simplicial complexes. (The only part of chapter eight which relies on chapter seven is the statement of the Whitehead theorem, which is not proven, at the very end of the chapter.)

Another complaint: His English does not always seem grammatically correct to me. (Maybe I'm sheltered living in the eastern US?) For example, "Since the arrows rotate 720 degrees as we go around the circle, so deg(f) = 2." does not sound like a full sentence to me. If I had only seen it once or twice I might mistake it for another typo, but this sort of sentence structure is all over the book. It really disrupts the flow for me.

The first half of this book covers point-set topology, the second half algebraic. If you want to read this book in full, knowing basic algebra is an absolute must. If you have familiarity with, for example, quotient groups, free groups, and the rank/nullity theorem from linear algebra you should be fine. If you only care about the first half, knowing naïve set theory and basic operations on matrices should suffice.

To summarize, the exposition is actually pretty good, but there are too many errors for me to recommend it.

IMPORTANT: Apparently, a lot of mistakes have been corrected in the most recent printing. Please read the comments to this review for details.

Topics are well motivated.

Theorems are proved in a rigorous yet intuitive style that one feels like it was an explanation rather than a dry proof typically found in the advanced math books.

Important key ideas are also sufficiently illustrated through examples and exercises.

If one finds it verbose, I'd recommend croom--a bit more like the typical math books but accessible.

The building blocks of topology are introduced in a very easy to understand manner. The prolific use of examples, some even entertaining, allows the reader to conceptualise abstract ideas and gain a better understanding of the theory.

We used this book in an introductory topology class I took. Some of the exercises are poor (e.g. counting the number of topologies) and the exposition wasn't anything to go crazy about. After a while, I found myself reading Munkres exclusively; it's much more comprehensive. Maybe this book is well suited for folks looking to get a flavor of topology but nothing super concrete.