Reference Request to Prepare for Hatcher's "Algebraic Topology"

Question

Hatcher himself has an excellent and always generously free set of notes on point- set topology:

http://www.math.cornell.edu/~hatcher/Top/TopNotes.pdf

It includes up to quotient spaces.

It seems like there is a vast gap between that and the beginning of his "AT."

I would appreciate any suggestions for covering the intervening material. PDF notes or reasonably priced books would be especially appreciated.

Thanks

Answer 1

In terms of actual formal prerequisites for Hatcher's book, basic algebra and those notes on topology should mostly cover what you need.

If you find that the notes aren't enough, it's most likely because you don't have enough practice using the theory that's presented there. The answer for that doesn't necessarily have to be another book on topology; if you're not in a rush to start Hatcher's book, the best thing to do could very well be to study a subject that constantly uses topology. For example, that could be Lebesgue integration, functional analysis or differential geometry (on manifolds).

An alternative might be a problem book in topology, as Hatcher's notes don't have a ton of exercises. An example would be the first three chapters of the book Elementary Topology: A First Course by Viro, Ivanov, Netsvetaev and Kharlamov.

EDIT: Here is a link for the last book. The online version is an abridged one, omitting the proofs and solutions. http://www.pdmi.ras.ru/~olegviro/topoman/index.html

Answer 2

Since I have asked the above question, I have been using two excellent sources that I think pave the way for a transition to "Hatcher" and comparable level texts:

Videos by N. Wildberger. Included in the series are several lectures on homotopy, the fundamental group, induced isomorphism, and covering spaces. These are full of examples, many pictures (I think are invaluable), and very clear explanations. The series includes many additional topics as well:

https://www.youtube.com/playlist?list=PL41FDABC6AA085E78

$EDIT$:

Thanks to a mention (I think in a comment by Mathemagician1234) I have switched to John Lee's "Intro to Topological Manifolds." It is extremely clear, and as a feature I especially like, he presents numerous cautionary examples. As well as lots of pictures.

http://www.amazon.com/Introduction-Topological-Manifolds-Graduate-Mathematics/dp/1441979395

(As a text, I had been using Gamelin & Greene "Intro to Topology", primarily Chapter III. It is very clear, especially in conjunction with the above lectures. And it is a Dover!)

http://www.amazon.com/Introduction-Topology-Second-Edition-Mathematics/dp/0486406806