I'm with Jonathan in that Hatcher's book is also one of my least favorite texts. I prefer Bredon's "Topology and Geometry."
For all the people raving about Hatcher, here are some my dislikes:
- His visual arguments did not resonate with me. I found myself in many cases more willing to accept the theorem's statement as fact than certain steps in his argument.
- He uses $\Delta$ complexes, which are rarely used.
- I would have preferred a more formal viewpoint (categories are introduced kind of late and not used very much).
- There aren't many examples that are as difficult as some of the more difficult problems.
In one place, the theory of fibrations, I feel tom Dieck's book is superior. Thus in the first edition of Spanier, in the proof of Lemma 11, of Chapter 2, Section 7, Spanier writes down an extended lifting function $\Lambda$, but he does not prove it is continuous. I did not manage to prove it was continuous, and in fact found the function was not well defined. I sent a correction of the definition to Spanier, and this appeared in the second edition, but I still do not know how to prove $\Lambda$ is continuous. Have I missed something?
On the other hand Spanier's ideas on the construction of covering spaces are still referred today by experts, with the notion of what they call the Spanier group.
I find Section 3 of Chapter 7 of Spanier on "Change of base point" rather hard work, and I feel it can all be much easier done, and more generally, by using fibrations of groupoids. But tom Dieck's book does not use this method either.
Spanier gives van Kampen's theorem for the fundamental group of a simplicial complex as an exercise, while tom Dieck's book does give the statement of the theorem for a union of two spaces. Hatcher gives a more general theorem, for a union of many spaces, but none of these mention the fundamental groupoid on a set of base points.
I tend to agree with 313's answer that readers should look around, and find what is easier for them, in different aspects.
My copy of Spanier, dated 1966, had a price of \$15, but when I checked for inflation that was equivalent to \$111 today.
March 13: I add that neither book develops the algebraic theory of groupoids. For a relevant discussion on this, see https://mathoverflow.net/questions/40945/compelling-evidence-that-two-basepoints-are-better-than-one/46808#46808
Best Answer
Warning : the following books are not algebraic topology books in the classical sens, but anyway it's impossible to get in AT without any background in topology : these two books will give you the necessary background and go after in the beginning of algebraic topology.
"Introduction to Topological Manifolds" by Lee is very readable and starts from scratch. It covers classic topology (4 chapters), one chapter on CW complex and classic topics in algebraic topology (fundamental group, covering space and groups) with emphasis on surfaces (with classification of compacts surface) with finally last chapters about homology and cohomology.
Another reference is the book of Munkres, Topology, which covers with lot (and lot) of details general topology, and also fundamental group and covering space.
For both, no background is needed (There is an appendix in Lee about group theory, and a chapter consacred to Set theory in Munkres !). For more advanced topics in algebraic topology (homology theory for example, or differentiable topology) I think calculus in $\mathbb R^n$, general topology and abstract algebra are highly recommended.