Learning Roadmap for Algebraic Topology

If you would like to learn algebraic topology very well, then I think that you will need to learn some point-set topology. I would recommend you to read chapters 2-3 of Topology: A First Course by James Munkres for the elements of point-set topology. If you would like to learn algebraic topology as soon as possible, then you should perhaps read this text selectively. In particular, I would recommend you to focus mainly on the following (fundamental) notions, reading more if time permits:

1. Topological space
2. Basis for a topology
3. The product topology (on a finite cartesian product; if time permits, you can read about the case of an infinite cartesian product but this is not urgently needed as far as algebraic topology is concerned)
4. Subspace topology
5. Closed set and limit point
6. Continuous function
7. Metric space
8. Quotient topology (this is a very important, but sometimes ignored, prerequisite for algebraic topology)
9. Connected space
10. Component and path component
11. Compact space
12. Hausdorff space
13. The separation axioms, Urysohn's lemma and the Tietze extension theorem (if time permits; these are very useful and inteteresting concepts but you can take the Urysohn lemma and Tietze extension theorem on faith if you desire)

I think that as far as algebraic topology is concerned, there are two options that I would recommend: Elements of Algebraic Topology by James Munkres or chapter 8 onwards of Topology: A First Course by James Munkres. The latter reference is very good if you wish to learn more about the fundamental group. However, the former reference is nearly 450 pages in length and provides a fairly detailed account of homology and cohomology. I really enjoyed reading Elements of Algebraic Topology by James Munkres and would highly recommend it. In particular, I think a good plan would be:

1. Learn the elements of point-set topology as outlined above.
2. Read chapter 8 of Munkres' Topology: A First Course to learn the rudiments of the fundamental group.
3. Read Elements of Algebraic Topology by James Munkres.

You will not need to know anything about manifolds to read Elements of Algebraic Topology but I believe that it is good to at least concurrently learn about them as you learn algebraic topology; the two subjects complement each other very well. I think a very good textbook for the theory of differentiable manifolds is An Introduction to Differentiable Manifolds and Riemannian Geometry by William Boothby (but this is a matter of personal taste; there are (obviously) many other excellent textbooks on this subject). The advantage of this textbook from the point of view of this question is that there is a flavor of algebraic topology present in one of the chapters.

I hope this helps!

I'm going to be controversial here, and suggest that you start with Spanier's Algebraic Topology, supplemented by Switzer's Algebraic Topology: Homology and Homotopy. These are very good and comprehensive books which have stood the test of time; books that present algebraic topology properly and algebraically. Many great algebraic topologists grew up on these books.

That having been said, I am also a fan of Munkres Elements of Algebraic Topology which works out examples very nicely using simplicial decompositions. Not to sound too old-fashioned, but I'm also a huge fan of Schubert's Topology as a book which works out the simplicial story properly. But, given your background, I strongly recommend that you dive straight into Spanier and Switzer, and enjoy their beautiful, classical, algebraic treatment of the material.

If your background in point set topology is insufficient, Munkres' Topology is a great book for foundations. Once you understand the basics in Munkres, you can move on to Armstrong's Basic Topology or Massey's Algebraic Topology: An Introduction (the former also contains all of the point set topology necessary to read it in its entirety) . Either of those should prepare you for Hatcher's Algebraic Topology, provided you have the necessary algebra background (which it sounds like you do). There will be some overlap between Hatcher and the other two, but attempting Hatcher without familiarity with the classification of surfaces and basic homotopy theory may be overwhelming.