A couple of months ago I asked a professor by e-mail to mentor me on topology during the summer. He advised me to study general topology (Hausdorff spaces, connectedness, compactness) and algebraic topology (from Hatcher) before we meet, as time permits.
Yesterday I finished chapter 2 of Munkres (I am planning to study up to chapter 3) but I have no knowledge of abstract and linear algebra. I took analysis and set theory, this is my background.
I consider myself as a beginner (if you look at my questions:-)). Although I study up to 5 hours a day, I feel I don't accomplish much. Also, I am particularly bad at finding counterexamples (perhaps because I have not taken calculus).
Coming back to my question, how much algebra do I need to cover to understand Hatcher? I have about 1 month to cover? Would you recommend some books/notes? How can I develop my skills at finding counterexamples?
I imagine you're either going to look at the section on fundamental group, or the section on homology. In either case, all I would recommend as background for algebra is an understanding of what it means for a subgroup to be normal and basic properties of homomorphisms (specifically the first isomorphism theorem).
I think using algebraic topology as a motivating factor for learning algebra is a good idea, but (in my experience) it's better to learn the concepts as you go. It's difficult to see what algebraic concepts are relevant beforehand. Just read about the appropriate topics as you see them in Hatcher.
As far as the general topology is concerned, the only other thing I would add to your list is to try to understand quotient spaces. A lot of the general topology can be swept under the carpet, but quotients are extremely important. Klaus Jänich's book simply titled "Topology" has some excellent examples.
Lastly, don't feel deterred by these feelings of "not accomplishing much". Learning mathematics is highly nonlinear: sometimes one can be stuck for days or weeks, and then suddenly a moment of clarity occurs and you jump ahead in understanding. The important thing is to be constantly working.