Struggling with Topology. Any advice? I'm a Junior Mathematics major at a small Liberal Arts college. I'm currently taking first semester topology (Munkres text). I feel like I'm barely able to tread water in this course. I was able to follow the material up until we began discussing limit points and convergence. I feel I should mention I have yet to take Real Analysis yet.  
I've taken two courses over Abstract-Algebra, one of which was an independent study over Ring Theory. I find Algebra much more intuitive and I understand it much better than general topology. 
The definitions seem to make sense to me, yet I find it difficult coming up with counter-examples and proofs for basic theorems. I really struggle with anything dealing with $R^{\omega}$. 
I really hope to attend graduate school, or at the very least be involved in mathematics in some way throughout my life. I feel like I have hit a road block. Does anyone have any study tips or advice for understanding basic general topology? 
 A: Don't beat yourself up over struggling so much that you doubt your ability to succeed in graduate school. Perseverance and hard work will go a long way. Don't give up. (But yes, consider how to learn.)
Be sure to understand the important things, and don't stress if some of it does not click in your first exposure. (In particular, I don't think the example you mention is central. Grasping continuity, compactness and connectedness is more important. Having a variety of examples that you can work with is also useful. For example, metric spaces are nice. When I'm thinking of a theorem, I often create subspaces of $\mathbb{R}^2$ in my mind to work with. I suggest you talk with your professor to see what he/she thinks is most important for your class.
Also, do you have the chance to work together with some of your classmates to talk about the material?
Although Munkres' text most certainly has its strengths, from my little exposure to it, I'd guess that I would not have liked my first introduction to the subject to be that text. Rather, I used a book by George Simmons, titled "Introduction to Topology and Modern Analysis." You should try it out when you have time and/or check out some other book.
A: Here is a good introductory text that takes you from calculus to topology and teaches you proof 
Elementary Point Set Theory by Andre Yandl
