How to master general topology for analysis? I started learning topology long ago. I first exposed myself to metric topology in Baby Rudin and Munkres Topology 2nd ed. Part I. Munkres is my most revisited book ever since. 
The first big challenge I faced is when approaching William Boothby's An Introduction to Differentiable Manifolds and Riemannian Geometry. I soon realized that I needed to learn some algebraic topology and differential topology, which I did much later. Nevertheless, everyday topology for me is still mostly general topology. I could say every bits and pieces of Munkres's Part I has its use in analysis, but hell, its really a lot to memorize.
I read the book through, or some chapters again and again. But somehow I still cannot memorize everything. So as a result, I had to come back to Munkres from time to time, the only difference being now I know what I am looking for. But I definitely cannot say I learn topology very well. This has puzzled me for a long time, because usually after I read a book three times, I can have a good feeling of at least the big picture. But with Munkres, its just less organized in my mind, not the big blocks (connected/ compactness/ countability/ separation/ compactification/ metrization/ completeness/ Baire space), but those small yet useful lemma/theorems/corollaries.
So, my question is: how to organize the huge body of general topology in one's mind for analysis's purpose (real/complex/functional/harmonic...on Euclidean space/manifold/Lie group)?
 A: A very good book for point set topology which emphasizes the connections with analysis and which is cheap is Albert Wilansky's ironically but appropriately titled Topology For Analysis.The book is somewhat more advanced then Munkres, it assumes the student has a good working understanding of the basic topology of Euclidean and metric spaces from undergraduate analysis. Wilansky's book begins with convergence,reviewing basic sequences and proceeding to develop general convergence via nets and filters. This sets the stage for developing all the topological machinery needed for functional analysis and operator theory, which I think is what you want. It discusses semimetrics and norms, separation axioms, compactification, function spaces and uniform spaces, as well as a number of topics that usually reserved for functional analysis courses, such as the weak topology,topological groups and the Gleason map. I think you'll find this book quite helpful for getting the topological structure of analysis mastered-and best of all, it's cheap. 
A: If you're interested by algebraic topology, the book of Fulton or the book of Bott and Tu are using differential forms and motivate some results by analytic approach. 
In the same spirit you can take a look to the fantastic book From Calculus to Cohomology.
