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)?