# 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)?

• Try an exercise in the new subject, fail miserably. Figure out what's missing, go read, do exercises. Try new exercise again. Talk with peers/professors/etc. Rinse and repeat...mental organization & abstraction comes with experience rather than precognition. – icurays1 Sep 17 '14 at 16:41
• @icurays1 Thanks, its one way to answer my question. – Troy Woo Sep 17 '14 at 17:59
• Try Lee's Introduction to topological manifolds. – mdg Nov 16 '14 at 9:25
• @G.S. I was going to do that. But thanks. – Troy Woo Nov 16 '14 at 9:26
• @mdg That's probably the best general introduction to topology that currently exists,but it's really designed to prepare students for advanced work in geometry and topology. So it doesn't really emphasize the parts of topology that are important in analysis. – Mathemagician1234 Mar 4 '15 at 23:55