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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)?

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    $\begingroup$ 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. $\endgroup$ – icurays1 Sep 17 '14 at 16:41
  • $\begingroup$ @icurays1 Thanks, its one way to answer my question. $\endgroup$ – Troy Woo Sep 17 '14 at 17:59
  • $\begingroup$ Try Lee's Introduction to topological manifolds. $\endgroup$ – mdg Nov 16 '14 at 9:25
  • $\begingroup$ @G.S. I was going to do that. But thanks. $\endgroup$ – Troy Woo Nov 16 '14 at 9:26
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    $\begingroup$ @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. $\endgroup$ – Mathemagician1234 Mar 4 '15 at 23:55
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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.

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  • $\begingroup$ I browsed the book, seems a decent one. Thank you. Do you mind also talk about algebraic topology references? I followed a course using chapters from Munkres PartII->Vick->Hatcher->Bott&Tu>ZHang Weiping, Lectures on Chern-Weil Theory and Witten Deformations. I kinda got lost when moving on to Hatcher. What do you think is the strategy then to approach algebraic topology (as a graduate student not as a researcher)? $\endgroup$ – Troy Woo Mar 5 '15 at 13:45
  • $\begingroup$ I guess you are not going to say something about algebraic topology. Here is your approval. Thanks for the recommendation. Good book indeed. $\endgroup$ – Troy Woo Apr 1 '15 at 19:26
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    $\begingroup$ What we really need for algebraic topology is a book that walks a middle ground between the 2 extremes of geometry (Hatcher) and a modern categorical approach (May,tom Dieck's books) .I'm happy to report there's a new book by Shastri that seems to do the best job of all the available books of walking this middle path-I'd heartily recommend .amazon.com/Basic-Algebraic-Topology-Anant-Shastri/dp/1466562439/… $\endgroup$ – Mathemagician1234 Apr 1 '15 at 19:33
  • $\begingroup$ Thank you so much sir, I didn't see your reply until today. I'm going walk through my course refs (Munkres->Vick->Bott&Tu->Hatcher->Zhang WP) while taking a look at this book. $\endgroup$ – Troy Woo Apr 11 '15 at 21:54
  • $\begingroup$ Hi, I just want to come back and thank you for recommending me Wilansky's book. I learned a lot from it and I realize Munkres is not so great a book in terms of explaining things...its just a pedagogical textbook. $\endgroup$ – Troy Woo May 17 '15 at 9:31
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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.

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  • $\begingroup$ Just want to say, I started reading From Calculus to Chomology yesterday. And I have to say it is a good book indeed, although not what I originally asked for. $\endgroup$ – Troy Woo Jan 20 '17 at 8:38

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