Prerequisites for John M. Lee's Introduction to Topological Manifolds? I currently have some questions regarding prerequisites, especially prerequisites from analysis before diving into Introduction to Topological Manifolds by John M. Lee.
I really liked the way author explained things from some parts I read and I've decided to use the book as a self-study book for topology. However, I am worried that my weak analysis background might hinder my progress after a few chapters.
The author states that basic set theory, group theory and advanced calculus are prerequisites in his book.
But I have a question regarding advanced calculus. Since I have taken set theory and abstract algebra 1 (covering groups and some parts of rings), I have no problem with that part. However, I am currently learning undergraduate analysis and I am not sure if I have sufficient knowledge for the book.
I am using Abbott and Lang as resources for learning analysis and I have barely scratched the surface, like I only just finished covering sequences and series part.
So my question is: what parts of undergraduate analysis are the ones used in the book or can I just jump into the book without much knowledge of analysis such as uniform convergence?
 A: The analysis prerequisites are summarized in Appendix B. My recommendation is to read that appendix carefully and try doing all the exercises. If you find that too challenging, then go back to an undergraduate analysis text and fill in some of the missing ideas and techniques.
A: If your undergraduate analysis includes an introduction to metric spaces, you should be fine. If not, tackling topological spaces might be slightly more challenging, although probably not insurmountable: Lee's exposition is very clear and thorough. A little introductory set theory and some feeling for curves and surfaces from multivariable calculus should go a long way in helping you with the early chapters. In fact, some topology books manage to be fairly gentle introductions while starting with the concept of a topological space before treating metric spaces (e.g. Introduction to Topology: Pure and Applied by Adams & Franzosa).
A: From the abstract itself:
A course on manifolds differs from most other introductory graduate mathematics courses in that the subject matter is often completely unfamiliar. Most beginning graduate students have had undergraduate courses in algebra and analysis, so that graduate courses in those areas are continuations of subjects they have already begun to study. But it is possible to get through an entire undergraduate mathematics education, at least in the United States, without ever hearing the word “manifold.”
