I think I have background in general topology on undergraduate (Munkres) level and also some background in functional analysis. Without Filters and Nets . However, there are still topics, which I wish to learn, because I keep hearing about them from other people, but don't have much experience with using them. Recently I decided to learn all this topics and decided to work them out from Engelking's book that seems to be the most complete book on the subject. At this moment I have climbed up to Stone-Cech compactification and Wallman extension. Which are useful and interesting concepts themselves, however I really wanted a different exposition (more modern, filter and category based). Moreover Engelking's exposition is overbloated and very terse. So I feel that working through his book, despite it being a great reference book, really slows me down.
Now I found this document by Dror Bar-Natan. It is very compact and contains only statements of main results, which I can try to prove myself. This format now seems optimal for my goal.But the problem is finding similar references for other topics. Here is a list of topics I wish to learn:
- Stone-Cech compactification as a functor (in filter setting)
- General Baire Spaces and Baire category
- Metric spaces with geodesics and Hopf-Rinow theorem (*),(!)
- Nagata-Smirnov Metrization Theorem
- Uniform spaces
- Baire category for metric or uniform spaces
- Basic topics in connectedness(*),(!)
- Basic topological dimension (*)
- General topological algebra
(*) - means that I have idea on what reference to use (!) - means that I should know this material but I still wont to brush it up
I would very grateful if you can share your experience about learning these topics and, if you know any, references to which are similar in spirit with the document above. It would be very valuable for me, if it uses category or sheaf theory, or applications to measure theory, functional analysis or metric or differential geometry.
I would appreciate any suggestions.