General topology reference Please suggest some books on topology which contains compactness , connectedness, completeness, denseness in C[a,b] , matrices spaces, function spaces and other topological spaces. A fruitful book or notes which completely devoted to these concepts in these particular spaces and which isolate the properties which  hold in real numbers spaces but not in these space. Thanks
 A: A lot of the results that hold for spaces of matrices or function spaces can be proved in the more general context of metric spaces by putting different norms on your spaces. You may also be interested in looking at some books on analysis that cover some of the more analytical properties of such spaces.
For topology you should check out Munkres's Topology, which covers most of the topics you have listed here.
For analysis, you should check out Rudin's Principles of Mathematical Analysis. I really enjoy the terse style that Rudin uses, and you might as well, especially after you have learned the basics of the subject.
A: I'm not an expert in this subject for sure but i can propose you a very nice book which is: 
Topology Without Tears
You can find it online.
This is  my favourite book in general topology.It has 12 chapters which some of them contain topology of metric spaces.It also covers my aspects of topology,from topological spaces to product and quotient topologies and metrization theorems.
This book is being updated and it is anticipated to have 15 chapters and more apendices.
The appendices of this book contain introduction to  dynamical systems and chaos theory, cardinal numbers,topological groups,Hausdorf measure and dimension,filters and nets.
Also there is a small chapter in the book for an intriduction to ordinal numbers.
It has big number of  great exercises that can help you to understand the theory. 
Many believe that this book is kind of an elementary treatment to general topology but i do not agree with that.I cannot call it elementary but neither extremely advanced like Munkres's or Kuratowski's books in topology.
I would recommend you to look at this book.
Also you can study Munkres's general topology book and if you want i can provide you a link of a video playlist in Youtube of some great lectures  in general topology.
Let me know.
A: I was very fond of Choquet's 1966 Topology when I was beginning  because it seemed to go straight to the essentials in a way that made everything very clear and intuitive.  It did not cover as much material as, say, Dugundji's 1966 Topology but it was a much better  book for learners.
