Can you recommend me a book about compactness (Real Analysis)? Can you recommend me a book about compactness  and connectedness where has examples, because Rudin have a few and are dificult for me. Thanks.
 A: The textbook Elementary Real Analysis (Thomson/Bruckner/Bruckner) has a large section on compactness arguments on the real line.  Rudin (I assume you mean baby Rudin) is excellent but not considered very friendly.  And if you are having trouble with the notion of compactness it is likely better to see thoroughly how the arguments are used in the setting of the real line before tackling them in a general metric space setting.
With the notion of compactness it is probably best to think of the concept more along the line of what arguments are available when working with compact sets, rather than "what they look like."  On the real line (or in Euclidean space) a compact set is merely closed and bounded.  In other spaces "compact" would have a characterization special to the space itself.  Hoping that more examples will make the idea clearer is, I think, misguided.
The real thing to learn here is what can you do with that.  If you learn fully about the Bolzano-Weierstrass theorem and the Heine-Borel theorem,  and see them in action then, when you encounter compactness in more abstract settings, these two should come to mind.
Connectedness, unfortunately, in our text came in only in an exercise.
The book is free for download as a PDF file.  This link will take you to the download page.
(For full disclosure, I should note that I am one of the authors of this book)
