Measure theory and topology books that have solution manuals
Perano, most textbooks on measure theory and topology are considered too high level to have solutions manuals in the usual sense-students at that level who need solutions manuals to get through their courses are considered doomed to failure. I don't agree with this thinking,I think all textbooks,regardless of level,should have complete solutions manuals. But most books at that level don't. But I do know a few exceptions and they're mainly problem course texts.
Ian Adamson's A General Topology Workbook covers all the main topics of point set topology-open and closed sets,subspaces, general convergence,etc.-through a series of beautiful exercises,all with complete solutions in the second half of the book.The only really "standard" textbook I know on measure theory that has a conventional solutions manual is Robert Bartle's A Modern Theory of Integration-which isn't really a conventional graduate course on measure and integration, but rather a development based on the Henstock-Kurtzwell integral. While I think this is a subject that's underused in teaching analysis and it's quite well presented in this book, it isn't really what you're looking for.
Lastly, there's a terrific problem course in measure and integration that comes with complete solutions-Problems in Mathematical Analysis III:Integration by W.J. Kaczor and M.T. Nowak. The exercises are immense, clear and not too difficult and come with complete solutions in the back. Since the book is so comprehensive and the courses in the subject have become so standardized-you may find all the solutions you need in the second half of this book. I'd also recommend getting the earlier 2 volumes in the same series-they provide great practice and additional training in real variables for the serious student.
Good luck!
René Schilling: Measures, Integrals and Martingales
The solution manual is not contained in the book, but available on the web page.
http://jianfeishen.weebly.com/uploads/4/7/2/6/4726705/solution-measure-integration.pdf .This contains some exercises and solutions.
The following has the solutions of the selected exercises of the book 'A First Look at Rigorous Probability Theory'. http://www.worldscientific.com/doi/suppl/10.1142/6300/suppl_file/6300-solutionsmanual_free.pdf.
Goodluck