The curriculum will consist of some parts of the following two books:
J. N. McDonald and N. A. Weiss, A course in real analysis, 2nd edition, Academic Press.
G. K. Pedersen, Analysis Now, 1989. Springer. ISBN: 978-0-387-96788-2. Corrected 2nd printing. Link to Springer's homepage for this book.
A note on Zorn's lemma and Tychonoff's theorem is available here.
The precise curriculum for the final oral examination is summarized here.