MATH 3220 § 1	FOURTH HOMEWORK ASSIGNMENT	Due Monday,
A. Treibergs		September 17, 2007.

You are responsible for knowing how to solve the following exercises. Please hand in the starred "*" problems.

Exercises from the text "Foundations of Analysis" by Joseph L. Taylor. 7.3 [ 4*, 6, 8, 9, 10*, 12* ] 7.4 [ 1*, 2, 3, 4*, 5 ] Additional exercises. A*   Let E be a subset of R. How many different sets can be obtained from E by taking closure or complementation? Prove your answer. If we denote closure by E- and complement by Ec, then the question is: at most how many different sets can occur in the sequence ..., E-c-, E-c, E-, E, Ec, Ec-, Ec-c,...   B*   Show that every open set G ⊂ Rd is the union of at most countably many open balls G = ∪n=1∞ B(ci, ri). [Hint. Consider balls whose centers have rational coordinates and whose radii are rational.]