- 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.
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,...
||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.]