 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 E^{c}, then the question is: at most
how many different sets can occur in the sequence
..., E^{c}, E^{c}, E^{}, E,
E^{c}, E^{c}, E^{cc},...

 
B* 
Show that every open set G ⊂ R^{d} is the union of at most countably many open balls G = ∪_{n=1}^{∞} B(c_{i}, r_{i}).
[Hint. Consider balls whose centers have rational coordinates and whose radii are rational.]

