Prove that there exists a with such that every member of is infinite, but is finite for all distinct .
Prove that for every there are disjoint subsets such that , is countable, and for no pair is the set nonempty and finite.
For the first one, let be an enumeration of the rationals. For every irrational number x, let be a sequence of rationals converging to x, and let . Then X will be the collection of all these sets .
For the second one, don't you just take Y to be the set of isolated points in X, or am I missing something?