(1) (Basically Exercise 4.4.1.) (a) Show that if S and T are chain-closed subsets, then the union of S and T is a chain-closed subset (b) Show that if S_i is a chain-closed subset for every i in I, then the intersection of all S_i (bigcap_{i in I} S_i) is a chain-closed subset (2) (Exercise 4.4.2.) Give an example of a subset S of D x D (the product cpo) that is not chain-closed, but which satisfies: (a) forall d in D, {d' | (d,d') in S} is chain-closed; and (b) forall d' in D, {d | (d,d') in S} is chain-closed. [Hint: consider D = D = Omega, the cpo in Figure 1.] (Compare this with the property of continuous functions given on Slide 15.) (3) Let P : D -> B_bot and g : D -> D be continuous. Define f : D x D -> D x D by f(d1,d2) = if( P(d1), (g(d1),g(d2)), (g(d2),g(d1)) ) Show that for fix(f) = (u1,u2), we have u1 = u2. (Use Scott induction.) (4) Prove that for f : D -> E monotone, if f^{-1} preserves chain-closed sets, then f is continuous, where "f^{-1} preserves chain-closed sets" means that, if S is a chain-closed subset of E, then f^{-1}(S) is a chain-closed subset of D (for all S). (5) Show that the collection of chain-closed subsets is not closed under arbitrary unions. (Hint: look at te Omega cpo.)