On a question of Herzog and Rothmaler

Herzog and Rothmaler gave the following purely topological characterization of stable theories. (See the exercises 11.3.4 – 11.3.7 in [2]). A complete theory T is stable iff for any model M and any extension M ⊂ B the restriction map S(B) → S(M) has a continuous section. In fact, if T is stable, taking the unique non-forking extension defines a continuous section of S(B) → S(A) for all subsets A of B, provided A is algebraically closed in T . Herzog and Rothmaler asked, if, for stable T , there is a continuous section for any subset A of B. Or, equivalently, if for any A, S(acl(A)) → S(A) has a continuous section . This is an interesting problem, also for unstable T . Is it true that for any T and any set of parameters A the restriction map S(acl(A)) → S(A) has a continuous section? We answer the question by the following two theorems.