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 Teq. 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(acleq(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.
Theorem 1. Let A be a subset of a model of T. Assume that the Boolean algebra of acl(A)-definable formulas is generated by
• some countable set of formulas,
• all A–definable formulas,
• all formulas which are atomic over acl(A).
Then S(acl(A)) → S(A) has a continuous section.
The conditions of the theorems are satisfied if, for example, L and A are countable, or, if there are only countably many non-isolated types over acl(A).
Theorem 2. There is a theory of Morley rank 2 and Morley degree 1 such that S(acl(∅)) → S(∅) has no continuous section.