The indiscernible topology: A mock Zariski topology (Q2732508)

Topology is often involved in the model-theoretic analysis of structures. A basic example concerns algebraically closed fields, where the Zariski topology is naturally related to the model-theoretic framework; in fact, the definable sets are just the constructible ones, i.e. the finite Boolean combinations of closed sets. More generally, topology plays a key role within geometric stability theory, in the program of classifying structures up to biinterpretability. Here Hrushovski's and Zilber's study of Zariski geometries emphasizes the importance of a topological approach via abstract Zariski topologies, for instance in recovering Zilber's Trichotomy for strongly minimal structures. NEWLINENEWLINENEWLINEThe paper under review is interested in determining some topologies intrinsically related to definable sets in arbitrary structures, extending or replacing these abstract Zariski topologies, and depending on the elementary equivalence type of a structure rather than directly on the structure itself. Accordingly, the authors deal with an arbitrary complete first-order theory \(T\), and work, as usual, in a big saturated model \(U\) of \(T\). The basic idea is to define, for every small subset \(X\) of \(U^n\), the indiscernible closure \(\text{icl}(X)\) of \(X\), as the set of all elements occurring in an indiscernible sequence of type order \(\omega\) containing infinitely many tuples of \(X\). This procedure is repeated to produce \(\text{icl}^{\alpha }(X)\) for every ordinal \(\alpha\) and eventually to build their union \(\text{icl}^{\infty } (X)\) when \(\alpha\) ranges over ordinals: \(\text{icl}^{\infty }(X)\) is the smallest indiscernible-closed set containing \(X\). It is not clear if \(\text{icl}^{\infty }\) preserves type-definability. So the previous construction has to be suitably rearranged (essentially at the limit steps) to produce \(\text{icl}^* (X)\), the smallest type-definable and indiscernible-closed set extending \(X\). By the way, it is observed that, for a stable \(T\), the procedure actually requires \(\leq |T|^+\) steps to stabilize and reach \(\text{icl}^*(X)\). NEWLINENEWLINENEWLINEThe authors show that, for every \(n\), the indiscernible-closed sets in \(U^n\) are just the closed sets of a topology \(\text{Ind}_n (U)\) of \(U^n\) -- the indiscernible topology. These topologies \(\text{Ind}_n (U)\) form an \(f\)-space on \(U\) in the sense of Hrushovski; they are invariant and locally Noetherian. Moreover indiscernible-closed sets include \(\text{acl}^{\text{eq}}(\emptyset)\)-definable sets. It is not clear whether type-definable indiscernible-closed sets form a topology, in other words whether an arbitrary intersection of them is again type-definable. However, for every small set \(A\) of parameters, the indiscernible-closed sets which are type-definable over \(A\) do form a topology (the indiscernible topology over \(A\)). NEWLINENEWLINENEWLINEThe authors study indiscernible topologies, and illustrate that the structure \(U\) and the theory \(T\) can be partly determined by them, and several model-theoretic properties are reflected by topological properties. In particular, the authors compare the indiscernible topology and the Srour topology, and show that the former is a refinement of the latter, but they coincide on definable sets. Particular emphasis is laid on stable theories \(T\); among other things, it is observed that, in some nice cases, the indiscernible topology behaves as the Zariski topology in algebraically closed fields.