Analytic Zariski structures, predimensions and non-elementary stability

The notion of an analytic Zariski structure was introduced in [1] by the author and N.Peatfield in a form slightly different from the one presented here. Analytic Zariski generalises the previously known notion of a Zariski structure (see [2] for one-dimensional case and [3], [4] for the general defini-tion) mainly by dropping the requirement of Noetherianity and weakening the assumptions on the projections of closed sets. This leads to interesting new phenomena, in particular, the family of closed-in-open subsets forms a hierarchy which starts with analytic sets and continues by induction to more complex ones, called in [5] generalised analytic sets (defined classically on the complex numbers and in the context of rigid analytic geometry). In [1] we assumed that the Zariski structure is compact (or compactifi-able), here we drop this assumption, which may be too restrictive in appli-cations. The class of analytic Zariski structures is much broader and geometrically more rich than the class of Noetherian Zariski structures. The main example