The Main Results

In an arbitrary space of signs X, generation of basic sets, stability indices, representation of functions by signatures and separation of closed sets reduce to the corresponding problems in the spaces of orderings V* associated to the subvarieties V of X. This general principle is proved in Sections 1, 2 and 3. Also, in Sections 1 and 2, we obtain criteria for a set to be basic open or to be principal open, and extend to X the inequalities among the invariants s, $\bar s$, t, $\bar t$, w and l, which were already known for spaces of orderings. Section 4 is devoted to the notions of real divisor and regularity in X. Using them we can bound from below s and $\bar s$. Moreover, we compare basicness of an open set and its closure, resp. of a closed set and its interior. Finally, Artin-Lang spaces are introduced in Section 5, jointly with the tilde operator: this is the notion that makes the abstract theory fruitful of geometric applications.