Sheaves of continuous definable functions

Extract

Let M be an o-minimal structure or a p-adically closed field. Let be the space of complete n-types over M equipped with the following topology: The basic open sets of are of the form Ũ = {p ∈ S_n (M): U ∈ p} for U an open definable subset of Mⁿ . is a spectral space. (For M = K a real closed field, is precisely the real spectrum of K[X ₁, …, X_n ]; see [CR].) We will equip with a sheaf of L_M -structures (where L_M is a suitable language). Again for M a real closed field this corresponds to the structure sheaf on (see [S]). Our main point is that when Th(M) has definable Skolem functions, then if p ∈ , it follows that M(p), the definable ultrapower of M at p, can be factored through M_p , the stalk at p with respect to the above sheaf. This depends on the observation that if M ≺ N, a ∈ Nⁿ and f is an M-definable (partial) function defined at a, then there is an open M-definable set U ⊂ Nⁿ with a ∈ U, and a continuous M-definable function g:U → N such that g(a) = f(a).

In the case that M is an o-minimal expansion of a real closed field (or M is a p-adically closed field), it turns out that M(p) can be recovered as the unique quotient of M_p which is an elementary extension of M.