Fix an o-minimal expansion of the real exponential field that possesses smooth cell decomposition. We prove smooth approximation of definable differentiable functions with respect to a topology closely related to the Whitney topology. As a consequence we obtain a strong version of definable smooth separation of sets, which we use to prove that definable smooth manifolds are definably affine.

The theory of subanalytic sets is an excellent tool in various analytic-geometric contexts; see, for example, Bierstone and Milman [1]. Regrettably, certain “nice” sets—like { (x, x) : x > 0 } for positive irrational r, and { (x, e−1/x) : x > 0 }—are not subanalytic (at the origin) in R. Here we make available an extension of the category of subanalytic sets that has these sets among its object...

Theorem 1. Let K = 〈K,+, ·, . . .〉 be an expansion of K all of whose atomic relations are definable in R. If K is a proper expansion of 〈K,+, ·〉 then the field R is definable in K. By “proper expansion of 〈K,+, ·〉”, we mean that there is a definable set in K which is not definable in 〈K,+, ·〉 (even with parameters). In particular, the theorem implies that there are no proper expansions of an al...

This work stems from an ongoing investigation into the distribution of rational points lying on particular sets in R. This is a question within transcendental number theory, but it appears that the model theory of o-minimal structures is a very suitable context in which to consider it. The origins of this current research lie in work of Bombieri and Pila ([1]), which established various estimat...

@0-categorical o-minimal structures were completely described in 4], and are essentially built up from copies of the rationals as an ordered set by`cutting and copying'. Here we investigate the possible structures which an @0-categorical weakly o-minimal set may carry, and nd that there are some rather more interesting (and not o-minimal) examples. We show that even here the possibilities are l...