Introduction

Up till now our definable sets were always given as subsets of an ambient space Rm, a very convenient restriction that has served us well. In this final chapter we want to break out of this restricted setting, and consider also, say, projective space, and its “definable” subspaces, more generally, spaces that are not given as subsets of Rm, but locally look like definable subsets of Rm. To stay within the context of “spaces of finite type” we require a covering of the spaces of interest by only finitely many “affine” definable patches. This idea is carried out in detail in Section 1. The main result of this section, theorem (1.8), generalizes a theorem of Robson for semialgebraic spaces, to the effect that a “definable space” (obtained by gluing finitely many affine definable sets) is isomorphic to an affine definable set if and only if the space is regular, a separation condition that is easily verified in many situations of interest.

In Section 2 we consider a related construction, namely that of taking the quotient space X/E of a definable set X by a definable equivalence relation E on X. (Definably gluing finitely many definable sets can be viewed as a special case of this construction, but is better treated separately, as we do in Section 1.) We ask when X/E can be realized as an ordinary definable set, and are particularly interested in the case that E is “definably proper” over X.