where closures of sets are defined to be their span. The closed sets are the linear subspaces of and the notion of dimension from linear algebra coincides with the pregeometry dimension. This pregeometry is homogeneous and modular. Vector spaces are considered to be the prototypical example of modularity. is locally finite if and only if is finite. is not a geometry, as the closure of any nontrivial vector is a subspace of size at least . The associated geometry of a -dimensional vector space over is the -dimensional projective space over . It is easy to see that this pregeometry is a projective geometry. Affine spaces Let be a -dimensional affine space over a field . Given a set define its closure to be its affine hull (i.e. the smallest affine subspace containing it). This forms a homogeneous -dimensional geometry. An affine space is not modular (for example, if and are parallel lines then the formula in the definition of modularity fails). However, it is easy to check that all localizations are modular. Field extensions and transcendence degree Let be a field extension. The set becomes a pregeometry if we define for . The set is independent in this pregeometry if and only if it is algebraically independent over . The dimension of coincides with the transcendence degree . In model theory, the case of being algebraically closed and its prime field is especially important. While vector spaces are modular and affine spaces are "almost" modular (i.e. everywhere locally modular), algebraically closed fields are examples of the other extremity, not being even locally modular (i.e. none of the localizations is modular). Strongly minimal sets in model theory Given a countable first-order language L and an L-structure M, any definable subset D of M that is strongly minimal gives rise to a pregeometry on the set D. The closure operator here is given by the algebraic closure in the model-theoretic sense. A model of a strongly minimal theory is determined up to isomorphism by its dimension as a pregeometry; this fact is used in the proof of Morley's categoricity theorem. In minimal sets over stable theories the independence relation coincides with the notion of forking independence. References H.H. Crapo and G.-C. Rota (1970), On the Foundations of Combinatorial Theory: Combinatorial Geometries. M.I.T. Press, Cambridge, Mass. Pillay, Anand (1996), Geometric Stability Theory. Oxford Logic Guides. Oxford University Press. Matroid theory Model theory