singletons and the closure of the empty set is the empty set. Independence, bases and dimension Given sets , is independent over if for any . We say that is independent if it is independent over the empty set. A set is a basis for over if it is independent over and . A basis is the same as a maximal independent subset, and using Zorn's lemma one can show that every set has a basis. Since a pregeometry satisfies the Steinitz exchange property all bases are of the same cardinality, hence we may define the dimension of over , written as , as the cardinality of any basis of over . Again, the dimension of is defined to be the dimesion over the empty set. The sets are independent over if whenever is a finite subset of . Note that this relation is symmetric. Automorphisms and homogeneous pregeometries An automorphism of a pregeometry is a bijection such that for any . A pregeometry is said to be homogeneous if for any closed and any two elements there is an automorphism of which maps to and fixes pointwise. The associated geometry and localizations Given a pregeometry its associated geometry (sometimes referred in the literature as the canonical geometry) is the geometry where , and For any , Its easy to see that the associated geometry of a homogeneous pregeometry is homogeneous. Given the localization of is the pregeometry where . Types of pregeometries The pregeometry is said to be: trivial (or degenerate) if for all non-empty . modular if any two closed finite dimensional sets satisfy the equation (or equivalently that is independent of over ). locally modular if it has a localization at a singleton which is modular. (locally) projective if it is non-trivial and (locally) modular. locally finite if closures of finite sets are finite. Triviality, modularity and local modularity pass to the associated geometry and are preserved under localization. If is a locally modular homogeneous pregeometry and then the localization of in is modular. The geometry is modular if and only if whenever , , and then . Examples The trivial example If is any set we may define for all . This pregeometry is a trivial, homogeneous, locally finite geometry. Vector spaces and projective spaces Let be a field (a division ring actually suffices) and let be a vector space over . Then is a pregeometry