A pregeometry (also called abstract pregeometry) consists of a set X, and a function cl (called clausure) which maps subsets of X to subsets of X, that is: cl:P(X) - - > P(X), and satisfies the following conditions, for all and all :

1. .
2. If Y Z, then .
3. cl(cl(Y)) = cl(Y).
4. (finite character) If , then there is a finite subset of Y, Y', such that .
5. (exchange principle) If , then . [here Ya is , similar for Yb].

A geometry is a pregeometry such that cl({a}) = {a} for all .

For example, let V be a vector space over a field, and, for , define cl(Y) to be the span of Y, that is, the set of linear combinations of elements of Y. Then the pair (V,cl) is a pregeometry, as it is easy to see.

In contrast, if X is a topological space and we define cl to be the topological-closure function, then the pair (X,cl) will not neccesarily be a pregeometry, as the finite character condition (4) may fail.

It turns out that many fundamental concepts of linear algebra --closure, independence, subspace, basis, dimension-- are preserved in the framework of abstract geometries.

Let (X,cl) be a pregeometry, and B,Y be subsets of X. We will say that Y is closed if cl(Y) = Y, and that B generates Y if Y = cl(B). Also we will say that B is independent if no proper subset generates cl(B), that is, for all , .

If B is independent and generates Y, then we will say that B is a base for Y. Equivalently, a base for Y is a minimal Y-generating set, or (by Zorn's Lemma) a maximal independent subset of Y.