Oriented matroids differ from rectilinear geometry by a tame topological deformation. This statement is given precise meaning by the Topological Representation Theorem 5.2.1. Sometimes this deviation allows for phenomena that cannot otherwise occur, and in this way oriented matroids provide a precise language for discussing the question: Which properties in space are truly geometrical, and which are essentially combinatorial?

To understand the Topological Representation Theorem it is very instructive to study the rank 3 case (the first non-trivial case), where visualization is easy. In the projective version, this identifies rank 3 oriented matroids with arrangements of pseudolines, a topic studied long before the advent of oriented matroid theory.

We will not attempt to give a comprehensive treatment of the basic results concerning arrangements of pseudolines. Grünbaum (1972) gives an excellent exposition of this material with many interesting examples, and his monograph is still the best place to enter the subject. This chapter is instead devoted to a rather detailed discussion of those aspects of the subject which are important from an oriented matroid point of view. Also, some general results that update the information in Grunbaum (1972) are given.

Arrangements of pseudospheres in low dimensions

The Topological Representation Theorem 5.2.1 assumes much simpler form when stated for oriented matroids of rank at most 3. Let us look at these cases in turn.