Plane Separation

Abstract

The plane separation axiom is a careful statement of the very intuitive idea that every line has “two sides.” Such an idea seems so natural that we might expect it to be a consequence of our present axiom system. However, as we shall see in Section 4.3, there are models of a metric geometry that do not satisfy this new axiom. Thus the plane separation axiom does not follow from the axioms of a metric geometry, and it is therefore necessary to add it to our list of axioms if we wish to use it. In this section we will introduce the concept of convexity, apply it to state the plane separation axiom, and develop some of the very basic results coming from the new axiom. In the second section we will show that our two basic models, the Euclidean Plane and the Hyperbolic Plane, do satisfy this new axiom. In the third section we shall prove Pasch’s Theorem, which gives an alternative formulation of the plane separation axiom in terms of triangles. This means that Pasch’s Theorem follows from the plane separation axiom and the plane separation axiom follows from assuming Pasch’s Theorem.