Chapter 18 - Pointless Geometries

Abstract

The focus of this chapter is on pointless geometries. The concept of point is assumed as the main primitive term for an axiomatic foundation of geometry. In pointless geometry, regions are considered as individuals, i.e. in the vocabulary of logic, first order objects, while points are represented by classes (or sequences), that is, second order objects. Obviously, expressions like ‘pointless geometry’ or ‘geometry without points' is understood as contractions of ‘geometry without the point as a primitive concept’. Complete lattices satisfying distributive laws are called frames or locals, if R = (R, ≤) is a frame, the elements of R are called regions and inclusion relation the relation ≤. The frames may be organized into a category FR by suitably defining morphisms. The definition is given in order to obtain a category extending that of topological spaces. If f is a continuous map from a topological space (X, T) into another (X’, T’), it determines, via f-1, a map from T’ to T preserving infinite joins and finite meets.