CHAPTER 18

Pointless Geometries

Giangiacomo GERLA

Dipartimento di Matematica ed Applicazioni, Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy

Contents

Introduction

1. Historical remarks

1.1. The first attempts

1.2. More recent proposals

2. Pointless topologies

2.1. Frames

2.2. Points of a frame

3. Connection structure

3.1. Whitehead's axioms

3.2. Grzegorczyk's axioms

3.3. The system of B.L. Clarke

4. The metrical approach

4.1. Pointless metrical spaces

4.2. Definition of diameters and distances

4.3. The points

5. Physical geometry

5.1. The first axioms

5.2. Points and Cauchy filters

5.3. Topological and metric structures

5.4. The basic theorems of physical geometry

References