**TEXTOS DE MATEMÁTICA - SÉRIE B - Nš 12**

THE REAL NUMBERS IN POINTFREE TOPOLOGY

*B. Banaschewski*

FOREWORD

These notes have their origin in the five-part series of seminar talks I gave at the Department of Mathematics of the University of Coimbra in April 1997. Although originally envisaged just as a written record of these talks, they eventually evolved into a much more extensive exploration of the area, taking into account new approaches and better results which I developed as a consequence of the work that went into preparing for this series as well as including some additional material that was not covered for lack of time. I am grateful to my colleagues at the University of Coimbra, and especially to Manuela Sobral, for giving me the opportunity to talk about this topic and thereby providing the stimulus for further thought, leading to a better understanding of this engaging subject.

September 1997 Bernhard Banaschewski

CONTENTS

Introduction 1

0. Background 3

1. The Frame of Reals 9

2. Continuous Real Functions 18

3. The Real Uniformity of a Frame 24

4. Function Algebras 32

5. The Cozero Map 41

6. Homomorphisms of Function Algebras 49

7. The Stone-Weierstrass Theorem 58

8. The Cozero Part of a Frame 64

Appendix 1. Frames and Propositional Theories 75

Appendix 2. f-Rings 79

Appendix 3. Cauchy Completeness of Uniform Frames 85

References 91