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