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