Many Valued Topology and its Applications

376 Pages·2001·33.154 MB·English

by  Ulrich Höhle (auth.)

MANY VALUED TOPOLOGY
AND
ITS APPLICATIONS
MANY VALUED TOPOLOGY
AND
ITS APPLICATIONS
Ulrich Hăhle
Fachbereich M athematik
Bergische Universităt, Wuppertal, Germany
~.
"
Springer Science+Business Media, LLC
Library of Congress Cataloging-in-Publication
Hohle, Ulrich.
Many valued topology and its applieations / Ulrieh Hohle.
p. em.
Includes bibliographieal referenees and index.
ISBN 978-1-4613-5643-1 ISBN 978-1-4615-1617-0 (eBook)
DOI 10.1007/978-1-4615-1617-0
1. Topology. 1. Title.
QA611 .H695 2001
51421--de21
2001020266
Copyright i.tI2001 by Springer Seienee+Business Media New York
Originally published by Kluwer Academic Publishers in 2001
Softcover reprint ofthe hardcover lst edition 2001
AH rights reserved. No part of this publieation may be reproduced, stored in a
retrieval system or transmitted in any form or by any means, meehanieal, photo
eopying, reeording, or otherwise, without the prior written permission of the
publisher, Springer Science+Business Media, LLC.
Printed on acid-free paper.
To Gisela and Felix
Preface
Many valued mathematical structures have grown out of the theory of lattice
valued sets from the mid-sixties of the last century and now form a new branch
of lattice theory interacting with different areas of mathematics. This book
develops some of these interactions between general topology and many valued
structuresaswell as some oftheresultingapplicationsto probability theory and
non-classical logics.
Most of the material of this book is a natural outgrowth of results and ideas I
presented at various conferences and workshops on category theory, topology,
functional analysis and probability theory during the last decade. Explicitly,
I would like to mention here the Northwest German Category Seminar, the
annual International Linz Seminar organized by E.P. Klement (Linz), and the
weekly Workshop on Functional Analysis held by R. Meise (Diisseldorf) and
D. Vogt (Wuppertal). The stimulating atmosphere of all these seminars and
workshops was an essential prerequisite to the preparation of this book and
played a crucial role in the process of rethinking and revising some aspects of
many valued topology. In particular, I am most grateful to B. Banaschewski
and H. Herrlich for their most valuable comments and hints on the theory of
frames and the history of topology. Furthermore my deepest thanks go to S.E.
Rodabaugh and A.P. Sestak for their enthusiastic support of this project and
for their profound influence on the treatment of many valued sobriety and the
stratification axiom. I alsogratefully acknowledge a .G.Smolyanov's comments
on topological problems related to the Minlos-Sazonov theorem, and the con
tinuingsupport I received from B.Schweizer concerning the development of the
theory of stochastic metricspaces.
In writing a book of this kind, it is very important that one can depend and
rely on the support of many talented and committed individuals. I wish to ex
press my sincere gratitude to P. Eklund, J. Gutierrez Garda,T. Kubiak, M.A.
De Prada Vicente, A.P. Sestak and M. Tidten for their indispensable help eli
minating misprints and polishing up the presentation. My deepest appreciation
goesto mywifeforher never-endingpatienceand selflesssupport accompanying
this work. Finally,Iwant to acknowledgeKluwer Academic Publishers for their
cooperation in bringing this project to completion, and particularly to Mr. G.
Folven for his capable assistance and patience concerning deadlines.
Wuppertal, November 2000 U. H.
Contents
Introduction 5
I Categorical Foundations 11
1 Categorical Preliminaries 13
1.1 (Epi, extremal mono)-categories . 13
1.2 Monads . 16
1.3 Problems . 25
2 Partially Ordered Monads 29
2.1 Partially ordered objects . . . . . . . . . . 29
2.2 Trace of partially ordered objects . . . . . 38
2.3 Axioms of partially ordered monads in C . 42
2.4 Some examples of ordered monads . . . . 48
2.4.1 Q-valued proper power set monad in SET . 49
2.4.2 <5-valued filter monad in SET 50
2.5 Problems . . 53
3 Categorical Basis of Topology 55
3.1 Topological space objects . . 55
3.2 Density and closed hulls . .. 60
3.3 The Hausdorffseparation axiom 67
3.4 The regularity axiom . 72
3.5 Compactness 89
3.6 Problems . 103
II Many Valued Topology 105
4 Quantic Basis ofFilter Theory 107
4.1 GL-moI1oidswith square roots 109
4.2 Q-valued filters . . . . . . . . . 120
4.3 Stratified <5*-valued filters . . . 125
4.4 Examples of stratified 15*-valuedfilters . 137
2 CONTENTS
4.5 Problems . . . . . . 143
5 Many Valued Topological Spaces 145
5.1 Motivating problem . 146
5.2 Axioms of many valued topologies ... . 149
5.3 Many valued spatiality and sobrifications 158
5.3.1 £-sobrification and spectral theory in Hilbert spaces. 159
5.3.2 lffi(n)-sobrificationand almost everywhere defined
random variables . . . . . . . . . . . . 165
5.3.3 Boolean valued topologiesand frames . 171
5.4 lB-valuedtopologies and convergence . 174
5.4.1 Heyting algebra valued topologies and limit structures 176
5.5 Stratified lB.-valued topologicalspaces . 180
5.5.1 Rigid, many valued real line . 188
5.6 Change of Basis. . . . . . . . . . . . . . . . . . . . 194
5.6.1 Spatial frames and the hypergraph functor 200
5.7 Problems . 202
6 Many Valued Convergence Theory 205
6.1 Density and closed subsets . 208
6.2 Limit points and the many valued T . 212
2-axiom
6.2.1 T -axiom and quantale valued topologies 215
2
6.2.2 The T;-axiom and stratified lB.-valued topologies 217
6.3 The many valued regularity axiom . 220
6.3.1 Regular Heyting algebra valued topological spaces 222
6.3.2 Star-regular, rigid lB.-valued topological spaces . 226
6.4 Many valued compactness . . . . . . . . . . . . . . . . . . 235
6.4.1 Compact, Heyting algebra valued topological spaces 236
6.4.2 Star-compactness and the stratification axiom. 243
6.5 Problems . 249
III Applications Of Many Valued Topology 253
7 Stochastic Metrics 255
7.1 Stochastic metric spaces . 255
7.2 Topological properties ofstochastic metrics 259
7.3 Problems . 277
8 Stochastic Processes 279
8.1 Outer measures and stochastic processes 279
8.2 Linear stochastic processes ..... 283
8.3 Uniformly continuous trajectories. 287
8.4 Continuous linear trajectories 290
8.4.1 Minlos' Theorem 299
8.5 Problems . 314
CONTENTS 3
9 Probability Measures 317
9.1 lBT -valued compactifications 317
m
9.2 lBT -valued continuous extensions 325
m
9.3 Problems 329
10 Topologies on M-Valued Sets 331
10.1 Local M-valued filter monad 335
10.2 Topological axioms for M-valued sets . . . . . . . . . . 350
10.2.1 Topologies on M-valued sets as sheavesover M 356
10.3 Problems 359
Appendix 361
A.l Regularity based on ortholattices 361
A.2 Topologization of Menger spaces . 364
Bibliography 367
Author Index 375
Subject Index 377
Introduction
Around the beginning of the 20th century General Topology arises from the
effort for establishing a solid basis for Analysis and is intimately related to the
success of set theory. Closed sets already appear in the work of G. Cantor; a
first axiom system of neighborhoods is presented in D. Hilbert's famous axio
matization of the plane (cr. pp. 234-235 in [39]); the notion of free ultmfilters
is anticipated by F. Riesz by his definition of ideale Verdichtungsstelle (cf. [88]).
One of the important contributions of F. Hausdorff consists in the freeing of
the neighborhood notion from its restriction to exclusive application to higher
dimensional manifolds (cf. [114]). As an illustration of this step we quote from
F. Hausdorff's famous book Grundziige der Mengenlehre (1914) the following
passage:
EineTheoriederraumlichen Punktmengenwiirdenun,verrnoge der zahl
reichen mitspielendenEigcnschaften des gewohnlichen Raumes,natiirlich
einen sehr spczicIlen Charakter tragen, und wenn man sich vornherein
auf diesen einzigen Fallfestlegen woIlte,so wiirde man fiir Punktmengen
einer Geraden, ciner Ebene, einer Kugel usw. jedesmal eine neue The
orie Zll entwickeln haben. Die Erfahrung hat gezeigt, daf man diesen
Pleonasmus vermeiden und eine allgemeinerc Thcoric aufstellen kann,
die nicht nur die genannten Falle, sondern auch noch andere Mengen
(Riemannsche Fliichen, Riiume von endlichen und unendlich vielen Di
mensionen,Kurven-und Funktionenmengenu.a.) umfafit (cf.pp.210-211
in [37]).
Even though Hausdorff is aware oftheimportant roleof neighborhoodsand the
significant interplay between neighborhoods and open: sets, he does not succeed
in identifying systems ofopen sets with neighborhood systems. As H. Herrlich,
M. Husek and G. PreuB explain in their note on the history of the notion of
topologicalspace (cf. [40]), Hausdorff's neigborhood systems are not topologies,
but open neighborhood bases according to the contemporary terminology. This
shortcoming is independently removed by C. Kuratowski 1922 (cf. [68])and H.
Tietze 1923 (cf. [111]) by including supersets of Hausdorff's neighborhoods. In
contrast to C. Kuratowski who only lays down the following formulation:
lThe name open set has been coined by C. Caratheodory (cf. p. 40 in [14]).
U. Höhle, Many Valued Topology and Its Applications
© KIuwer Academic Publishers 2001
6 INTHODUCTION
Ondit que E est un entourage du point p,lorsquepest situe it l'interieur
de E (cr.p. 189 in [68]).
H. Tietze gives a full axiomatization of this extended concept of neighbor
hoods and formulates that system ofneighborhood axioms which later has been
adopted by standardtextbookson generaltopology. At thesametime H.Tietze
introducesan axiomsystemforopensubsetsand proves theequivalencebetween
the axiomsofopen subsets and neighborhoodaxioms. Up to a minimal tighten
ingby P. Alexandroff(cr. p. 298in [2]) Tietze'saxiomsystemofopensubsetshas
been accepted byN.Bourbaki (1940)as afoundationofgeneraltopology. In this
sense H. Tietze finishes the debate on the axiomatic foundations of topological
spaces. A characteristic of this debate is the fact that intuitive interpretations
of neighborhood axioms or axioms of open subsets play only a marginal role.
It is interesting to see that neither the work of Hausdorff nor the important
contributions of Kuratowski and Tietze contain any relevant remark referring
to an intuitive understanding of topological spaces. Hausdorff only motivates
topological spaces by metric spaces and defines a topological space as a set E
in which certain subsets U(x) are related to points x of E satisfying a certain
collection ofaxioms (cf.p. 211-213in [37]). Referring to Hausdorff's definition
of topological spaces, in 1923H. Tietze feels the necessity to add in a footnote
that these relations by the way are quite arbitrary assignments (d. p. 292 in
[111]). Up to the author's knowledge there exists only one vague comment on
the intuitive nature of topological spaces given by A. Appert in 1951:
Pour qu'un ensembleE d'objects de nature quelconquc puisseetre con
sidere commerepondant auconcept intuitifdespatialite, ilest necessaire
de s'etre donne non seulement les elements au points de I'ensemble E,
maisaussid'avoir precisecertaines relations deproximiteau desituation
decespoints lesuns par rapport aux autres. C'est seulement quand ces
relations auront ete fixees quel'on pourra dire qu'une topologie est intro
duite dansI'ensembleE, ouencoreque E est un ensemble topologique (cf.
p. 10 in [3]).
After the emergence of categor'y theory in the mid-forties of the 20th century
this lack of intuitive understanding makes the categorical axiomatization of
topological spaces difficult and therefore leads to the following problem:
What are the categorical axioms of topological spaces ?
In contrast to general topology it is immediately clear that algebraic structures
can be internalized in certain categorical frameworks - e.g. group theory can
be done within any category with finite products (cf. [22]), or monoids can
be defined in any monoidal category (cf. [74, 75]). On the other hand, there
have been attempts to do topology in a categorical framework which lead to
the concept of topological categories. Essentially, these categories share the
important property of existence ofinitial and final structureswith the category
of ordinary topological spaces (see e.g. [1]). In this sense categorical topology
does not make any contributionto thesolutionoftheabove-mentioned problem.