Axiomatic characterization of physical geometry

Lecture Notes in Physics Edited by 1. Ehlers, Mtinchen, K. Hepp, Ziirich R. Kippenhahn, Mtinchen, H. A. Weidenmtiller, a...

Author: Heinz-Jurgen Schmidt

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Lecture Notes in Physics Edited by 1. Ehlers, Mtinchen, K. Hepp, Ziirich R. Kippenhahn, Mtinchen, H. A. Weidenmtiller, and J. Zittartz, K61n Managing Editor: W. BeiglbGck, Heidelberg

Heidelberg

111 H.-J. Schmidt

Axiomatic Characterization of Physical Geometry

Springer-Vet-lag Berlin Heidelberg

New York 1979

Author Heinz-Jiirgen Schmidt Fachbereich 5 Naturwissenschaften/Mathematik Universittit Osnabriick Postfach4469 D-4500 Osnabrtick

ISBN 3-540-09719-8 ISBN o-387-09719-8

Springer-Verlag Springer-Verlag

Berlin Heidelberg New York New York Heidelberg Berlin

Library of Congress Cataloging in Publication Data Schmidt, Heinz-Jtirgen, 1948. Axiomatic characterization of physical geometry. (Lecture notes in physics; 111) Bibliography: p. Includes index. 1. Geometry. 2. Axiomatic set theory. I. Title. II. Series. QC20.7.G44S35 530.1’5162 79-23944 ISBN 0-387-09719-E

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 5 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 0 by Springer-Verlag Printed in Germany

Berlin

Printing

Beltz

and

2153/3140-543210

binding:

Heidelberg Offsetdruck,

1979 Hemsbach/Bergstr.

PREFACE This book will deal with the basis of a theory, which can be considered as the most ancient part of physics, namely Euclidean geometry.

For

about 100 years there has been a debate on the physical space problem, especially stimulated by the creation of tivity.

(non-Euclidean)

General Rela-

In spite of this, contrary to the impression generated by some

textbooks on physics,

the topic is far from being in a final form. The

problems of interpretations often neglected,

and definitions of physical concepts are

partly because methodological rigor is

replaced by physical intuition,

and partly because these problems are

inherently difficult and inextricably intertwined. situation in mathematics, pre-Bourbaki millenium.

(successfully)

In contrast to the

the foundations of physics are still in their

I think, however, G. Ludwig has made an impor-

tant step toward an adequate understanding of physics,

and this book

may be viewed as a partial realization of one point of his program. A large class of physical applications of Euclidean geometry concerns constructions with rigid bodies.

Thus geometry yields propositions

about the behaviour of these bodies and is, in this sense, an emperical theory. This standpoint was adopted by H. v. Helmholtz

[HELl and

A. Einstein, who wrote: "Feste K~rper verhalten sich bez~glich ihrer Lagerungsm~glichkeiten wie K~rper der euklidischen Geometrie von drei Dimensionen;

dann enthalten die SMtze der euklidi-

schen Geometrie Aussagen ~ber das Verhalten praktisch starter K~rper." Consequently,

([EIN] p. 121)

G. Ludwig suggested

[LUD2] going one step further and

formulating geometry explicitly as a theory of possible operations with practically rigid bodies, and "transport".

using as basic concepts

"region",

"inclusion"

IV In 1977 I started carrying out this program in detail. completed by connecting mathematical

the theory of regions and transports with the

results on the Helmholtz-Lie

a second part dealing with mobility this approach was presented [SCHI]. Following conference, completed

problem

in Osnabr~ck

to the Fachbereich

schaften der Universit~t

Osnabr~ck

and accepted

in November

1978.

In conclusion

I should

at this

"Zum physikalischen

5, Mathematik/Naturwissen-

as the author's

like to thank K. B~rwinkel,

Habilitationsschrift

J. Ehlers, A.

Hartk~mper,

A. Kamlah,

couragement

and interest have been of great value to me. Further I

express my gratitude

G. Ludwig,

D. Mayr and G. S~Bmann, whose en-

to T. and M. Louton

for revising the translation

of my manuscript.

I have also much appreciated

Frau A. Schmidt's

rapid and accurate

August

1979

Frau P. Ellrich's

typing of the manuscript.

~ "

~'~

1977

with rigid bodies, which

The German version entitled

was presented

by chains,

in November

generated by the discussions

I added a chapter on operations

Raumproblem"

[FRE]. Together with

and distance measurement

at a conference

suggestions

this work.

One part was

~'-~" ~

and

CONTENTS

I

I. I n t r o d u c t i o n

17

2. O p e r a t i o n s w i t h rigid bodies 2.1 General e x p l i c a t i o n

17

2.2 C o n s t r u c t i o n of regions

26

2.3 C o n s t r u c t i o n of transport mappings

48

3. Regions and t r a n s p o r t mappings

61

3.1 4 Axioms

61

3.2 Points

66

3.3 Regions as point sets

71

3.4 C o n g r u e n t mappings

79

3.5 Chains I

83

3.6 C o m p l e t i o n of the group

91

3.7 Chains II

I O0

4. The H e l m h o l t z - L i e p r o b l e m

108

4.1 I m p l i c a t i o n s of the theorem of Yamabe

108

4.2 M o b i l i t y and distance m e a s u r e d by chains

118

4.2.1 Proof of "(i) =>

(ii)"

4.2.2 Proof of "(ii) => 4.2.3 Proof of "(iii)

=>

122

(iii)"

126

(i)"

132

4.3 T i t s / F r e u d e n t h a l c l a s s i f i c a t i o n

149

5. C h a r a c t e r i z a t i o n of E u c l i d e a n geometry

152

5.1 D i m e n s i o n

152

5.2 C u r v a t u r e

154

5.3 E u c l i d e a n r e p r e s e n t a t i o n

155

6. R e f e r e n c e s

160

7. N o t a t i o n s

163

1.

INTRODUCTION

This book p r e s e n t s

an a x i o m a t i c

p hysic a l

This will

geometry.

exploring

1.1.

some problems

Geometry,

spacetime), theory

contains

very general the

geometrical

identification

occuring

reduced

- via E. N o e t h e r ' s concepts

theorem

of space

groups.

p hysic a l

theories,

Geometry

is the m a i n m e d i u m

Another

aspect

If We d e s c r i b e

traced back

to a g e o m e t r i c a l

relation

"pre-theory" [LUD 3]).

The

statements

physical

in terms

precise

theory, "data"

can be of the

by d i f f e r e n t

between

these

theories.

This

measurement

notion

manner,

theory

is o f t e n

using

PT I, w h i c h

of "tracing the interis a

PT2, under c o n s i d e r a t i o n in

can be

(see

PT 2 consist of t h e o r e t i c a l

PT I. These statements in turn are

of basic

construction

of theories

Moreover,

and the c o r r e s p o n d i n g

every p h y s i c a l

of a p h y s i c a l

another

experiments of physics

and p r e - t h e o r i e s

in

PTI, and so forth.

would

where

consist

geometry

of a

is located

the outset. Therefore,

an a x i o m a t i c

some

momentum,

theories

role of g e o m e t r y

measurement.

in a more

experimental

a systematic

hierarchy

: (almost)

of the p r e - t h e o r y

interpretable Thus,

r.e.

physical

theory).

such as energy,

spacetime)

(resp.

of such a connection.

as follows

theoretical

of

for the m o m e n t

the same nature

fundamental

can be r e s t a t e d

space

Every

or q u a n t u m

there must be a c o n n e c t i o n

of the

intention

- to the i d e n t i f i c a t i o n

(resp.

formulated

back"

concepts

of

space.

(ignoring

in d i f f e r e n t

s ymmet r y

the

role in physics.

concepts

of p h y s i c a l

momentum,

different

physical

of t h e r m o d y n a m i c s

angular

with

as the theory of physical

a constitutive

versions

to the f o u n d a t i o n s

be d e v e l o p e d

dealing with

understood

plays

approach

formulation

of g e o m e t r y

as a p h y s i c a l

at

theory

is of c o n s i d e r a b l e

especially has no

for a t h e o r y w h i c h

pre-theory

Geometry,

interest

(at least

understood

for m e t h o d o l o g i c a l

is a p r e - t h e o r y in the sense

as a p h y s i c a l

for all others

indicated

theory,

research, but

above).

presents

two p r i n c i p a l

questions: I. How can the g e o m e t r i c a l sense may g e o m e t r y 2. W h e r e

do we know

If it is p o s s i b l e second

question

validity

of a p h y s i c a l

course,

this

For

instance,

as "logical

investigated

There

in

true,

in c o n n e c t i o n

for i n s t a n c e

(see

reference

whose

cases

the results

with

each

criteria

of the

a theory

is a c c e p t e d

of experiments.

as it appears

"conflict"

Moreover,

other opinions those w h i c h

which is t h o u g h t

[BOH]). This

could

theory

on the

Of

sur-

not be u n t e r s t o o d is at m o s t should

of v a l i d i t y

occupy

of the e m p i r i c a l

w.r.

assume

be

(see the dis-

brings

geometry

being

be d i s c u s s e d

us back

the r e m a i n d e r content

to the v a l i d i t y

to be the a-prior i base

o p i n i o n will

to e x p e r i m e n t s

a n s w e r will

problem

with

speaking,

the

[LUD 3]).

"protophysics", physics

Roughly

theory,

question

and the role of a p p r o x i m a t i o n

are n e v e r t h e l e s s

geometry,

The

in most

in what

(if at all)?

as a p h y s i c a l

is not as trivial

contradiction".

approximately

is "true"

to the general

theory.

or,

"real things"?

geometry

not c o n f l i c t

statement

be i n t e r p r e t e d

to

that g e o m e t r y

is r e d u c i b l e

if it does

cussion

be a p p l i e d

to f o r m u l a t e

as true

face.

concepts

a part of of e m p i r i c a l

briefly

to the initial

of this book.

of geometry,

of

below.

question,

To a p p r o a c h

3 scales

of d i m e n s i o n

n e e d to be d i s t i n g u i s h e d : The m i c r o s c o p i c

(~),

the m a c r o s c o p i c

the

(or "!aboratory") (L) and the

astronomic

(A) dimension.

restricted

to the l a b o r a t o r y

perception

arises

Moreover,

we will

operating

with

physical

for example

(e.g.,

our g e o m e t r i c a l

to that part of L - g e o m e t r y rulers

and compasses,

since we feel,

of g e o m e t r i c a l

aspects

or

that this g e o m e t r y

optics

is doubtful

relativity.

(e.g.,

whether

independently

as well

is

as of other

utilizing

Clearly

there

is a close

The e x p e r i m e n t s

which

a small

take place

by L - g e o m e t r y

- e.g.

permit

of a u n i v e r s a l

occur,

pretheory

On the other hand, of these more

scale

structure, interpret could

imply

that

formulate

or general

the L - g e o m e t r y

the nature

should

theories,

as a theory

of

encounter

theories has

geometries. space

processes,

are d e s c r i b e d

Hence L - g e o m e t r y

and only L - g e o m e t r y as m e n t i o n e d

the various

Such p r o c e s s e s

of such

must

in w h o s e

the

be context

characteristics

above.

be p o s s i b l y

viewed

as a limit

not only due to its m a t h e m a t i c a l

but also due to its rules L-geometry

theory

ultimately

"L-theories".

L-geometry

extensive

It

of q u a n t u m

us to deduce

as one of the p r e - t h e o r i e s

~- and A - g e o m e t r y

theories.

can be f o r m u l a t e d

between

in the L-dimension.

and other

and

mechanics.

connection

or large

of certain

geometries

to this we shall

classical

by means

or telescopes)

the c o r r e s p o n d i n g

In c o n t r a s t

and the a s t r o n o m i c

indirectly

using m i c r o -

of such theories

without

on e i t h e r

of the m i c r o s c o p i c

can only be e x p l o r e d

L-experiments

this

is

geometries.

dimensions

viewed

approach

it works well.

ourselves

and joists),

The g e o m e t r i c a l

which

confine

axiomatic

dimension, from w h i c h

and in w h i c h

rigid bodies

building-stones a pre-theory

The present

of interpretation.

of the a s s e m b l a g e

If we

of r i g i d bodies,

general r e l a t i v i t y t o g e t h e r w i t h equations of matter, respectively,

q u a n t u m theory of solid state,

p o s s i b i l i t y of a s s e m b l i n g

c e r t a i n bodies,

or,

could p r o v i d e the

thus revealing

e u c l i d e a n s t r u c t u r e of space in l a b o r a t o r y dimensions. of such a p r o b l e m of c o n s i s t e n c y

pre-theory

=

\

the

The solution

s y m b o l i z e d by the d i a g r a m

r e s t r i c t e d theory

/

m o r e e x t e n s i v e theory

w o u l d l e g i t i m i z e and e x p l i c a t e the a f o r e m e n t i o n e d

i d e n t i f i c a t i o n of

various c o n c e p t s of space in d i f f e r e n t theories.

E v e n in the case of l a b o r a t o r y g e o m e t r y the r e l e v a n t concepts - "point",

"line",

(from now on just geometry) "plane",

"angle" - have no d i r e c t p h y s i c a l meaning. r e p r e s e n t e d by "small" e x p l a i n in p h y s i c a l

spots or m a r k i n g s

"distance"

and

W h e r e a s points may be

it is m o r e d i f f i c u l t to

terms w h a t a line or a d i s t a n c e b e t w e e n two

points is. Of course,

it is p o s s i b l e to c o n s i d e r c e r t a i n p r o c e d u r e s

p r o d u c i n g s t r a i g h t edges or for c o m p a r i n g d i s t a n c e s

and to "define"

the c o r r e s p o n d i n g c o n c e p t s o p e r a t i v e l y by these procedures.

Basically, proceeds

the p r o t o - p h y s i c a l

in this way.

a p p r o a c h of the E r l a n g e n - K o n s t a n z

They f o r m u l a t e standards

group

for m e a s u r i n g devices

and s o - c a l l e d "principles of h o m o g e n e i t y " ,

from w h i c h they seek to

derive a Euclidean geometry

83 ff.).

(see [BOE]

p.

This a p p r o a c h seems to d e p r e c i a t e the e m p i r i c a l basis of g e o m e t r y favour of a n o r m a t i v e basis.

However,

one can argue,

e m p i r i c a l c o n t e n t of g e o m e t r y is then m a n i f e s t e d

that the

in the tacit

in

a s s u m p t i o n of p r a c t i c a b i l i t y of the standards or w o r k a b i l i t y of the procedures. When one tries to s t r i n g e n t l y analyze the conditions of g e o m e t r i c a l p r o c e d u r e s one must translate the p r i m i t i v e g e o m e t r i c a l operations into a m a t h e m a t i c a l

language and formulate the conditions of

w o r k a b i l i t y as m a t h e m a t i c a l axioms:

If,

for example,

rods,

the goal of the present volume.

one compares distances by t r a n s p o r t i n g m e a s u r i n g

these rods must not be deformed during transport.

s a t i s f a c t o r y to claim:

It is not

"experience shows that they are not deformed",

b e c a u s e d e f o r m a t i o n would need to be m e a s u r e d by other n o n - d e f o r m e d m e a s u r i n g rods. A universal d e f o r m a t i o n is not detectable. and does not exist,

Hence it is m e a n i n g l e s s

says the operationalist,

m e a s u r e d by t r a n s p o r t i n g m e a s u r i n g rods.

d i s t a n c e is what is

In p r i n c i p l e we agree,

w o u l d still try to improve on this argument at two points. conditions of the w o r k a b i l i t y of the p r o p o s e d operations e x p l i c i t e l y formulated. distance

but

First,

the

should be

One apparent condition in the c o m p a r i s o n of

is, that two m e a s u r i n g rods made from different m a t e r i a l s

have the same length before some transport, the same length after the transport as m a t h e m a t i c a l

axioms,

if and only if they have

(see ( 2 3 1 8 ) ) .

these conditions make it p o s s i b l e to define

the concepts under c o n s i d e r a t i o n as m a t h e m a t i c a l f o r m a l i z e d physical

When f o r m u l a t e d

theory.

terms w i t h i n the

This has the additional advantage that

we are now no longer r e s t r i c t e d to one specific method of m e a s u r i n g a quantity. theory

Each appropriate theorem of the m a t h e m a t i c a l part of the

(e.g., on the e q u i v a l e n c e of two definitions)

another possible corresponds

"operational definition",

to the same physical concept.

now yields

which n e c e s s a r i l y

This is the second i m p r o v e m e n t of an o p e r a t i o n a l i s m w h i c h does not take into a c c o u n t the de facto p l u r a l i s m of m e a s u r i n g methods. (Admittedly it w o u l d be n e c e s s a r y to study other p h y s i c a l theories such as g e o m e t r i c a l optics and their c o n n e c t i o n to rigid body g e o m e t r y in order to c o n s i d e r the full p l u r a l i s m of g e o m e t r i c a l measurements.)

In short,

the above is the s t a n d p o i n t of G. Ludwig

w h i c h we adopt.

Given a m a t h e m a t i c a l

(see [LUD 1,2,3]),

f o r m u l a t i o n of a p h y s i c a l theory,

c e r t a i n sets and r e l a t i o n s of the theory play the role of p h y s i c a l l y i n t e r p r e t a b l e terms;

the i n t e r p r e t a t i o n being either direct or

J d e r i v e d from s p e c i f i c pre-theories.

H o w e v e r the d e v e l o p m e n t and

s u b s t a n t i a t i o n of the m e a s u r i n g p r o c e d u r e s

for the n o n - i n t e r p r e t e d

terms is a c h i e v e d by a p p r o p r i a t e m a t h e m a t i c a l

constructions within

the t h e o r y . If it is p o s s i b l e to derive all terms and t h e o r e m s of the theory from the i n t e r p r e t e d terms by means of some axioms expressible

in these terms,

one has reached the a x i o m a t i c basis of

the theory.

T u r n i n g back to geometry, relations)

we have to decide w h i c h terms

of E u c l i d e a n g e o m e t r y of the 3 - d i m e n s i o n a l

s u i t e d as i n t e r p r e t a b l e terms,

(sets,

space E 3 are

i. e. w h i c h terms are as close as

can be p o s s i b l e to the p h y s i c a l a p p l i c a t i o n s of geometry.

Following

the a p p r o a c h of G. L u d w i g

([LUD 2] II and IV), w h i c h in some aspects

is due to H. v. H e l m h o l t z

(see [HEL]), we choose a class of subsets

of E 3, called

(spatial)

by c o n f i g u r a t i o n s

regions, w h i c h are e x p e r i m e n t a l l y r e a l i z a b l e

of fixed bodies.

The inclusio__nn of regions w o u l d

c o r r e s p o n d to the r e a l i z a t i o n of "sub-bodies" w i t h i n these configurations.

F i n a l l y the group of c o n @ r u e n t m a p p i n g s

lations and proper rotations

(and their products)

formed by transcould be inter-

p r e t e d as d e s c r i b i n g the t r a n s p o r t of rigid bodies.

The

formulation

considering

of an a x i o m a t i c

an a b s t r a c t

given by a r e l a t i o n relations

T on R

is subject physical These

over

points

class

structure,

basis

We will

regions

of b o u n d e d

namely

tell

open

subsets

of as

(R,<,T)

can be

above.

(Being

for some kind of

(R,c,T).)

us about the i n t e r p r e t a t i o n

define

(= m i n i m a l

but

(R,<,T)

of r i g i d bodies.

indicated

(R,<,T),

structure

the triple

in such a way that

points

as c e r t a i n

Cauchy prefilters).

of E 3, the points

points

in R, see

is i n v o l v e d

could be d e f i n e d

(3313)

ff.).

of

systems

of

If R = R is the

of E 3 are r e c o v e r e d

and since the c o r r e s p o n d e n c e entities

is at best

§ 6), one could

(and must)

represent

Although translate

this was each

ning regions

clear,

pluralism"

theorems

does

by small

of idealizareal things (see

[LUD 3]

regions.

an i n s t r u m e n t

into a statement

more

apparent

By dint of the axioms P becomes

the c o n s t r u c t i o n

not lead

"points"

points

becomes

the space of points

However,

an a p p r o x i m a t i o n

of regions

to

concer-

mappings).

of distances.

factor).

"limits"

between

we now have

concerning

(and t r a n s p o r t

"operational

postulated,

previously

statement

the m e a s u r e m e n t

as minimal

In any event a process

and m a t h e m a t i c a l

ment.

Further,

the

to

and a set T of

and t r a n s p o r t

for

amounts

construction.

(= atoms

This

of regions")

w h i c h may be thought

fixation

not hold

the a x i o m a t i c

Equivalently,

tion

the

thus

endowed w i t h

the space E 3 in the m a n n e r

and distances?

by this

of axioms,

this does

"contracting"

"regions"

mappings").

to be choosen

of this

does

of g e o m e t r y

("inclusion

("transport

have

precise,

completion

What

< on R

laws c o n c e r n i n g

represented more

set R of

to a series

axioms

basis

immediately

a metric

used to d e v e l o p e to a m e t h o d

when we c o n s i d e r w h i c h will space

be

(up to a

the e x i s t e n c e

of d i s t a n c e

measure-

One p o s s i h i l i t y reached

equipped

form, with

the axioms centric

so to speak,

an i n c l u s i o n

balls

in the work

method

[LUD 2],

congruent

regions,

points

should

book.

measuring

methods

guaranteed

In addition, basic

terms

derived

try to give

"body",

leading

from single

spatial

regions.

regions

which

The w h o l e

approach

in a recent

can be found

[BOE]. of o v e r l a p p i n g of a c h a i n w i t h

to connect

become

two points.

connecting

smaller

is a m a i n

necessary

to point

the a p p l i c a b i l i t y

and e s t a b l i s h formulate

terms

two pairs

The of

distances

and smaller. topic

of this

out that d i f f e r e n t

of w h i c h

is

Congruence

the i n t e r p r e t a t i o n

a pre-theory

associated

account

bodies

fixed

body".

with

In this

of the process at c e r t a i n

is i n t r o d u c e d

can be r e p r e s e n t e d

construction

Proving

This

of the c o r r e s p o n d i n g

of chains

"reference

an e x p l i c i t

naturally

of the theory.

R,<,T we will

from f u n d a m e n t a l

These

set of con-

distances

of links

of chains

of the chains

to f a c i l i t a t e

"inclusion",

number

to the q u o t i e n t

are p o s s i b l e

the

reals.

consist

are n e c e s s a r y

it is only

balls,

"addition".

allows

of)

points w h i c h

"lengths"

by the axioms

fixed point.

and s i m i l a r l y

of W. B~chel

study of the t h e o r y

For the m o m e n t

[TIT]

(proportions

class w h i c h

links

system

can be

[B&K].

and the m i n i m a l

converge

as the c o n g r u e n t A thorough

of J. Tits

between

of the m i n i m a l

and a law of

the n o n - n e g a t i v e

and A. K a m l a h

"chains"

fixed c o n g r u e n c e quotient

onto

IV or in the essay

It c o n s i d e r s

a single

a s y s t e m of c o n c e n t r i c

extensive

of m e a s u r i n g

sets of points w h i c h

containing

relation

to be m a p p e d

of W. B a l z e r

Another

regions

of an s o - c a l l e d

can be found

in

be to c o n s i d e r

by t r a n s p o r t e d

sets will

p aper

would

by the

works within

in w h i c h

R,<,T

the c o n c e p t s pre-theory

of the

"process",

we will

of a b s t r a c t i o n

reference

bodies

as a r e l a t i o n

to

between

same body.

some a r b i t r a r i l y

are

chosen,

but

fixed

reference

ference

system,

bodies.

the r e l a t i o n

i. e. a c e r t a i n

It remains

between

It is c o n v e n i e n t

"inertial

system",

already

exists

speaking,

which

within

systems

regions

does

§ 2). One

to m e n t i o n

is usually

not d e p e n d

can o b s e r v e

"steel-congruent"

details

see

section

theory,

tion

theorem

work

on c e r t a i n

history

and

to a n a l y z e reference

the concept

to be purely

of p h y s i c a l

within

Roughly

the re-

relation (see

deformation have

of an

kinematic,

geometry.

bodies

one w o u l d

of

[LUD 2] II of bodies

to d i s t i n g u i s h

"plastic-congruent".

of a p h y s i c a l

in most

of complexity,

quantum

thought

of re-

For more

2.1.

Axiomatization

degree

that

that the c o n g r u e n c e

therefore

say

requires

here

a material-dependent

between

above

of d i f f e r e n t

on the r e p r e s e n t i n g

systems;

class

theory

s y s t e m may be c h a r a c t e r i z e d

by the p r o p e r t y

in n o n - i n e r t i a l

1.2.

geometries

the f o u n d a t i o n s

an inertial

ference

a task of a k i n e m a t i c

different

systems.

equivalence

cases when

and c o n c e r n

is n o r m a l l y

geometry.

mathematical

in the sense

a representation

even provable.

for example,

in p r o j e c t i v e

theory

ideas

the a x i o m a t i c

theorem

discussed

of a high

The a x i o m a t i z a t i o n based

We wish

of

on the c o o r d i n a t i z a -

to base

and results

the p r e s e n t

which

characterization

have

a long

of a class

of

geometries.

It is not by chance,

that this

H. v. Helmholtz,

in 1866 f o r m u l a t e d the idea of the i n t i m a t e

connection

between

free m o b i l i t y axiomatic Starting

who

the p o s s i b i l i t y

of m e a s u r i n g

characterization with

and b r o u g h t

tradition

S. Lie,

rods

of m e a s u r i n g

and d e v e l o p e d

of the classical

1890,mathematicians

them into a rather rigorous

of c h a r a c t e r i z i n g

a class

goes back

of g e o m e t r i e s

to

a

distances from this

geometries

have form.

physicist,

then

bv p o s t u l a t e s

idea an

[HEL].

improved Since

and the

on his

ideas

the p r o b l e m

of m o b i l i t y

on

10

the group of automorphisms,

k n o w n as the H e l m h o l t z - L i e p r o b l e m has

b e e n t r e a t e d by a great n u m b e r of m a t h e m a t i c i a n s . in [FRE] is quite complete.) become somewhat

As one can imagine,

(The b i b l i o g r a p h y the p r o b l e m has

i n d e p e n d e n t of its p h y s i c a l origin,

w h i c h this w o r k will

a condition

(try to) r e m e d y .

The m o s t general results on the H e l m h o l t z - L i e p r o b l e m have been o b t a i n e d by J. Tits

[TIT] and by

1954. They used L i e - a l g e b r a i c

H. F r e u d e n t h a l

[FRE] in 1953 and

t e c h n i q u e s and the t h e o r e m of Yamabe

in order to drop the a s s u m p t i o n of a m e t r i c as well as earlier a s s u m p t i o n s of d i f f e r e n t i a b i l i t y . The basic c o n c e p t s now are: a set of points,

a topological

point t r a n s f o r m a t i o n s .

(121)

s t r u c t u r e on it and a group of

The axioms read in the F r e u d e n t h a l version:

A c o m p l e t e group of u n i f o r m h o m e o m o r p h i s m s t r a n s i t i v e l y on a connected, Hausdorff

locally c o m p a c t / u n i f o r m

space such that the stability s u D g r o u p g e n e r a t e s

at least one orbit w h i c h d i s s e c t s

This d e t e r m i n e s

operates

the space.

a rather small class of g e o m e t r i e s and groups

(see

s e c t i o n 4.3). The p h y s i c a l r e l e v a n c e of the postulates, i m m e d i a t e l y clear.

For example,

however,

should the

is not

c o m p l e t e n e s s of the

group or the local c o m p a c t n e s s of the space be regarded as a m a t t e r of purely m a t h e m a t i c a l c o n v e n i e n c e or as a law of nature?

Our a x i o m a t i c

formulation,

s t a r t i n g with the

partial answer to this problem.

(R,<,T)-theory yields a

The p o s t u l a t e s of

by means of m a t h e m a t i c a l d e f i n i t i o n s

(121) are o b t a i n e d

and constructions,

for inStance

by c o m p l e t i o n of the group of t r a n s p o r t m a p p i n g s T. So far this is a

11

q u e s t $ o n of m a t h e m a t i c a l convenience.

But these c o n s t r u c t i o n s

only be p e r f o r m e d if certain axioms w i t h i n the assumed.

can

(R,<,T)-theory are

S i m i l a r l y some of the F r e u d e n t h a l axioms are e x p r e s s i b l e as

"pre-axioms"

in the

(R,<,T)-setting.

the p h y s i c a l core of the p o s t u l a t e s

Thus our a x i e m a t i z a t i o n reveals (121). Moreover,

this m e t h o d

shows some u n e x p e c t e d connections between properties, w h i c h seem c o m p l e t e l y d i f f e r e n t at first glance.

In this context we draw the

readers a t t e n t i o n to the double i n t e r p r e t a t i o n of a x i o m R4 (317))

as a p r o p e r t y of "local p r e - c o m p a c t n e s s "

A n o t h e r example is t h e o r e m

(see

or "pre-transitivity".

(325), which shows the intimate c o n n e c t i o n

b e t w e e n the p o s s i b i l i t y of m e a s u r i n g distances by means of chains and the R i e m a n n i a n structure of space. Hence,

our main task consists of finding a pathway w h i c h leads from

the rather general structure of regions and transport m a p p i n g s the F r e u d e n t h a l

1.3.

to

system.

The following is a brief outline of the structure of this

work.

In section 2 we develop a p r e - t h e o r y As m e n t i o n e d above, basic concepts mapping",

PT I w.r. to the g e o m e t r y PT 2.

our aim is to give an i n t e r p r e t a t i o n of the

in PT2, namely

"regions",

"inclusion",

"transport

by means of concepts which are closely related to

e x p e r i m e n t a l situations,

namely

"process",

"inclusion

(of processes)",

"body" and "reference body".

The concept of a "process" as a p r e - r e q u i s i t e

could be avoided and really serves only

for later studies of kinematics.

Moreover,

section 2 may be c o n s i d e r e d as an analysis of the axioms RI, R2 in

PT 2 (i.e.:

(R,<) is a w e a k l y d i s t r i b u t i v e lattice and T a subgroup

of Aut R) in terms of the p r e - t h e o r y

PT I.

12

In section mappings.

3 we f o r m u l a t e the theory

PT 2 of regions and t r a n s p o r t

The group T is m a d e into a t o p o l o g i c a l group

natural way:

a t r a n s p o r t is "small",

(T,t)

in a

if a c e r t a i n region meets the

t r a n s p o r t e d region. A x i o m R3 is the s i m p l e s t way to insure c o n t i n u i t y of m u l t i p l i c a t i o n axiom of "rigidity". p r e f i l t e r s on

in

"Points"

(T,t). It r e s e m b l e s F r e u d e n t h a l ~ s

are now i n t r o d u c e d as m i n i m a l C a u c h y

(R,<). By a x i o m R4, a c o v e r i n g

(except the least region O)

"contains"

of points P becomes a complete,

law, each region

at least one point.

uniform,

locally compact

The set

(by R4)

H a u s d o r f f space.

Regions may be r e p r e s e n t e d by open, and t r a n s p o r t m a p p i n g s

r e l a t i v e l y compact subsets of P

induce u n i f o r m c o n t i n u o u s h o m e o m o r p h i s m s of

P, w h i c h are c a l l e d " c o n g r u e n t mappings". in general

faithful,

is c o m p l e t e d w.r.

g e n e r a t e s the lattice

This r e p r e s e n t a t i o n ,

not

(R,c) and the group T.

to the t w o - s i d e d u n i f o r m i t y

induced by t, thus

g e n e r a t i n g a group T, also c o n s i s t i n g of a u t o m o r p h i s m s of the uniformity on P. Similarly,

R is dense

in the lattice R of all open,

r e l a t i v e l y compact subsets of P.

The next steps p r e p a r e the proof of F r e u d e n t h a l ' s ly he p o s t u l a t e s

completeness

compact convergence, uniformity.

"pre-connectedness"

later.

theorems

to a finer in this

To this end we p o s t u l a t e axioms R5, a kind of of T, w h i c h is m a t h e m a t i c a l l y m o r e than we need,

but appears p h y s i c a l l y

"chains"

Unfortunate-

to the u n i f o r m i t y of

whereas we had to c o m p l e t e T w.r.

Hence we m u s t prove F r e u d e n t h a l ' s

d i f f e r e n t setting.

We define

of the group w.r.

axioms.

reasonable.

and compile some simple p r o p e r t i e s to be used

Chains of m i n i m a l

d i s t a n c e s b e t w e e n points

length may serve to define a p r o p o r t i o n of ("chain quotient"),

if c o n v e r g e n c e may be

19

assumed.

For this

(R,<,T)-chains

it proves

or

(R,c,T)-chains.

to f o r m u l a t e

an e q u i v a l e n t

This

in section

is done

mobility

is e s s e n t i a l l y

quotient. type,

In this

elliptic Cayley

space

corresponding

version

whether

one c o n s i d e r s

it is then p o s s i b l e

of axiom R6 on the

(R,<,T)-level.

4. We show that F r e u d e n t h a l ' s

a x i o m of

equivalent

of the chain

to the c o n v e r g e n c e

in the c l a s s i f i c a t i o n isomorphic

over

numbers.

Therefore

case P is a R i e m a n n i a n

as is shown

P is e s s e n t i a l l y

to be i r r e l e v a n t

either

the real,

theorem

of e q u i v a l e n c e metric

may be viewed

as an exact

or q u a t e r n i o n s

the fifties,

a new insight

and R i e m a n n i a n

structure

reformulation

special

or hyperbolic,

up to some known

since

provides

dance b e t w e e n

to an affine,

to the I d - c o m p o n e n t

group of isometries

this has been known

of a very

of T i t s / F r e u d e n t h a l .

or coraplex,

T is i s o m o r p h i c

Whereas

manifold

or

or

of the exoentions.

the above m e n t i o n e d into

the

interdepen-

of p h y s i c a l

space

of the c o r r e s p o n d i n g

and

idea of

H. v. Helmholtz.

It is not d i f f i c u l t an isotropic more

Riemannian

involved.

invariant

First,

subgroups

P/N a C~-manifold. over

to c o n s t r u c t

in this

setting.

tive

transformations

if the s t a b i l i t y

[YAM]

that T/N

operate

subgroup

distance,

if c o n n e c t e d

operates locally

would

a chain

along

direction

geodesics

in

of the proof

is

is a Lie group

exists

and hence

by rotations "almost"

consisting

is

a canonical

carried chart

as translations.

of long,

in a "forbidden"

in

and the transi-

not operate t r a n s i t i v e l y

to a point

small

of the chain q u o t i e n t

Further there

subgroup

chains

one can find a r b i t r a r i l y

The c o n v e r g e n c e

the s t a b i l i t y

be k i n k e d

by

The other

N of T such

which

of con s t a n t

space.

minimal

thin

Now,

on the sphere links w o u l d

direction

and the

14

o~

'I

forbidden

direction

/

.> fig.

minimal Other

length

chains w i t h

restriction the

of these

chains w o u l d

"globular"

of m o b i l i t y

form of the links,

means level.

Plastically

A Riemannian

minimal

each

is k n o w n

chains.

of section

3-space

on

from

if one p o s t u l a t e s

in s e c t i o n

of a m e t r i c

on the space

3 by

(R,<,T)can be

properties.

could be lined w i t h

4 building

to be flat is half

4 this

is done

depend

of convergence.

Euclidean

can be f o r m u l a t e d

region

line of two sides

By the results

3. This

to a

could

the a s s u m p t i o n

by T i t s / F r e u d e n t h a l

covering

that m a x i m a l l y

space

the c h a i n q u o t i e n t

the d i m e n s i o n

by c e r t a i n

a manner

bisecting

obtained

that

speaking,

could not be s e n s i t i v e

to s e p a r a te

R7 and R8 w h i c h

It is k n o w n

characterized

such

matter

to be O and d i m e n s i o n

of axioms

increase.

contradicting

list of g e o m e t r i e s

curvature

links

and thus

It is a s t r a i g h t f o r w a r d the

(131)

stones

stonework

touch each other.

iff in each

triangle

as long as the t h i r d

is e x p r e s s i b l e

the

side.

as a p r o p e r t y

of

in

15

1.4.

How can these results be applied and what sort of new problems

arise from them?

First the a c h i e v e m e n t of an axiomatic physical g e o m e t r y furnishes an example to test some m e t h o d o l o g i c a l d e f i n i t i o n s p r o p o s e d in [LUD 3] § 10.5.

It concerns,

points and d i s t a n c e s

among others,

the question,

whether

for example may be viewed as "physically

r e a l i z a b l e facts ~' (reale S a c h v e r h a l t e

) in the sense d e f i n e d therein,

and w h i c h limit p r o c e s s e s need to be c o n s i d e r e d for this purpose. E q u a l l y i n t e r e s t i n g , t h e p r o b l e m of the a p p r o p r i a t e u n i f o r m i t y of physical u n c e r t a i n t y has not been tackled.

Further,

it w o u l d be d e s i r a b l e to extend the present a p p r o a c h to

space-time-geometry.

Here it is p o s s i b l e to p r o c e e d in various

d i r e c t i o n s . One way w o u l d be to consider a lattice of spatiotemporal regions

(i.e.

"processes")

and a group of "reproductions"

as basic notions and to use the results on the g e n e r a l i z e d H e l m h o l t z Lie p r o b l e m leading to p s e u d o - R i e m a n n i a n geometries has r e c e n t l y been a c h i e v e d by D. Mayr

[FRE 2]. This

[MAY]. A n o t h e r p o s s i b i l i t y

is

to obtain s p a c e - t i m e - g e o m e t r y by pasting together the g e o m e t r i e s of d i f f e r e n t inertial frames by means of coincidences.

The author is

c u r r e n t l y w o r k i n g on this aspect [SCH 2]. If we had insights as to, why spacetime is a smooth manifold,

per-

haps we could b e t t e r u n d e r s t a n d the physical reason of the a s s u m p t i o n of d i f f e r e n t i a b i l i t y in classical m e c h a n i c s

(see [LUD 2] p.

Besides the g e o m e t r i c a l m e a s u r e m e n t s of rigid body perimentalists

use a wide variety of other methods

392).

(RB) L - g e o m e t r y , e x from e l e c t r o n

m i c r o s c o p y to radio a s t r o n o m y in order to explore g e o m e t r i c a l extensions of objects.

From a m e t h o d o l o g i c a l point of v i e w this is

a q u e s t i o n of indirect m e a s u r e m e n t s

of g e o m e t r i c a l entities by means

18

of p h y s i c a l of c e r t a i n

laws of the c o r r e s p o n d i n g basic

introduced). analyzed

The

in terms

by a p r e - t h e o r y

(here L - g e o m e t r y

simplest

theory

these

in the

image plane

infer

some g e o m e t r i c a l

below

the

Euclidean

geometry

on a small

of an a x i o m a t i c

More

would

theoretical,

(non-relativistic) unitary yields cance

of such

transports But

others

the a p p a r a t u s

the total

a representation

not d e f i n e

by m e a n s cross

in terms

a theory

believes,

has a l r e a d y that the

from the E u c l i d e a n really

Euclidean

been

space(time) one,

(see

on a very

scales.

accuracy.

theories),

on a m i c r o s c o p i c

If a it is

e. g.

[LUD small

scale

by the fact

form of q u a n t u m

but we do not even know

on atomic

apparatus.

of an experiment,

(= p r e p a r a t i o n

achieved

signifi-

scale b e c a u s e

should be f a c i l a t e d

of the general

devices

which

experiment.

structure

experiments

group,

by c o n s i d e r i n g

an a r b i t r a r y

(and other

in a s c a t t e r i n g

Usually

by some

The p h y s i c a l

on a m i c r o s c o p i c

theory

under-

optics.

to g e o m e t r y

and m e a s u r e m e n t

within

of

entities.

establ{~hed

10 -15 m is the result

analysis

of m a c r o s c o p i c

apparatus)

the g e o m e t r y

of space(time)

of s c a t t e r i n g

an a x i o m a t i c

observable.

far

in case of q u a n t u m

of the Galilei

of p r e p a r a t i o n

of q u a n t u m

section

is c o u p l e d

are

the

of g e o m e t r i c a l

of g e o m e t r i c a l

is well

c a n n o t be a d j u s t e d

The r e c o n s t r u c t i o n

that

the p o s i t i o n

of for i n s t a n c e

calculated

theory

we may

thus be p a r t i a l l y

analysis

(up to a factor)

and m o v e m e n t s

this does

length

quantum

The p a t t e r n

plane w h i c h

the v a l i d i t y

formulation

measurements

representation among

scale could

be

can be i n t e r p r e t e d

(otherwise

instance

be the a n a l o g o u s

indirect

optics.

in the o b j e c t

For

is could

laws of lens r e f r a c t i o n

of R B L - g e o m e t r y

stood by m e a n s difficult

be g e o m e t r i c a l

structures

be useless).

relations

of a m i c r o s c o p e

By the

scale of a c c u r a c y would

in w h i c h

probably

of RBL-geometry.

microscope

and an i n t e r p r e t a t i o n

concepts

in detail w o u l d

of i n t e n s i t y

theory

theory

as

and m e a s u r e m e n t I]. N e a r l y scale

everybody

is d i f f e r e n t

to what

extent

it is

2.

OPERATIONS

2.1.

GENERAL

The p r i m i t i v e inclusion,

WITH

RIGID BODIES

EXPLICATION concepts

transport

of the theory

mapping

seem to be a p p r o p r i a t e regions

are o c c u p i e d

of regions

by solid,

mappings

are thought

occupied

successively

A closer

inspection

are t h e o r e t i c a l

that this attempt

however

process

process

reveals

to found a p h y s i c a l

Therefore

it will

formulation mappings

depends

operations

with

on certain

"primitive"

would become

in w h i c h a vicious

to give an explicite,

conditions, These

concepts

It may even be s u s p e c t e d

The c o n s t r u c t i o n

rigid bodies.

two regions

from the level of rigid bodies

geometry

of this process.

between

the use of geometry,

be n e c e s s a r y

and t r a n s p o r t

(or its copies).

of abstraction.

presupposes

Spatial

the i n c l u s i o n

of "sub-bodies"

that these

obtained

at first glanoe,

"at rest",

the t r a n s i t i o n

by the same body

and,

3 - region,

interpretation:

r i g i d bodies

to r e p r e s e n t

in section

intuitive

physical

to the i n c l u s i o n

constructs

via an i n t r i c a t e

- are very

for direct

corresponds

developed

appear

are the axioms

the

circle.

mathematical

of regions

which

case

and transport:

as the laws of

~I,

~2, BI to B8

and ]I. In this

section we will

now c o n s i d e r e d

as p r i m i t i v e

leading

to the formal

"Rigid"

bodies

Under w e a k abrasion.

or a pile of straw

exerted

on a body

compasses

stones,

of w a t e r

illustrate

walls,

or honey,

restricted

also may be r e g a r d e d t o g e t h e r with

boards,

deformation

a cloud of steam, If the

in order

as rigid,

a steel ball

are

2.2 and 2.3.

nails,

perceptible

which

the ideas

of part

are not rigid bodies.

are c a r e f u l l y

or a table

of the concepts

and axioms

they do not exhibit

a droplet

this body

further,

definitions

a chain

tions,

the m e a n i n g

and,

are for example:

forces Thus

explain

bricks. or a rope,

forces

to avoid d e f o r m a -

for example

a pair of

lying on it. This

18

concept

of r i g i d i t y

experience.

We are not c o n c e r n e d

body

concept

body

in relativity.

Neither which

in c l a s s i c a l

~re we

allows

looking

the

geometry

the degree possible

not a n a l y z e

of a c c u r a c y

to e x c l u d e

metry,

say w i t h

needed

for the

these

they

do not play

deformations

edges

effects;

step

shapes.

After

to c o n s t r u c t

the c e n t r a l

[REI]

they

Finally,

are p r o d u c e d

or s p h e r i c a l

forces, § 18). within

in c o n s t i t u t i n g

but of course

which

of geometry.

it is p o s s i b l e

by small

rigid-body-geometry

first

rigid

of the r i g i d body

(see for i n s t a n c e

The

all r i g i d b o d i e s

foundations

set up,

etc.

of the rigid

(non-existing)

of this kind of geometry.

straight

been

the

definition

as elastic

theories.

of e v e r y d a y

idealization

or w i t h

as an issue of r e f i n i n g

of richer

does

concept

the

for an a b s o l u t e

magnetostriction

is r e g a r d e d framework

with

mechanics

for all effects

heat expansion, This

is a p r e - t h e o r e t i c a l

it w o u l d be

are not

the c o m p l e t e

role as in the a p p r o a c h

of

restrict

u s i n g geo-

These

such s y m m e t r i c

the

theory

objects,

has but

"protophysics"

[BOE]. N e x t we turn to the n o t i o n only be d e f i n e d w i t h kinds are

parts

rest

thus

"rest"

"inclusion".

to d i f f e r e n t

But this

cannot

criterion

for the body b e i n g

we

to d i s t i n g u i s h

order

bodies"

("Ger~ste"

only

in

[LUD2]

p.

operations

We will

of

in

are to be

there m u s t be a

the transport. which

as

"rigid bodies"

we c o n s i d e r

Hence,

as r e f e r e n c e s

26 ff.)

Two b o d i e s

to be t r a n s i t i v e

in terms

a subset of bodies

the state of rest.

can

exist different

of reference.

fails

at rest after

to serve

"rest"

they can be c o n s i d e r e d

every body m u s t be moved.

the e x p e r i m e n t s to define

relation

Obviously

and there

systems where

If the t r a n s p o r t i n g

almost

during

bodies

be d e f i n e d

performed,

suggest

"at rest".

to o t h e r

in a s i t u a t i o n

of a r i g i d body.

reality, and

respect

of rest a c c o r d i n g

in r e l a t i v e

of b o d i e s

To this

are not be m o v e d

for other bodies

call them

and say that

in

"reference a body

is at

end

19

rest,

if it is fixed with

respect

is part of a larger

rigid body

the r e f e r e n c e

Consider

body.

of a s p a c e - s h i p large that

cabin.

the small

configurations,

a set B

(for body)

(for inclusion) fulfilling

axioms.

is a rather

theory

k £ B, w h i c h may be w r i t t e n conceived call k,

as a t r a n s l a t i o n

is a rigid body"

mentioned

before).

together,

forming

a body b has been

of the sentence

bodies,

so that

consider

bodies)

B is not the

"set of all

B is a term w i t h i n

An a s s e m b l a g e

such as

is a d d i t i o n a l l y

"this

if one screws

it is e n t i r e l y

we will

object,

which

to the n o n - f o r m a l i z e d

artificially

the

given by a r e l a t i o n

theory,

a body b, one may write

Usually

notions

object.

[BTS].

down in this

Similarly,

or l o o s e n i n g

(for r e f e r e n c e

ill-defined

(according

fixing

or

their movement.

it clear,

in the sense of

contains

bodies m u s t be so

the structure

To make

also

if it

reference

of our basic

R c B

which

i.e.,

of a l a b o r a t o r y

do not affect

description

body,

the walls

w h i c h occur w h e n

on B and a subset

which

a mathematical

of these

endowed with

several

rigid bodies"

for example

forces,

For the m a t h e m a t i c a l

(a "configuration")

The mass

practically

to some r e f e r e n c e

two boards

criteria

bl, b 2

"b I ~

b and b 2 ~ b".

composed

of its m i n i m a l

clear w h i c h bodies

I will

satisfy

sub-

"k ~ b" and

"not k ~ b". "Not k ~ b" means part of it; actually

that the body b is c o n s t r u c t e d

it does not mean

decomposed.

(This

of ~ to the i n c l u s i o n relation

~ does

two parts. different

Sticking

compound

of p r o c e s s e s

together

point

bodies

decide w h e t h e r

in this

granted

given below.)

if for example

from the o r i g i n a l

The crucial

that the body b w h i c h

is, by the way,

not apply,

without

the Darts

using

contains

k is

by the r e d u c t i o n

Moreover,

a piece of w o o d

constitutes

k as a

a new

the is cut into

body

piece of wood.

approach

is the p o s s i b i l i t y

to pieces and to reassemble. he has r e p r o d u c e d

of taking

The g e o m e t e r

has

to

the same body with w h i c h he started

20

(without

using

recognize

We will

a sequence

introduce

"process" present with

context,

the p o i n t s

event, region.

Again,

A body

do not b e l o n g

Thus

our

but once,

that

to this

is c o n v i n c e d

an - a d m i t t e d l y

is as follows:

We w i l l

now discuss

in terms note

In principle,

the p r o b l e m

of c o n f i g u r a t i o n s

that the d e f i n i t i o n

reference bodies:

w h i c h may be d e f i n e d

two

principle,

reference

depend

1 6 B k does

theory

The

spatial

the same

as p r e s e n t e d

(In spite of that Moreover,

the as

on ~.

regions

can be d e f i n e d

the body

on the c h o s e n

k £ B. First

"frame"

class

of

of r e f e r e n c e

frame ~ 6 ~ could,

"position"

not change

of ~ .

could be r e p r e s e n t e d

as an e q u i v a l e n c e

rigidly.

to sub-bodies

geometry,

terms.

- structure

bodies w i t h i n

be c o n n e c t e d

a configuration

physical

1 6 B k containing

will

of its

the set ~ is e n d o w e d w i t h

in these

how

the stages

in terms

some kind of abbreviation.)

complicated

inclusion

- the sections

can be d e f i n e d

that any p h y s i c a l

very

diagram

(see

by a s n a c e - t i m e

namely

(223)).

to use

e.g.

of

of s p a t i o - t e m p o r a l

subset

c o u l d be f o r m u l a t e d

stages,

are examples

be r e p r e s e n t e d

a space-time

in this book,

this g e o m e t r y

in the sense of an individual

concept

of bodies ~

in the

in some c o n f i g u r a -

to a subset of processes,

(C,B,R).

author

of a b o d y

corresponding

formulation

prefer

for c o n n e c t i n g

two d i s t i n g u i s h e d

which would

k c ~

of a

speaking, n e c e s s a r y

fixing

notion

the s t r u c t u r e

we w o u l d

the c o n c e p t

the s u b - p r o c e s s e s

the i n c l u s i o n final

The

is a natural

or - u s i n g

Note

strictly

of return of a pendulum,

now c o r r e s p o n d s

w orld- t u b e .

concept,

is c o n v e n i e n t

use this

who m u s t

a second time.

of a body b e t w e e n

there

of its history,

Formally,

is not,

theories.

We will

happening

to the m a t h e m a t i c i a n

fundamental

but w h i c h

physical

"processes".

~.

another

the m o v e m e n t

between

analogous

of symbols

p 6 z, w h i c h

other

tion,

geometry),

in

of a body k w i t h i n

if some o t h e r body

is r e m o v e d

21

from

1 or a d d e d

alterations 11 ~

12 ~

to the

to i. C o n s i d e r i n g

we obtain 13 E

same

...

"chains"

~IN,

"position"

u////

of c o n f i g u r a t i o n s

in this

case

the

region

crucial

different from

bodies

represents

exept

occupy

tions.

Two

figuration.

in o u r

of the b a s i c

axioms

notions

(BI to B4).

number

say

said

to b e l o n g

can be

already

some

The

of s u b - b o d i e s ,

substituted

and

respect one

The

raises

should

of c o n f i g u r a -

to the c o n s i d e r e d in each

that we some

ob-

difficulties

clear

contains

con-

by m e a n s

in fact

assumptions

in a n a t u r a l

infinitely

hardness-

property

can be d e f i n e d

intuitively

a body

m a y be

this

for a n o t h e r

proof

~ of b o d i e s

Whereas

a region

But

context

in this w a y

simple

to a b s t r a c t

of c o m p o s i t i o n

with

of

- as c o l o u r s m a s s ,

extension.

provided.

two p o s i t i o n s

A 6 R. We h a v e

in terms

inclusion

~ of regions.

whether

of the b o d i e s

in this

body)

(21])

region

equivalent

relation

to f o r m u l a t e

an i n c l u s i o n

are

"substitution"

an e q u i v a l e n c e

we h a v e

same

approach

if they A

the

of g e o m e t r i c a l

positions

constructions

of k,

i i are

the r e f e r e n c e

is to d e c i d e

properties

the p r o p e r t y

be e x p r e s s i b l e

tain

problem

the m a t e r i a l

all

of c o n s t r u c t i v e

i

fig.

Now,

number

of k.

///~

(The h a t c h e d

and

a finite

only

way

as

induces

a finite

divisible

and

(in an

22

appropriate sequen c e Regions proved

mathematical

of p o s i t i o n s must

that

sense)

(R,,)

since

containing

be c o n s i d e r e d

in a strict

postulate

model),

it may be r e p r e s e n t e d

smaller

as c o n n e c t e d

(although

and t h e r e f o r e

be a lattice.

it w o u l d

To this

additional

"formal"

regions

A

6 R. Let < be the c o r r e s p o n d i n g

insures

that

other

l

(R,<)

properties

distributivity

regions

and smaller

is a w e a k l y

required

in section

3) r e p r e s e n t e d

logical

closure

depends

on a c e r t a i n

by p o i n t

illustrates

of t r a n s p o r t

if they

can be r e p r e s e n t e d

basic

regions

Therefore

positions

implicitely

that

assumed

in colour,

instance

define

the b l o c k is l o o s e n e d the table by an angle will

even

full (in

the same topobut d i s t r i b u t i v i t y

(The c o u n t e r - e x a m p l e

k,

be 7 2 .

mapping

30 ° . Then

is not

of A, not only by depend

explicitely

~ = T(~1,z2,k).

It is

small i n h c m o g e n u i t i e s , symmetry

B I 6 R, o p e r a t i o n a l l y .

around

a transport

assignement

will

it seems

for

transport

is unchanged.

fixed on a table

say,

determine

also by a s y m m e t r y

exhibits

not

At first g l a n c e

any n o n - t r i v i a l

BI,

2.3 does

are said to be "congruent"

of the body k:

and rotated of,

in section

if its region

T(~1,~2,k)

an iron b l o c k

new p o s i t i o n

some

are to be

physically,

(AI,A 2) should

each body

such that

a new position,

N O W we will

on R. A x i o m B5 (besides

if regions

same body.

the t r a n s p o r t

on the r e p r e s e n t i n g

yields

to c o n s i d e r

We do not p o s t u l a t e

~A I = A 2. But this

~A = A is s a t i s f i e d

the identity.

lattice

Two regions

by the

~ : R ÷ R such that

instance

mappings

concepts.

a pair of c o n g r u e n t

since

to

this point.)

further

unique

inclusion

choice of r e p r e s e n t a t i o n .

require

mapping

not be

sets of d i s j o i n t

two sets with

cannot be d i s t i n g u i s h e d

The c o n s t r u c t i o n

that

sets,

this will

not be r e a l i s t i c

finite

2.3).

reason:

sub-bodies.

end one needs

distributive

for the f o l l o w i n g

section

(329)

a 6 R, n a m e l y

by a

Imagine

in some p o s i t i o n

71 . N o w

some axis p e r p e n d i c u l a r it is again

fixed

for

to

and its

28

For s i m p l i f i c a t i o n from

we will

first

assume,

~I and thus may be r e p r e s e n t e d

the table

"simultaneously"

that the region B I is d i s j o i n t

by another b l o c k

i,

say lying on

with k.

.f'"o.°.o.... "..% ..."""':"

i ~.. "',2"

..°.~°°..°~°°'~'°'"°" m °°°.oo°~t°°°°

°,oO.b~°°~°°°°°~°°

~ °-"'" i°';.°.

.°°°o.O°..°..°°.°'" °°"""

m

|°.°"".°]

fig. Now we connect thus

k and 1 rigidly,

constructfng

manner

that

position At this

will point

I. O b v i o u s l y

represent several we have

the region

questions

Enlargement this

frames"

tion o c c u r i n g mobility,

not simply (see

in n o n - i n e r t i a l

as e x e m p l i f i e d

1 in its new

B I.

of the f o l l o w i n g

two positions

form:

~I and ~2'

its mobility. but

rather

We c o n s i d e r

as a c r i t e r i o n

28). A m a t e r i a l - d e p e n d e n t

frames w o u l d hinder

in the figure

it can be

as part of a larger body m.

as an axiom, p.

frames,

arise:

considered

[LUD2]

n2" Then

= T(~1,~2,k) def

of a body does not restrict

assumption

"inertial

B2

used an a s s u m p t i o n

in the same w a y when

and clamp

1 and then we move m in such a

k will be in p o s i t i o n

if a body k can be m o v e d b e t w e e n moved

say by an iron band

a new body m ~ k,

subsequently

(212)

below.

for

deforma-

the r e q u i r e d

24

® fig. Of course, effect

in an a p p r o x i m a t e l y

small by using

entities

rigidity,

nevertheless geometrical geometry. and the

inertiality

experiments

further

development

to show that

T = T(~1,~2,k) not depend

representing

from the r e g i o n

struction

twice.

proofs

concerning

We will

But

of p u r e l y

a strict

inertial

limit

to such frames.

on the rigid c o n n e c t i o n

B I. In general,

from both

B] will

case we have

and p e r f o r m

First,

forming m

not be to c o n s i d e r

an

the above con-

the result m u s t be i n d e p e n d e n t simple

to

frames ~ 6 ]

: R ÷ R is w e l l - d e f i n e d .

are b a s e d on 3 additional,

task

is to show that

in this w a y

same holds

the 3

of C 1. The

axioms

B6,

B7,

B8

configurations.

The r e m a i n i n g obtained

by m e a n s

forms

exist

that

intertwined.

is r e s t r i c t e d

of 7 1 and in this

region C 1 d i s j o i n t Again

there

of g e o m e t r y

auxiliary

various

that

shows

are

inertiality

and n o n - i n e r t i a l i t y

the region B 2 = ~B I m u s t

disjoint

This

and a c c ur a c y

to test

to p o s t u l a t e

and on the p o s i t i o n

system one could keep this

rigid materials.

it is p o s s i b l e

We have

2. We have

very

inertial

(2]3)

forms

a g r o u p of

for the c a n o n i c a l

finish

this

section

the class

some

mappings

(~,~)-automorphisms.

extensions with

of t r a n s p o r t

on

Then

(R,<).

remarks

on the o b j e c t

and

the

T

25

~ n t e r p r e t a t i o n s of these constructions.

Of course,

there is no need for

a x i o m a t i z a t i o n of this e l e m e n t a r y field of e x p e r i e n c e caused by d i f f i c u l t i e s of a p p l y i n g geometry to the c o n s t r u c t i o n of buildings, furniture or physical apparatus.

The only point is to show how the

foundations of g e o m e t r y could in principle be accomplished.

Therefore

the axiom system p r e s e n t e d is not analyzed in view of consistency, completeness

and independence.

Surely not all p r o p o s i t i o n s w h i c h are

i n t u i t i v e l y true for operations with rigid bodies can be proved f r o m our axioms.

Only those p r o p o s i t i o n s which are used to derive the

basic concepts and axioms RI, R2 of section 3 are c o n s i d e r e d and c o m p r e s s e d into the 71 stated axioms. E u c l i d e a n geometry,

and at least one axiom,

suited for compact spaces However,

Obviously,

(spherical,

a model w h i c h satisfies o u r

we had in mind

(2234)

(iv), is not

elliptic geometry). axioms and thus shows their

c o n s i s t e n c y w o u l d be desirable for another reason. We have or indirectly)

p o s t u l a t e d a great variety of regions.

there m u s t exist a great number of configurations, temporal processes

(directly

That means that

hence of spatio-

and it is not clear a priori w h e t h e r o r d i n a r y

space-time is large enough to contain the r e q u i r e d variety of processes.

2S

2.2 C O N S T R U C T I O N

OF R E G I O N S

We now turn to the m a t h e m a t i c a l Its

species

(221)

of s t r u c t u r e

principal (C,B,R)

base

and an a x i o m a t i c (The t e r m i n o l o g y The el e m e n t s part of Q". identified cesses:

~, s t r u c t u r a l

(i)

x pp~

agrees w i t h

[LUD

the set of its m o v e m e n t s ,

I] p. P~

81 ff.)

Q is read as "P is a A body

hence w i t h

is a p a r t i a l

ordering

R is a n o n - e m p t y

a set of pro-

bodies

("GerOste").

k,l

can be rigidly

the r e l a t i o n of ~,

the

between

following

on ~,

of d i s j o i n t

(and i) w i l l be called

non-empty

subsets

of ~,

subset of B.

affixed a part

in order

of m: k,l ~ m.

~ and ~ a n d

example

to form a new body m.

has been provided:

....."-°-i~-~o--°3

5

ii

time

7

k

1

space fig.

In o r d e r

to m o t i v a t e

I

T

k 6 B is

~I:

(iii)

nition

or

to B8,JI.

of B are c a l l e d bodies.

B is a family

clarify

~I,~2,BI

[BTS]

(ii)

k

x pp~,

relation:

The e l e m e n t s

~

Two bodies

PT I.

term:

P £ ~, are c a l l e d processes;

with

"basic geometry",

k c ~. R c B is the subset of r e f e r e n c e

(222)Axiom

Thus

of the

is g i v e n by

set:

E p(~x~)

details

(223)

to

the e x a c t defi-

27

The

"minimal"

represented

processes

I,...8

as s u b s e t s

identified

with

the

and

their

of 2 - d i m e n s i o n a l

"unions"

13,

space-time.

135,

The

etc..,

3 bodies

are are

subsets:

k = {1,3,5,7,13,35,57,135,357,1357}, 1 = {2,4,6,8,24,46,68,246,468,2468}, m = {12,34,1234}.

The

body

contains This

k is a p a r t a subprocess

leads

(224)

to the

(225)

Q 6 k, n a m e l y

to be a p a r t i a l

axiom

Axiom

(226)

Proposition:

Proof:

Clearly, E

is r e f l e x i v e

assume

etc . . . .

inductively

k and m are clude:

(227)

QoC

Po,

on B, p r o v i d e d

many

either

the

that

the

set

ordering

and

further

construct

PI -'] QI -] P2 ~

finitely

~13.

t h a t Q C-p.

some

Hence

disjoint

PI

o

There

6 k such

by a x i o m

Pn = Qn for

or i d e n t i c a l

In o r d e r

6 k.

a countable

.... which,

steps.

on B.

transitive.

k E m and m E k and P

that

after

1234

P 6 m

is finite.

~ is a p a r t i a l

Qo 6 m s u c h

P o ' ~ Qo ~

process

1 2 - 7 I, 34 ~ 3 ,

ordering

Let be P 6 7, then {Q 6 ~IQ ~ P }

We

each

holds.

~2:

antisymmetry

that

Let be k, m 6 B.

<=> VP 6 m BQ 6 k such def

be p r o v e d

following

sense

following

Definition: k E m

Ecan

of m in the

to p r o v e exists

that

PI C

the

some Qo'

infinite

sequence

72, w i l l

be c o n s t a n t

some

(by a x i o m

n 6 ~, 71

and

(ii)),

since we con-

k = m.

Definition:

L e t k 6 B.

M(k)

will

denote

M(k)

= def

{ 1 6 B

the set of m i n i m a l

sub-bodies

I 1 ~ k and Vm 6 B, m

of k,

E 1 :> m = i}

that

is:

28

The f o l l o w i n g

(228)

proposition

Proposition:

is immediate.

For each k 6 B,

{i 6 BII ~ k} is finite, M(k)

(229

is finite

Definition: k A 1 <=> def In this

(2210

and non-empty.

Let be k,

there

1 6 B.

exists

no n 6 B such that n ~ k and n E 1.

case k and 1 are

called E - d i s j o i n t

or simply:

disjoint.

Definition: (i)

BR

= def

{i 6 Bl3r

is called (ii)

the

set of c o n f i g u r a t i o n s .

Let ~ be the e q u i v a l e n c e relation

E on B R. The

F = BR/~, def (iii)

6 R such that r ~ i}

is c a l l e d

Let be k 6 B. B k

generated

set of e q u i v a l e n c e

by the

classes,

the set of frames.

: def

{i 6 BRIk E 1 and Vr 6 R,

r E 1 => k A r} is c a l l e d (iv)

relation

the set of c o n f i g u r a t i o n s

of k.

Let be k 6 B,~ 6 F, and generated valence

~ the e q u i v a l e n c e r e l a t i o n k,~ by the r e l a t i o n E on B k N ~. The set of equi-

classes

of p o s i t i o n s

Pos

(k) = B k N ~/ U is called def k,~

of k in the frame ~. T h e

the set

canonical

surjection

B k n <0 ~ B k n ~/ k,~ will

be d e n o t e d

by 1 ~ pos(1,k),

the

configuration

l".

the

"position

(The frame ~ is u n i q u e l y

of k w i t h i n determined

by 1 and n e e d not be indicated.)

(v)

Pos

=

{ (n,k)Ik

6 B,

~ 6 Pos

(k)}

def is c a l l e d Further,

the

set of p o s i t i o n ~

pos(l,k)

(in the frame ~).

= (pos(1,k),k). def

29

(vi)

Let

(~i,ki)

6 Pos

(~I,ki) r-- (~2,k2) (or,

a configuration

reference

body.

relatively ration

containing

reasons

k must

constructive remains

which same

on Pos

based

on a

can be r i g i d l y

tied,

are

frame.

each

.

construction

Of

course,

k is a c o n f i g u r a t i o n from

~1

reference

in a c o n f i g u r a t i o n

a configu-

of k. body.)

of k,

(For t e c h n i c a l If we m a k e

the p o s i t i o n

of k

constant.

mentioned

in m o s t

bodies

identical

ambiguity

frame

~ E F is k e p t

fixed

and w i l l

not

be e x p l i c i t l y

cases.

with

the

same

since

the

sub-bodies

however,

(2211)

in the

a sub-body

alterations

the

If,

lie

ordering

to be some

configurations and

~2 c

N ~2 # ~)"

a partial

be E - d i s j o i n t

in s e q u e l

Two

71

is t h o u g h t

Two

at r e s t

r-is

= 1,2.

<=> k I E k 2 and def

equivalently,

Clearly,

Thus

for i

set of m i n i m a l can be

the

sub-bodies

are

should

disappear.

To this

Axiom

BI:

M(ml)

: {nl,...nk},

fixed

sub-bodies

combined

nj C

n o t he

in v a r i o u s

in d e f i n i t e

manners.

positions,

this

end we p o s t u l a t e :

L e t be i i 6 Bm. , i = 1,2, 1

¥j = 1...k,

need

and

further:

m 2 and p o S ( l l , n j ) =

pos(12,nj).

T h e n m I E m 2 holds.

(2212)

Proposition: M ( m I) = M(m2)

Proof:

By

(2211)

If two p o s i t i o n s

we

If,

in a d d i t i o n

holds,

of

(2211

m I E m 2 and m 2 E m I •

coincide

partially,

they

[] should

coincide

totally:

(2213)

Axiom

B2:

pos(ll,n)

Let

,

then m I = m 2.

conclude

of a b o d y

to the a s s u m p t i o n s

be I i 6 B k,

= pos(12,n).Then

i = 1,2, pos(ll,k)

n ~ k and = pos(12,k)

holds.

30

We now turn section

2.1,

the same

two b o d i e s

spatial

position,

(2214)

to the d e f i n i t i o n

of " s p a t i a l

(in their

region,

if,

regions".

respective

for each

one can be s u b s t i t i t u t e d

As i n d i c a t e d

positions)

configuration

will

belonging

in

determine to their

by the other.

Definition: (i)

(ii)

Let

1 6 Bk,

C(1)

= {m 6 Bil £ Bin]. def

Hence

C(1)

= {m 6 B l m E 1 and Vr 6 R, r C 1 => r A m}.

C(l,k)

= {m 6 C(1)Im ~ k or m A k}. def

A substitution

is d e f i n e d

to be a 6 - t u p l e

( 1 1 , k 1 , ~ 2 , a , 1 2 , k 2) satisfying: a) i i 6 Bk.

for i = 1,2,

1

b)

n2 = P ° S ( 1 2 ' k 2 ) '

c) a: C(ll,k I) + C ( 1 2 , k 2) is a s u r j e c t i v e

mapping,

d) e k I = k2, e) Vm 6 C(ll,kl),

m A k I =>

(am = m and pos(ll,m)

= pos(12,m),

f) ~m ~ an <:> m C n. By f),

~ is i n j e c t i v e .

In order

(2215)

to i l l u s t r a t e

the above

Example:

/JJ

//I// •

o

d e f i n i t i o n , we

/ /

provide

the f o l l o w i n g

JJ

/ I /

.

.!..,;: :

Q::': : :'.,. 11

~2 C(ll,kl)

= {n, m, kl,

klm,

klmn,

mn}

C(12,k2)

= {n, m, k 2, k2m,

k2mn,

mn}

12

3~

(2216) Lemma:

L e t be 11 6 Bml,

substitution.

Then

= ~IC(ll,ml) def

k I ~ m I and

C(ll,ml)

and ~ 2

( 1 1 , k 1 , ~ 2 , ~ , 1 2 , k 2) a

c C(ll,kl)

holds.

= P°S(12'm2)' def

If m 2

= ~ml, def

then

~

( 1 1 , m 1 , ~ 2 , ~ , 1 2 , m 2) is a s u b s t i t u t i o n .

Proof:

L e t be m 6 C(ll,ml). mAr

mI C

m implies

kI E

m and m A ml,

6 R i m p l i e s m A k I, r. H e n c e m 6 C(ll,kl).

since for all m 6 C(11):

~ is s u r j e c t i v e ,

am ~ m 2 = ~m I => m ~ m I = > m

6 C ( l l , m I)

and am A m 2 = ~m I => m A m I => m 6 C 11,mi). The o t h e r d e f i n i n g

properties

of a s u b s t i t u t i o n

are i m m e d i a t e l y

clear.

(2217)

Definition: T = (i)

Let S =

(11,k1,~2,e,12,k2)

( 1 2 , k 2 , ~ 3 , B , 1 3 , k 3) be s u b s t i t u t i o n s T 0 S

= def

(ll,k1,~3,Se,13k3),

and ~, c o n s i d e r e d (ii)

S -I

= def

(12,k2,n1,~

of the r e l a t i o n

-I

groupoid

and n e u t r a l

,11,ki),

where

are g i v e n by

We o m i t the proof,

which

(2219)

L e t ~i 6 Pos(ki)

Definition: (i)

is s t r a i g h t

~I ~ ~2 <=> V11 def such

of

a

-I

is the i n v e r s e

, w h e r e i is the i d e n t i t y

of s u b s t i t u t i o n s

to the m u l t i p l i c a t i o n

elements

B~ is the p r o d u c t

~.

T h e class

w.r.

= P°S(11'kl)" def

as r e l a t i o n s .

=(ll,k1,n1,1,11,kl) def on C(ll,kl).

Proposition:

and ~I

where

(iii) E(ll, k l )

(2218)

and

has the s t r u c t u r e

defined (2217)

(ii) and

forward.

for i = 1,2.

6 ~I Hl 2 6 ~2 Ha

that

{Vn 6 B, (nil I and nAkl)

above.

=> n ~ k 2} =>

The

of a

inverse

(iii).

32

(ll,k1,~2,~,12,k2) We will called (ii)

say: the

~I N ~2 ~1 a n d

n2 can be

<:>

(n1'ki)

(2220)

Lemma:

equivalent

the

result

of

6 Pos

for

Proof:

n of

L e t m 6 B be a minimal

It is r e a s o n a b l e only

fixate

and v i c e

complication Therefore (i.e.

(2221)

has

we want

bodies

sucht

with

the

Let

exists

mentioned omitted,

without

interested

the

considered

when

loss

in t h e s e

s.c.

for m i n i m a l

same

there

properties.

substitutability that

no p a r t s

s.c.

in

in some

(n'k')

exists Hence

for

of k 2 are u s e d

(2219) (i).

of the

equivalent

This

following

sub-bodies

with

to

trivial proofs. "copies"

and p o s t u l a t e :

E Pos

such

that

subset. "~ ~ w'

Then

and

=> n { L).

serve

These

the

6 Pos

a position

relation above.

mutual

hence

(w,k)

lemmata

~2

[]

positions)

there

equivalence

¢ k 2.

equivalen~

B3:

following

m

the

the b o t h e r s o m e

Axiom

~I ~

and m A k I. T h e n

to r e p l a c e

(Vn 6 B, n r-k'

The

to c h e c k

enjoying

condition,

versa,

to be

and

If

~ ~2"

t h a t m E 11

consequently,

under

be

11 .

to p o s t u l a t e

positions

~1

will

positions

(2227)).

i = 1,2,

sufficient

n E B, n ~ m,

n ¢ k 2 and,

k]

write

{...}

~2 ~ ~I"

called

also

~I"

(s.c.)

be

It is a l w a y s

sub-bodies

for

condition

~I ~ ~2 a n d

v2 w i l l

we wLll

substituted

substitution

(anticipating ~i =

is a s u b s t i t u t i o n .

one

are of

to p r o v e takes

technical

continuity,

details.

(2227),

into

namely

account

in n a t u r e by the

the

t h a t ~ is an complication

and thus

reader

who

may

be

is n o t

33

(2222)

Lemma: ~I

Let

i 6 Bk2,

6 Pos(ml),

k I r- k2 ,
<7 ~ ~I'

m 2 = ~ k 2 and

(l,k1,~1,~,l',m1)

P2 = P ° s ( l ' ' m 2 ) "

Then

for

be

i = 1,2,

a substitution,

<2 N ~2 h o l d s .

//////

~

k2

6Q fig.

(2223)

Proof: I.

"K2

~ P2""

Let

~ be

Vp

E B,

Note, 1.1.

We

a configuration (p ~ ~ a n d

that

assert:

follows:

kI E Vp

~ 6 K 2 and

p A k 2)

k 2 implies

6 B such

that

the

s.c.

be

satisfied:

=> p ~ m 2. mI = ~ kI E

~ k 2 = m 2.

p is m i n i m a l ,

p C ~ and

p A kI

p ~ m I.

Proof: 1.1.1.

Consider

the

case

p A k 2. N o w

p ~ m 2 by

the

above

s.c.,

hence

P ~ m I• 1.1.2.

1.2.

1.3.

In t h e

ease

p E k 2 we

hence

p ~ mI.

Since

K I = Pl

there

exists

We

assert:

Proof:

and

the

infer

s.c.

a substitution

from

p A k I that

is p r o v e d

p = ~p A ~ k I = m I,

for minimal

(~,k1,~1,~,~,mT).

8k 2 = ek2 (= m 2 ) .

We write

M(k2)

= M ( k I)

~ R and

conclude

p

(see

(2220)),

34

M(~k 2) = M(m I) ~ R = M(Sk2) Further, i,i'

pos(~,m)

1.2.

= pos(llm)

£ ~1" The latter

by definition By virtue

of

since ~m 6 R, am = Bm = m.

implies

(2210) (2216)

and pos(~,ml)

(iv). By

V n 6 M(ml) , pos(l,n) (2212),

there exists

(T,k2,~2,B,l,m2) , which 2.

~2 ~ <2 is analogously

(2223)

Lemma:

= pos(llml)

completes

assume

= pos(l~m)

~k 2 = Bk 2 follows.

a substitution the proof

of <2 ~ ~2"

proved.

Let nl,k I 6 B be such that n I is minimal

further

because

and n I ~ kl,

substitutions

(111,n1,~2,~,112,n2), (112,k1,K2,B,122,k2), (122,n2,~1,Y,121,nl)-

B

111

112

122 fig.

Then

the following

(i)

C(111,ki)

c C(111,nl)

C(121,k2)

c C(121,ni).

(ii)

holds: and

= C(112,k I) n C(112,n2), def

and D 2

= C(122,k 2) n C(122,n2). def exist mappings

commutative:

121

(2224)

Set D 1

There

Y

~, B, ~ making

the following

diagram

35

C(111,ki )r" % C ( 1 1 1 , n i )

v DI

C(112,ki)~

~

~_C(112,n 2)

~--

> C(122,n 2)

I--

,6 "~ D ~

C(122,k2) ~

f_

!

~T

IT

%

C (121, k2) C--~. C (121 ,n I )

(The a r r o w ~ uniquely

- denotes

the i n c l u s i o n

f

determined

embedding.)

by this p r o p e r t y

and thus

a,

are

6, Y are ~-isomor-

phisms.

(iii)

Let

~

= ~ 0 ~ 0 ~, def

pos(111,n1).

Then

<2

= p o s ( 1 2 1 , k 2) and p o s ( 1 2 1 , n I) = def

(111,k1,K2,g,121,k2)

is a s u b s t i t u t i o n .

Proof:

(i)

Because

n I is m i n i m a l ,

C(111,ni)

= C(111

we have

D C(111,ki)

and,

analogously,

C(121,n I) ~ C(121,k2). (ii)

1. We w i l l

prove:

It holds: The

latter

~[C(111,ki)]

= D I.

k I & n I => ak I = k I and k I & n 2. follows

from the i m p l i c a t i o n s :

k I = ak I 6 C(112,n2) k I & n 2 or k I ~ n 2 ~k I = k I ~ n 2 = ~n I => k I ~ nl,

in c o n t r a d i c t i o n

to

kI A nI. 1.1.

1.1.1.

We assume

m 6 C(111,ki)

and will

Clearly,

am 6 C(112).

Consider

the case m ~ k I. H e n c e

showam

6 C(112,ki).

am ~ ~k I = k I and

86

a m 1.1.2.

In

6 C(112,ki). case

m

A k I assume:

3 h r-am,

a k I.

It

h F" k I, k I A n 2

follows:

(see

I.

h A n2 h 6 C(112,n2) i r-m

such

~i F" ~mt i ~ m, Hence 1.2.

that

ak I

k I in c o n t r a d i c t i o n am

A mk I = k I and

We~assume

First

s E C ( 1 1 2 , n 2)

that

s = at.

It remains

r 6 C(111)

1.2.1.

If

s " l k I , ~r

1.2.2.

If s A k I, a s s u m e h 6 C(111)

(iii)

~ak

and

s = mr.

s 6 C ( 1 1 2 , k I)

3h such

(see

r 6 C(111,ki).

6[D I] = D 2 a n d

We

have

a)

a n d b)

Let

to

to

show are

m 6 C(111)

pos(111,m)

I. A s s u m e hence

m

points

s "lk I or

to

f)

c),

are

in t h e

d),

f)

s A k I.

r 6 C(111,ki).

infer to

proved

analogously.

definition

follow

s A k I.

from

(2214)(ii). (ii) ; it

e). and

m

A k I. W e

either

m

A n I . It f o l l o w s ,

m = 6m.

a)

such

k I. N o w

and we

¥ [ D 2] = C ( 1 2 1 , k 2)

= pos(121,m)

n I is m i n i m a l ,

implies

F ak I = k I i n c o n t r a d i c t i o n

immediate;

show

h P- r,

(i))

Thus

~ r 6 C(111,ni)

r ~ k I and

that

ah

3.

thus

r 6 C(111,ki).

I = k I, h e n c e

= C ( 1 1 1 , n I)

r A k I and

and

to s h o w :

~h E mr = s a n d

remains

Since

that

show:

such

note

A k I.

mm 6 C(112,ki).

s 6 D I and will

Clearly,

and

to m

3 r 6 C ( 1 1 1 , k I)

that

2.

h = al

Moreover,

have

to d e r i v e

6m = m a n d

. A n I or m that

m

pos(111,m)

"I n I h o l d s .

A n 2 and m = am = = pos(112,m)

Bm = 7m,

= pos(i22,m)

=

37

pos(121,m). 2. N o w

consider

is d i s j o i n t it can be

from

ml, .... mk,

~m ~ which From

hence

$-2m ~

(2211).

substitutions

Lemma:

of

according

in 1.1.

we

conclude event

=

6m ~ nl, of m

# ~m, we

sequence

to

which

(228).

we

infer

Hence by

~m = m.

(2213):

[]

expresses sub-bodies,

a sort

of c o m p o s a b i l i t y

can be e x t e n d e d

for

to the case

sub-bodies.

L e t kl,

ni

k, w h e r e

n i A k I for i = 1 , . . . , k . are

A k I. As

....

of d i s j o i n t number

m

pos(111,nl)

In the

infinite

= pos(112,n1)

lemma,

because

Any m i

6m. = m. and l l

(ii)

= pos(112,m).

preceding

(2225)

6n I : n I and

is i m p o s s i b l e

of a f i n i t e

latter

= {nl,ml,...,mt}.

By a s s u m p t i o n ,

~m ~ m by

pos(111,nl)

let M(m)

V i : 1,...,k,

a countable

~-Im ~

pos(111,m)

The

Using

obtain m G

that

= pos(112,mi).

pos(112,n1).

and

n I and k I, the

shown,

pos(111,mi)

would

the case m ~ nl

Further,

the n i are m i n i m a l the

following

and

substitutions

assumed:

©©

I©©I

@ @ fig.

(2226)

38 SIo

=

(110,n1,~,~1,111,n~) !

!

$11 = (!11,n2,~2,e2,112,n2)

$I ,i-I = (ll,i-1'ni'~!'~i'll li'n!i) !

S1,i

l

= (ll,i,ni+1,vi+1,~i+1,11,i+1,ni+1)

!

$I , L-I = (11,L_1,nL,v L',~L,II

L,nL )

SI,L

= (11,L,k1,K1,6,12,L,k 2)

S2,L

= (12,L,n~,vh,Yt,12,L_1,nL)

$2,i+I

= (12,i+1,nL+1,gi+1,Yi+1,12,i,ni+1)

S2,i

= (12i,n~,vi,Yi,12,i+l,n i)

$21

= (121,n~,v1,Y1,120,n I)

If V i=I,..

,L, pos(12,i_1,ni)

6

= YI' .... YL 6 ~L,...,~I def

<2

= P°S(12,t'k2)' def

then

= pos(ll,i_2,ni) , C(11,0,k I)

and

(11,0,k1,<2,~,12,0,k 2) Is a substitution.

Proof:

First,

(2223)

is applied to the inner triple of substitutions

SI,L_ I, SI,L, $2, L. This gives a substitution T I = (11,L_1,k1,<,~1,12,L_1,k2). Again, desired

(2223)

is applied to SI,L_2,TI,S2,L_I,

substitution

~ is an equivalence

Proof:

~ is symmetric

remains

until the

is constructed.

(2227) Theorem: Clearly,

and so forth,

to show, that nl =

relation on Pos (p.

and reflexive

by definition.

~2 and ~2 = ~3 implies

I. To this end assume zi = P°s(li'ki)

for i=I,2,3

71 = and

~3"

It

39

{nili=1 ..... L} = { n 6 B [ n ~ l l , n A k l , n ~ k 2 and n is m i n i m a l } , w h i c h def the set of all m i n i m a l s u b - b o d i e s of 11 w h i c h h i n d e r the substitution

of k 2 for k I. F u r t h e r ,

let N I

= def

is

{ n 6 B l n ~ l I or

n~l 2 or n~13}. If ~I

= pos(ll,nl), def ~I ~ v~ and V n E n{, finiteness

there exists

a

n ~ N I. This

follows

of N I . v1 = ~I' implies:

(v~,n~)

6 Pos,

by

(2221)

such that and the

V 11 6 ~I 3 111 6 ~

B ~I such

that $IO Vn,

= ( ~ 1 , n 1 , ~ , ~ 1 , 1 1 1 , n {) is a s u b s t i t u t i o n , if the s.c. def (nE~ I and nan I) ~ n ~ n{ is s a t i s f i ed. It is i n d e e d s a t i s f i e d

for the 11 and n~ c o n s i d e r e d

above because

n F i I ~ n 6 N I ~ n ~ n~. This p r o c e d u r e bodies

w i l l be

n i, i = 1 , . . . , L ,

P°S(ll,i'ni+1) V nE

(h-1)-times

are c o n s e c u t i v e l y

U { n 6 B I n ~ n i _ I}

w i l l be $I, i =

substitutions i=O,...,k-1.

S i n c e k I A n i we h a v e

pos(11,L,kl)

~. Claim:

Vn,

Proof:

replaced

by n~i' w h e r e

and

The c o r r e s p o n d i n g

( l l , i ' n i + 1 ' ~ i'+ 1 ' ~ i + 1 ' l l , i + 1 ' n'i+1 ) for

p o s ( l l , i _ 1 , k I) = p o s ( l l , i , k l ) Hence

such that the sub-

= v i + ~ v'i+1 = pos(ll,i+1,ni+1)'

= N i-I ~ N i def

nisn

r e p e ated,

~i kl = kl and

(see(2214) (ii)

d) and e)).

= ~I"

(n~ll, L and nAk I) ~ n ~ k 2.

Obviously,

it s u f f i c e s

to s h o w this

for m i n i m a l

V i 6 {I...L}

such t h a t n C n!l' or V i 6 {I...L},

n is m i n i m a l .

In the f i r s t case,

above

construction.

Therefore

we can

n is i n v a r i a n t Because

Thus

n A kl,

a s s u m e n d n! for all i = 1 . . . L l

n ~ k 2 would

to n C 11, t, s i n c e

construction.

This c o m p l e t e s for ~I = ~2"

is the a s s e r t i o n .

and c o n c l u d e ,

that

~i" Thus n ~ 11, L ~ n E 1 1 •

imply n 6 {nili=1...L}

contradiction

is j u s t the s.c.

' because n A n i,

n E n~1 immlies~ n ~ N i m N I by the

n ~ 12 and n ~ k 2, w h i c h

u n d e r all m a p p i n g s

n. E i t h e r

iI, t c o n t a i n s

in

no n i by its v e r y

the p r o o f of the a b o v e

claim,

which

40

3. T h i s p r o v e s

the e x i s t e n c e

of a s u b s t i t u t i o n

S~, L = ( 1 7 , L , k 1 , ~ 2 , 8 2 , 1 ~ , L , k 2 ) . The construction Vn,

(nCl]

modified

also

insures,

and nAk I) ~ n ~ k3, w h i c h we may assume,

~ n ~ k 3. H e n c e

~2 ~

73 y i e l d s

I

~!i' if the c o r r e s p o n d i n g

d i n t of ~i reduce

for n i C

n i ~ n k for some k > i, w h i c h

4.2.

n. C l

imply by c o n s t r u c t i o n

4.4.

n i A nk,

k 3. T h i s w o u l d

under

n¶3' k3

violate

which are

since this w o u l d

s.c.

nI 6 NI. for ~2 ~

n3"

In this c a s e n i is i n v a r i a n t

83 . H e n c e n i A k 2 is in c o n t r a d i c t i o n

k2•

$IO

= pos(12,i_1,ni)

... SIL , S2L,...

( 1 1 , k 1 , ~ 3 , 6 , 1 3 , k 3) and t h e r e b y

w e can a s s u m e

need only apply (2228)

by

since n k is m i n i m a l .

however

the a b o v e

for all k > i,j.

5. We h a v e p o s ( l l , i _ 1 , n i ) sequence

impossible,

n I• { N 3. m NI,

all Yk' k > i a n d

to n i C

are s a t i s f i e d ,

is i m p o s s i b l e

n[ for s o m e j ~ i is also 3

ni C

is p o s s i b l e

13,i:

4.1.

4.3.

s.c.

This

to ni ~ 13, i. S i n c e the n i are minimal, t h e r e

4 possibilities

In sequel,

a substitution

substitutions

$3, i : 1 3 , i , n i , ~ i , Y i , 1 3 , i _ 1 , n i) for i=k...].

only

the

(i'2 , k , k 2 ,~3,~3 '13, k ,k3).

4. Set Sl, L = S~, k 0 S~, L a n d c o n s t r u c t

simply

implies

s.c.

Vn, (nCl~, k and nAk2) $3, k :

t h a t the s.c.

R(~)

spatial regions. written

to the

~1 ~ ~3"

is a l w a y s

satisfied

u s e d in the p r e v i o u s

since we

proof.

= PoS(
The canonical

as a + reg(~)

also w r i t e

(2225)

This y i e l d s a s u b s t i t u t i o n

proves

t h a t the s.c.

the t e c h n i q u e

Definition:

$21.

and can a p p l y

and, w i t h

reg(pos(l,k),k)

surjection

~

: Pos ÷ R w i l l be

an a b u s e of l a n g u a g e ,

: reg(l,k)

for 1 6 B k N

we will

41

(2229)

Definition:

Let A, B 6 H.

A < B

~ H l , k A , k B 6 B such t h a t A = r e g ( l , k A ) , B def and k A C k B.

(2230)

Axiom

B4:

Let 1 6 Bkl

D Bk2 N ~

= r e g ( l , k B)

such that k I A k 2. T h e n

r e g ( l , k l) % r e g ( l , k 2)

(2231)

Lemma: AI

Proof:

Let

1 6 Bk2 and k I ~ k 2. T h e n

= reg(l,kl) def

~ r e g ( l , k 2)

C l e a r l y A 1 • A 2 by

M ( k I) ~ M ( k 2 ) , equivalent

m'

p o s ( l ' , k I) =

reg(l,m)

(2221)),

= reg(l',m')

h e n c e pos(l,m)

(see

Proof:

Clearly,

•

and

R e p l a c e m in 1 by an

i' 6 Bk½

and

= reg(m,nA),

h e n c e A = r e g ( l ' , ~ k A) n A = ekA,

be the em' = m'

reg(l",m)

= reg(l",m')

in

[]

ordering

on R.

N o w a s s u m e A • B and B • A,

B = reg(l,kB)

: reg(m,nB),

of a s u b s t i t u t i o n

= r e g ( l ' , n A)

and

Further,

(2230).

4 is r e f l e x i v e .

(2212),

and p o s ( l , k 2) = ~ = p o s ( l " , k 2 ) ,

conclude:

is a p a r t i a l

n B C n A. By the e x i s t e n c e

(223]):

(2214) (ii)e)).

by c o n s t r u c t i o n

to m ~ m'

Proposition:

from

thus o b t a i n i n g

: pos(l",m).We

(2232)

A = reg(l,kA)

but m { M(kl).

= ~. Let ( l ' , k 1 , ~ , ~ , l " , k 2 ) def s u b s t i t u t i o n . F r o m m' A k I it f o l l o w s

= pos(l",m')

contradiction

N o w a s s u m e A 1 = A 2. By

p o s ( l , k 2)

corresponding pos(l',m')

(2229).

say m 6 M(k2), (see

= A 2. def

(l,kB,

i.e.

k A ~ k B and

pos(m,nB),~,l',nB) ,

and n A E n B = ek B ~ ~k A we i n f e r

n A = nB, A = B.

N e x t a s s u m e A • B ~ C, i.e.: A = r e g ( l , k A) B : reg(l,kB)

= reg(m,nB)

C = reg(m,nc)

and k A ~

As above,

we c o n s i d e r

kB,

n B C n c.

the s u b s t i t u t i o n

(m,nB,pOs(l,kB),~,m',kB).

It

42 follows

from

(2222),

k A E k B = ~n B ~

(2233)

(2234)

~n C i m p l i e s

Definition:

(i)

t h a t p o s ( m ' , e n C)

L e t A,

B C 6 R such

(ii) A A B

~ def

not

(AraB),

L e t A,

t h a t A,

(ii)

A m B ~ the

~-supremum

(iii)

{c6RIC
contains

V C 6 R 3 D 6 R

(v)

V C 6 R,(AmB

rigid bodies.

is a s p a t i a l

postulates

of

A v B exists,

a finite

for

(2234)

we

such

number

4-maximal

of

t h a t C A D.

and A,BAC)

~

(AvB)

A C.

(2235)

regions

Hence,

region,

B < D,

regions,

(iv)

spatial

B").

("A a n d B a r e d i s j o i n t " ) .

3 D 6 R such

Intuitively,

("A m e e t s

B 6 R.

fig.

The

t h a t C ~ A,B,

(i)

spatial

by

[]

B 6 R

~ def

B5:

Hence

A ~ C.

A m B

Axiom

~ pos(m,nc).

are

connected,

do n o t p o s t u l a t e

it c a n

since that

they the

are

represented

infimum,

A A B

indeed be disconnected.

basically

mean

that one

can produce

and

43

fixate very

a sufficient

complicated

Mathematically, tive

lattice

variety

to e x p r e s s (2234)

of r i g i d these

enables

R consisting

of

bodies.

laws

on t h e

us to e m b e d

finite,

Although,

be

(B,C,R)-level.

R into

disjoint

it w o u l d

a weakly

unions

distribu-

of s p a t i a l

regions.

(2236)

Proof

Lemma :

L e t C, A. 6 R, i = 1 . . . n , 1

the

supremum

A =

Y i=1...n

V i=1...k,

C 4 B, A k + I a n d A A is the

D

Le£ D

~ A k + I, D

(2237)

Then

The

assertion

is t r i v i a l

( k < n) C < A. a n d B = Y l i=I...k

for n = I.

A. 6 R. W e h a v e l

> A. f o r i

i=1...k+1.

We conclude

D > B and,

that

from

~ B V A k + I = A.

relation

V A,

o n n):

Definition:

Therefore,

C 4 A.. 1

= B V A k + I £ R b y ( 2 2 3 4 ) ( i i ) . L e t us s h o w def of all Ai, i = 1 . . . k + 1 . C l e a r l y , A > A i for

supremum

i=1...k+1.

V i=1...n,

A. 6 R e x i s t s . 1

(by i n d u c t i o n

Assume:

and

Let a be any

on a generated

la/~J

subset

B At,...

R and

oo t h e

equivalence

b y m.

= I is e q u i v a l e n t

B 6 a H n 6 ~

of

to:

A n 6 a such

t h a t A I = A, A n = B a n d

(V i = 1 . . . n - 1 , A i m A i + 1 ) .

(2238)

Lemma:

Let

a c R be

finite

and

l a / ~ i = I. T h e n Y A 6 R A6a

exists. Proof

(by i n d u c t i o n

assertion

on

is t r i v i a l .

Pal): Now

If a is e m p t y

assume,

far = k + I, A 6 a a n d c o n s i d e r disjoint

union

k + I elements. I~il

S k.

induction

o f its ~ Say

that

la'/~l

it h o l d s

a' = a \

-equivalence

or a singleton

{A}.

classes,

= n and consider

for

the

lap ~ k. L e t

a' is the which

set,

finite,

have

~. 6 a ' / c o l

Set B i = Y D f o r i = 1 . ° . n a n d A i = B i y A. D6e. 1 hypothesis and A i by B i m A.

less

than

where B i exists

by

44

NOw,

C =

exists

by

(2236)

and

is e a s i l y

shown

to b e

the

y a. DEa

supremum

(2239)

y A. 1 i=I . . .n

Lemma:

Let

further Then

a, b c R b e

finite

and

la/~o I =

[b/so I =

I,

V A 6 a V B E b, A A B.

(y A) A6a

A

(y BEb

B)

holds.

Proof: I.

L e t B E b.

We will

1.1.

For k = 2 the

assertion

1.2.

Assume

for a l l

(2239)

prove

B A

is j u s t

a such

and write Y A = Y A6a i=1...n of

(2238).

hypothesis the

Because

I~iI

can be

applied

(VAEa, (VBEb,

Proof: the

A~B)

~ B &

BA Y A), A6a A

lal.

Ai =

Consider

(Y D) ~I- A DE~.

as

Iai : k + I in t h e p r o o f

1

~ k - I a n d n ~ k, consecutively

A.

Now

the

induction

in o r d e r

to d e r i v e

set B =

A,

a = b and use

AEa

which

h a s b e e n S h o w n in I.,

( ' ~ B). B6b

to c o n c l u d e :

[]

Definition: (i)

R

= { a c R l a is f i n i t e a n d V A , B 6 a , A = B o r A A B } . N o t e : def E R. T h e e l e m e n t s a E R w i l l b e c a l l e d r e g i o n ~.

(ii)

Let

a, b 6 R.

Proposition:

a < b

~ def

V A E a ~ B E b such

R is p a r t i a l l y

Reflexivity

corresponding

assume

on k =

proved:

(y A) AEa

(2241)

induction

lal ~ k.

AEa

(2240)

by

(2234) (v).

that

Ai,

A)

result.

We have

2.

(y A6a

ordered

and transitivity

properties

a < b a n d b < a,

of

i.e. :

(R,<).

are

that A

< B.

by <.

irmnediate c o n s e q u e n c e s

In o r d e r

to s h o w

of

antisymmetry,

45

V A

6 a 3 B

Hence V A of

there

6 a, R.

above

(2242)

A

Now

£ b

such

are

two

4 ySA. A

<

BA

A

~ y~A thus

Theorem:

A <

B,

mappings

Thus

assertions

the

that

= A

any

V B 6 b

: a ÷ b

A yBA

is

and,

imply

For

infimum

~

and

y

impossible

by

2 regions w.r.

b ~ a,

a,

b

to

the

6 a such

: b ÷

a such

and

=

antisymmetry

a c b and

a ^ b exist

and

3 A'

A

yBA

by

definition

o f ~ , BA = A a : b.

the

supremum

parital

B < A'.

that

hence

6 R,

that

= yA.

The []

a v b and

ordering

< .

Proof: Definition

I.

of

a v b.

/ Consider 6

a v b

= def

a

6 R.

v b

We

have

that

relation

(2238),

c(e)

co

= Y A6~

show

that

spatial

c(~)

v b

regions.

£ c(B),

6 e V B

a

a <

By

A Ha A

6 a

A

whenever

6 6, A

6 a.

~ B.

e,

But

B 6

6 R exists.

It

that

~ c(~)

=

Y C6~

1.3.

Equivalently,

1.4.

a,

b < d ~ c(~)

A

this

is

follows:

6 ~ 6

(aUb)/co

c 6 a v b.

b < (avb)

(avb). < d.

6 a v b,

~ 6

of

a finite of

(aUb)/Go

U b

such

Assume

a U b and

definition

(avb).

Assume

A

consists

(2239).

7 .2.

on

any We

define

{c(~) I e 6 ( a U b ) / o g } .

to

disjoint

V A

equivalence

(aUb)/co . B y

[

7.1.

the

(aUb)/co .

just

number

a v b, are

the

this

of means,

different,

assertion

of

i.e.

46

1.4.1.

Let

A,

that

1.4.2.

B 6 ~ and

A

Now

A

Let

F,

< DA, m B ~

be

both

DA

m DB,

i.e.:

of

H i + 1 6 b.

Hi,

D.

Hi+ I <

By

D

is

exist

H i , i=1...n,

c(~)

=

F)

~ D

A A

or

of

i we

hence

define

for

C I, C 2 6 A ~

A,

B and

C I v C 2 6 R exists C 1,

This

2.2.

Now C I,

C 2,

we

have

A

A B

proves

and

B

for 6 d

and

F,

F = H I,

Hi+ I cannot

instance: such

that

regions

G ~

in d a r e

D.

< d.

spatial by

assume

regions

of

(2234) (iii).

C I m C 2.

satisfies

By

(2234) (ii),

( C I V C 2) <

A,

B.

By

maximality

= C 2.

6 R.

a ^ b

= [~ A A B is a f i n i t e s u b s e t o f R. L e t def A6a.B6b C 2 6 a ^ b. T h a t m e a n s C i 6 A i ;% B i, A i 6 a, B i 6 b f o r If A I = A 2 a n d

Assume

for

case:

< a,

a ^ b

(aAb)

2.4.

c < a,

b ~

c < a,

b means:

there

b

is

c <

B I = B 2,

2.1.

A I A A 2 , then

instance

2.3.

Then

Hi

that

6 R:

C 1 = C 1 "¢ C 2

i=I,2.

any

such

B ~ D B.

such

spatial

infer

6 d

a ^ b.

= the set of maximal def { C 6 R I C A , B } . A A B is f i n i t e

of

the

(avb)

B

Let

Thus

H D

H DA

D A = D B-

Assume

since

on

6 d,

b.

shows:

unique,

conclude

First-we

i=1...n-1.

1.4.1

we

2.1.

that

there

induction

of

such

a < d,

d 6 R,

1.4.2.

Definition

H DB 6 d

Since

since

From

(y F6~

m B.

and,

a,

H i 6 a and

2.

B 6 b A

H i m Hi+ I for

elements

disjoint.

1.4.3.

6 a,

analogously:

G 6 ~,

G = H n and

A

shows:

Ci ~

C I = C 2 or

A i implies

C I A C 2.

C I A C 2. A t

6 R.

clear

by

definition.

a ^ b.

exists

V C

6 c -3 A

a maximal

6 a, M

B £ b

6 A A B

such such

that that

C ~ C <

A, M.

B.

47

M 6 a ^ b shows

(2243)

Proposition:

c < a A b.

The

order-preserving Proof:

i

: R ÷ R,

i(A)

= def

{A},

is an

injection.

immediate.

(2244)

Definition:

A

called

distributive,

¥ a, By

mapping

weakly

b,

induction

(2245)

c 6 R,

one

Lemma:

Let

Each

distributive

(see

eounterexample

Proof:

~

(R,^,v)

(R,A,v,O)

~

lattice

is

smallest

element

0 will

be

iff

a n d b^c=O)

proves

ai^c=O)

with

~

(avb)

A C = O.

the be

a weakly

ai

distributive,

n 6 ~.

Then

^ c = O.

weakly

distributive,

but

not

conversely

(329)).

Theorem: R

(a^c=O

easily

(gi=l...n,

(2246)

lattice

(R,A,V,¢)

is a w e a k l y

distributive

lattice.

{~).

By

(2241)

and

(2242)

it r e m a i n s

to

show

the

weak

distributi-

vity.

Thus V C D

assume

a ^ c = b ^ c = ~.

6 c V D 6 a v b,

is o f

the

form

D =

It

is e n o u g h

to

show:

C A D = ~. y D.6~

D i,

~ 6

(aUb)/~

. Each

spatial ~

region

D

is i

1

an

element

of

a or of b,

follows

from

R

follows,

% {~}

(2234) (v)

if w e

by

hence

C A

D i = ~.

induction,

anticipate

that

Axiom

31

Analogously C A

to

(2245)

( Y D i) = C A i=1...n

(2311),

which

it

D = ~.

implies

48

2.3

CONSTRUCTION

OF T R A N S P O R T

According

to the ideas

transport

mapping

F i r s t we w i l l We r e c a l l

(231) A x i o m

3 additional

C(1)

B6:

developed

V 1 6 B R V D c C(1)

simply,

c o u l d be e i t h e r

axioms

2.1 we h a v e to d e f i n e

configurations.

(2214) (i).

~ m 6 B R such that D = C(m)

= pos(m,k).

that those parts

omitted

the

of the same body.

concerning

: { m 6 B I I E B m} by def.

and V k 6 D, pos(l,k)

This means

in s e c t i o n

i n d u c e d by a p a i r of p o s i t i o n s

state

that

MAPPINGS

of 1 w h i c h

or r e p l a c e d by p a r t s

are not c o n t a i n e d

of r e f e r e n c e

in D

bodies.

J////I

D =

C (i) : {p,q]

{p,q}

fig.

(233)

A x i o m B7:

Let kl,

(232)

k 2 6 B,

i, m 6 Bkl

p o s ( l , k i) = p o s ( m , k i) for i=I,2.

Then

9 M 6 ~

such that

9 11,...

V ~ 6 {I,...M-I}

1M 6 Bkl

R Bk2

i v ~ 1 + I or 1

of p o s ( l , k i) = pos(m,ki)

of c o n f i g u r a t i o n s

from

the e x i s t e n c e

possesses

normal

and

of a "joint"

Definition: C(1)

Bk I

Bk 2

~-greatest

configurations.

there

separately.

exist

such

sequences

A x i o m B7 p o s t u l a t e s

sequence.

A configuration a

1 = 1 I, m = 1 M and

~ 1 + 1.

By the d e f i n i t i o n

(234)

n Bk2 be such that

1 6 B R w i l l be c a l l e d element

normal

iff

i. BRN is the set of all

49

////////%z/

/.////-/- /-.//

11 ~ BRN

12 6 BRN, i2=mn

fig. (236) Definition:

(235) C(1) c C(m) and

Let i, m 6 B R. 1 { m def

V k 6 C(1), pos(l,k) = pos(m,k). (237) Axiom B8: Iv =C

Let 1

or I v = ~ .

6 B R for v=1 . .N .and .[I II. [2 I2 [3"

IN'

Then H i~ 6 BRN, v=1...N, such that

iv { iv and i I I] 12 I2 i3"''iN"

JJ/~

YI

/ j j"

12

///////>~P

/////////

l I "

Y3

r-

12

11

fig.

(238)

13

50

(239)

Definition:

The

f r a m e ~ 6 F w i l l be c a l l e d

iff the f o l l o w i n g that k ~ m,

property

holds:

F o r all k, m,

In this c a s e w e w i l l w r i t e s u b s e t of i n e r t i a l

pos(12,k) (k,m, ll,n)

= pos(n,k). ÷ 12 .

jwl///////

,ni!

m

m

11

12 fig.

Axiom

J1:

T h u s we p o s t u l a t e the

further

arbitrary The

(2310)

J % ~. that

inertial

developement

inertial

following

a configura-

f r a m e s w i l l be d e n o t e d by J c F.

@iI (2311)

frame,

11, n £ B such

11 6 Bm N ~ and n £ B k D ~p,there e x i s t s

t i o n 12 6 Sm n ~ s a t i s f y i n g

The

an i n e r t i a l

frames

can be f o u n d and w i l l

of p r e - g e o m e t r y

within

some

perform

f i x e d but

f r a m e ~ 6 J.

lemma

is an i m m e d i a t e

consequence

of the d e f i n i t i o n

(239). (2312)

Lemma:

(i)

k ~ n E m and

(ii)

1 6 B m and

(iii)

(k2'm'h1'q)

(k,m,h,p)

(k,m,h,p) ÷ q1'

÷ q implies

q £ Bm a n d k I ~

p o s ( q , k I) = p o s ( q l , k l ) . (Use a x i o m B2.)

+ q implies

(k,n,h,p)

(k,m,l,p) m implies

÷ q

÷ q.

51

(iv)

(k,m,h,p)

÷ q and pos(p,k)

(k,m,h,p)

÷ q.

(v)

h [-k

and

(2313)

Definition:

(i)

A

(k,m,l,p)

(positional) =

(2210) (vi)).

(ii)

~ is c a l l e d

(iii)

Two chains

11

(~2,k2)

cyclic

danger

12 ...

iff

of p o s i t i o n s

where

11

(~2,k2)

12

T ~ (Pl,kl)

11

(P2,k2)

12 ...

=

...

two p o s i t i o n s

I

for

is e i t h e r f " o r

of c o n f u s i o n

--I

we may write:

(~N,kN).

congruent,

{ (~i,kI)

, such that

(nN,kN).

(nl,kl)

d, ~ are c a l l e d

In p a r t i c u l a r ,

÷ q.

(~ ,k ) 6 Pos

I v (~v+1,k +i), Without

implies

(h,m,l,p)

is a s e q u e n c e

(~1,kl;~2,k2;...~N,kN),

~ (nl,kl)

(iv)

~ q implies

N-chain

~=I...N-I I (~v,kv) (see

= pos(p,k)

~ [] ~, iff

(~N,kN)

and

(PN,kN). (~i,ki),

(~2,k2)

are c o n g r u e n t

iff

kI = k2. (v)

An N - c h a i n

~ =

transportable an N - c h a i n Clearly, (vi)

Any

A cyclic N - c h a i n

2-chain

(2214) Proof: (2315)

Proposition:

and each

is:

V

~ (~1,k])

T

6 Pos

there

and s a t i s f i e s

satisfying

transportable.

exists

o ~ T.

iff it is w e a k l y

(v) will

frames

(239)

This can easily

is w e a k l y

be cyclic. is j u s t : be g e n e r a l i z e d :

transportable.

[] Any

(~],k]), 11

V (p,k)

transportable

of inertial

By induction.

That

is c a l l e d w e a k l y

(using a x i o m B3).

Any N-chain

Proposition:

(~N,kN)

(p,k)

a is c a l l e d

property

is w e a k l y

...

contains

T is unique

the d e f i n i n g

11

iff V ~ 6 {I...N}

Y which

transportable Thus

(~I,ki)

two p o s i t i o n s (~5,k5)

(~2,k2)

...

can be joined by a 5-chain.

6 Pos (~5,k5).

there

exists

a 5-chain

52

Proof: Aldef=

By

(2234) (i) there exists

reg(~i'ki)

<

D

a region D 6 R such that

for i=1,2 and, by

(2234) (iv), a region C 6 R

such that C A D, hence C A Ai; and further B I, B 2 6 R satisfying substitutions AI~

B i ) Ai, C for i=I,2.

the sequence

appropriate

of regions

by some sequence

(~i,ki) ~- (B1,b I) -~ (y,c) ~ is the required

The crucial

point

transportability

Lemma:

Proof:

5-chain.

of cyclic

of transport

chains.

of cyclic

Consider

(k1'si'hi'P)

of positions

(B2,b 2) -~ (~2,k2).

in the theory

the transportability

(2216)

After

BI ~ C < B2 > A2

may be represented

This

there exist regions

The following

bodies

s I, for i=I,2,

+ qi" Then pos(ql,k2)

= pos(q2,k2) .

(kl,Sl,hl,P)

implies

(kl,s2,hl,p)

÷ q1'

(2312) (ii)

implies

(kl,S2,h2,P)

÷ q1"

(kl,s2,h2,P)

÷ q2"

= pos(p,k I) = pos(q2,kl). hence pos(ql;k2)

(2317)

m,n 6 B

Let k i ~

(i=I,2)

Now axiom B2

(2213) yields

= pos(q2,k2) .

h £ B m, 1 6 B n

pos(h,k i) = pos(l,ki),

(kl,m,h, p) ÷ q,

follows

= pos(r,k2).

that pos(q,k2)

h i 6 Bs. and 1

+ q1'

pos(ql,s 2) = pos(q2,s2),

Lemma:

imply

4-chains.

(2312) (i)

Hence pos(ql,kl)

lemma will

ki ~ s2 E

By assumption,

By assumption,

is the

(kl,n,l, p) ÷ r. Then it

53

..-I///////

?

%%%%~%

f l'/

\\

",,\ )7 fig.

(2319) Corollary: Proof: form

}

#

,pt '#

(2318

Any cyclic 5-chain Is transportable.

Without loss of generality we may consider a 5-chain of the (~i,ki) r-- (n2,m) --i (~3,k2) r- (~4,n) --I (~i,ki).

Let h 6 Bm, 1 6 B n , z2 = pos(h,m), of generality)

which is just the claim of

the transport:

(2317).

o

(231 7) :

By virtue of pos(h,k i) = pos(1,k i) and axiom B7 sequence of configurations

corresponding

such that

). Let hv 6 BRN be the

sequence of normal configurations

(237) and sv the greatest element of C(h v)

(k2'sv'hv'P)

(233) there exists a

h~ 6 BkID Bk2, v:1...k,

h = hl 11 h2 I2 "'" IL-I hL = 1 (Iv = C o r b

according to axiom (see (234)). We put

÷ qv for v=1...L. We infer: kl,k 2 6 C(hv), hv { hv

k I ,k 2 E C(h v) ~ k 1,k 2 • s v. Since s lemma

(without loss

(k2,n,l, p) + r. Thus we have to prove:

pos(q,k I) = pos(r,kl),

B8

and

p 6 Bk2 be the position determining

(k2,m,h, p) + q,

Proof of

74 = pos(l,n)

(2316) and conclude pos(qv,kl)

(~) pos(ql,kl)

= pos(qk,kl) .

I v sv+ I (238) we may apply = pos(qv+1,kl).

By induction,

54

From

(k2,m,h, p) ÷ q, m ~ s I and

(2312) (iv), (i) that

(k2,m,hl,q)

p o s ( q l , k I) = pos(q,kl). with

(~) this proves

(2320)

Theorem:

Proof:

Without

and the N - c h a i n

being

÷ ql and by

Analogously:

pos(q,kl)

Any c y c l i c loss

(k2,Sl,hl, p) ÷ ql it follows

p o s ( q L , k I) = pos(r,kl).

N-chain

of g e n e r a l i t y

we may

[]

assume

N = 2L + I, t 6 ~ ,

ot the form ..

(~2L,k2L) -~ (~i,ki), Further

and

÷ n2i

(k2i_1,k2i,12i,n2(i_1)) chain.

Together

is transportable.

z2i = P ° S ( 1 2 i ' k 2 i ) for i=1...L.

congruent

that

= pos(r,kl).

(~i,ki) r- (~2,k2) -~ (~3,k3)r-

(~)

(iii)

We have

p o S ( n o , k I) = pos(n2L,kl)

to show that

we may

assume

for i=i...L

where Pl = P°S(no'kl)

defining

it is cyclic,

the

i.e.

.

f

~2i+I !

by

!

C sS

d

fig.

(2321)

55

Set A = iL .IV... =

reg(~2i,k2i),

After some appropriate

B ~ A and C > A, B (see axiom B5

substition we will obtain bodies b E c i 6 B

together with configurations

m2i 6 B R (i=1...L)

w2i = pos(m2i,k2i) r- pos(m2i,c2i), and pos(m2i,b) (~)

= C, reg(m2i,b)

= B

B. Now let

÷ P2i

We claim:

(232Oa) pos(P2,b) Proof:

such that

reg(m2i,e2i)

= B for some constant position

(k2i'c2i'm2i'n2i)

for i=1...L.

(2234)).

From

= pos(P4,b).

(k2,c2,m2,n 2) + P2 and

(k3'c2'm2'n2)

+ P2" Analogously:

pos(n4,k 4) = pos(n2,k3)

and

(2312) (v) follows

(k3'c4'm4'n4)

(2312) (iv):

may state the assumptions of lemma

÷ P4 and by dint of

(k3,c4,m4,n 2) ÷ P4" Now we

(2317) in the following form:

Let k N

= b. Then def k i E c 2, c 4 (i=3,N), m 2 6 Bc2, m 4 6 Bc4, pos(m2,k i) = pos(m4,ki), (k3'c2'm2'n2)

Hence

÷ P2 and

(k3'c4'm4'n2)

÷ P4"

(2317) yields: pos(P2,k N) = pos(P4,kN).

D

By the same method pos(P2,b)

= pos(P4,b)

(~)

(b'C2L'm2k'P2)

From

(~l,k1)m-(~2,k2),

= ... = pos(P2t,b)

is proved, hence

÷ P2L" (~2L,k2L)

and pos(m2i,k2i)

we infer pos(m2,k I) = 71 = pos(m2L,kl). since k i E c 2, e2L(i=I,N);

Again

= w2i for i=1,L

(2317) may be applied

m 2 6 Bc2, m2L 6 Bc2k,

pos(m2,k i) = pos(m2k,k i) and (kN'C2'm2'P2)

÷ P2

(kN'C2k'm2L'P2)

(trivial),

+ P2L

(~)"

Now pos(P2,k I) = pos(P2L,kl) pos(P2,k 2) = pos(n2,k 2) by pos(no,kl),

using

follows, (~)

and thus pos(P2,k 1) = pos(n2,k I) =

(~). Similarly,

from which pos(no,kl)

further

= pos(n2L,kl)

pos(P2L,k I) = pos(n2t,kl) is concluded.

D

holds,

56 Now c o n s i d e r will

define

(2322)

a triple

~(71,~2,k)

: Pos

Let

6 Pos

(P1,1)

Since

=

We will

The

I~

inverse

(~,i{)

11

= def

exists

a

sat/

=

(2314),

(P1,1). there

exists

a

...

(65,15)

= (85,1) .

assignement

is w e l l - d e f i n e d :

(65,1)

5-chain

I~

...

(B~,l~)

=

(B~,I).

of ~ and ~v o b t a i n i n g

I41(~4,14) ...

...

(~,l~)

congruent

:

9-chain

65 = B~.

(~i,11)

the cyclic

9-chain.

=

(P1,1). ~ is cyclic

by

(2320).

Hence

[] of the form

set of all t r a n s p o r t s

(2322), will

(2323)

Proposition:

(i)

Let

(ii)

~(p2,P3,1)

(iii)

T I, T 2 6 7 ~ ~I o <2 6 T,

(iv)

T(~],71,k)

(v)

T(~1'72'k)-1

• 6 7. T(P1,1)

£ T,

will

be d e n o t e d

(2324)

(vi)

(~5,15)

there

chain

(~,11)

corresponding

Mappings

...

(2315)

(o I , i),

the f o l l o w i n g

another

= (~5,15)

(71,k I) =

and

11

(51,11)

(P1,1)

and its t r a n s p o r t e d

the

to

transportable

show that

end a s s u m e

=

.

(~1,k)

(ei,11)

=

~(~i,72,k)

z (P1,1)

We

5-chain

T ~ (72,k)

We c o m b i n e

k 6 B and 71 , 72 6 Posp(k).

. According

~ is w e a k l y

congruent

~' s (w2,k)

÷ Pos

~ joining

d ~ (71,k)

To this

where

a map

5-chain

(2323)

(71,~2,k)

=

(02,1)

o ~(~i,P2,1)

be c a l l e d

by 7.

~ ~ : ~(TI,~2,1).

= ~(Pi,03,1).

= ipo s, = ~(~2,71,k) .

(71,k) r- (p1,1)

~ <(71,k) rr T(P1,1).

transport{,

the

57

(_2325) Corollary:

T is a s u b g r o u p

bijections Def. Proof of (i)"~"

on P o s - a n d

of Bij(Pos)

operates

freely

- the g r o u p (isee [BGT]

of all

III§

4.3

2) on Pos.

(2324):

follows,

since

cyclic

"~" Let T = T(i~1,~2,k) be j o i n e d

5-chains

and

by a c y c l i c

joining

these

(P2,1)

and

13-chain

T joining

(~1,m)

are t r a n s p o r t a b l e 6 Pos arbitrary,

13-chain

positions

~, such that

pairwise.

(2320)

According

there P2'

exists

~2 and some p o s i t i o n

[2319) = a I , ~I'

~ contains

~I can 5-chains

to T(~1,~2,k) (P1,1)= a congruent

cyclic

o 2 6 Pos(m).

This

proves (c2,m) (ii)

= ?(~i,~2,k) (~1,m)

Both m a p p i n g s (~1'k)

(iii)

yield

T(~1'~1'k) (PI'I)

(v)

Follows

(vi)

Let T = T(o1,~2,m).

from

joins

Next we have

to some

(ii),

equivalent

(2326)

Lemma:

Let

and

(~1,k). =

joining

(~1,m)

The c o r r e s p o n d i n g

(~2,k) r- (P2,1)

transports

by

(i),

to spatial

(P1,1)

congruent

= ~(P1,1).

regions~

map e q u i v a l e n t

and

Obviously

positions

we

(.see def.

positions.

(~i,k),

~(~],~2,k) (P1,1)

Then,

T I o T 2 = ~(pl,P3,1).

5-chain

that t r a n s p o r t s

onto

(P3,1).

(iv).

Any

~(~1,k)

(2219))

Corollary:

and

(~1,m)

to e x t e n d

need the p r o p e r t y

by

=

= (P1,1) .

(ii)

contains

and T I (P2,1)

and,

(iv)

chain

if a p p l i e d

6 POS.

T I = T(p2,P3,1)

(2327)

the same p o s i t i o n

Let T 2 = ~(pl,P2,1)

also

= ~(pl,P2,1) (~1,m)~

=

(Pi,l) (P2,l)

6 Pos

and ~1

Zl ~ Pl ~ ~2 ~ P2"

(i=I,2) p]

such that

Then

~2

P2 holds.

58

Proof:

According to (2219) let P2 6 72 be given such that the

substitution

condition holds. Then we have to construct a substitution

(P2,k,P2,e,r2,1). (STEPI)

This will be accomplished

By axiom B8

in 5 steps.

(237) there exists a p2 6 BRN such that P2 {

The position of all bodies from C(P2)

remains unchanged.

P2"

Let

C(~2) = {clc~2}. (STEP2)

Apply T(72,71,k),

Pl 6 71 . Thus pos(Pl,k)

i.e. consider

= pos(Pl,k)=

(k,P2,P2,Pl)

÷ Pl where

71 .

P2 6 72

/

F / /

T(~ 1 ,~2 ,k)

Q /

®I

/ / / / /

/

Jh //

/ / /

/~ ...~

T(PI,P2,Z)

<

•

?2

®

r 2

/

fig. (2328)

/ / /

59

(STEP3)

We may take the s u b s t i t u t i o n

Pl for granted. auxiliary

(Otherwise

substitution

(Pl,k,P1,B,r1,1)

this could

as in s e c t i o n

Now consider

(STEP5)

By a x i o m B6

C(r 2) = BC(P2,k)

(236)

U {clc~l}

It is c l e a r

that,

restriction

of ~, enjoys

2.2.).

there exists

Hence

by some

a substitution

s 2 6 P2"

an r 2 6 B R such

and all bodies

of this

that

set have the same

= BIC(P2,k). def

by c o n s t r u c t i o n ,

C(r2,1)

= eC(P2,k)

and ~, as a

the properties (2214) (ii) c), d),

for a s u b s t i t u t i o n .

Let m 6 C(P2,k)

to ~I ~ Pl and

be a c c o m p l i s h e d

(l,BP2,rl,s 2) + r 2, w h e r e

as in ~2" We set ~

are r e q u i r e d

w~r.

exists.

(STEP4)

position

condition

and m 6 C(P2,k)

It remains

to show

f), w h i c h

(2214) (ii)

e).

and m A k. We set

~2

= P°s(P2'm) = P°s(P2'm) and c o n c l u d e def (~1,m) = p o s ( P l , m ) = T(~2,~1,k) (u2,m). F u r t h e r def pos(Pl,m) = p o s ( r l , m ) , since B is a s u b s t i t u t i o n pos(r2,m) hence

= pos(r2,m)

pos(r2,m)

The e x t e n s i o n lemma,

which

(2329)

Lemma:

= T(pl,P2,1)

can be p r o v e d

a map

Finally, = (u2,m),

Q is now a c o n s e q u e n c e

of the f o l l o w i n g

immediately.

Let be ~ an e q u i v a l e n c e

and f : P ÷ P

map.

0 ~(~2,~i,k) (u2,m)

= pos(P2,m). of t r a n s p o r t s

Bm = m and

relation

defined

on a set P

satisfying

V p,q 6 P, p ~ q ~ fp ~ fq. Then there

exists

a unique

: P / ~ ÷ P/~ such that the d i a g r a m p

f>p

pi t

?..,> p/~

commutes

(~ d e n o t i n g

the c a n o n i c a l

If g : P ÷ P is a n o t h e r

map,

surjection).

we have

f o g = ~ 0 g and

map

60

= ip/~.

We will

so is ~ a n d ~

I = ~-I

write

~(~I,~2,k) mappings

(2330)

If f is b i j e c t i v e ,

= T(~1,~2,k) obtained

in this

Theorem:

Proof:

By

: R ÷ R

and

that A < B implies immediately

Finally

we

are

to

T is a s u b g r o u p

from

lift

set of t r a n s p o r t

of A u t ( R , < ) .

TA ~ ~B for all

follows

by T the

way.

T is a s u b g r o u p

(2329),

denote

of Bij(R).

If r e m a i n s

T 6 T and A,

B 6 R. B u t

(2324) (vi).

transport

to show, this

[]

mappings

to the

lattice

of r e g i o n s

(R,<) .

(2331)

Definition:

(i)

Let

T 6 T and

a =

{ A I . . . A n}

£ R,

Ta

= {~A I. ..TAn}. S i n c e def ~A i A ~Aj. H e n c e ~a £ R and ^

6 Aut(R,<),

A i A Aj

implies

^

T : R + R. ^

(ii)

T

= (T(~6T}. def mappings.

(2332)

Theorem:

Proof:

Let

be

a, b 6 R, that

of T are

called

transport

of A u t ( R , < ) .

a < b and A (

also

~ £ T.

a < b means:

B.

that

V TA 6 Ta B TB 6 Tb s u c h the c l a i m

elements

T is a s u b g r o u p

V A 6 a 3 B 6 b such It f o l l o w s ,

The

follows

from

that TA < TB. H e n c e ~a < Tb. A (aOT) = a 0 T and IR = IR"

The

remainder

of

3.

REGIONS

We will

AND T R A N S P O R T

formulate

by a p r i n c i p a l

MAPPINGS

a theory

base

PT 2 whose

species

set R, the structural

of s t r u c t u r e

term w h i c h

is given

is c h a r a c t e r i z e d

by (<,T)

6 P(RxR)

and axioms

PP(RxR)

x

RI to R6.

(Later in section

5 the axioms

R7 and R8 will

be

added.)

PT 1 in the p r e c e d i n g

The theory theory"

of PT 2, w h i c h

section

establishes

is to be v i e w e d

the physical

as a "pre-

interpretation

of the

elements: a 6 R as regions, T 6 T as ~ r a n s p o r t m a p p i n g s a < b as an i n c l u s i o n More precisely, theory

PT~

embedding

O f regions.

we state,

results

from

(Einbettung),

In oth e r words: We prefere

using

PT I

the t e r m i n o l o g y

by r e s t r i c t i o n

PT~

where

is PT 2 w i t h o u t

that the structures

mapping"

are highly

"inclusion"

been d e v e l o p e d which

refer

"transport"

3.1 (.311)

via a process

theory

the

and

R3 to R8. within

PT I .

thus a d m i t t i n g

"region",

idealized

of a b s t r a c t i o n

to in the concrete and

axioms

that

pre-theeries.

It should be emphasized, and

[LUD 3],

as theorems

PT 2 as an i n d e p e n d e n t

of other

of

(Einschr~nkung)

already proven

RI and R2 are

to formulate

the p o s s i b i l i t y

and of

observable

"transport

constructs

from simple

entities,

which

have

concepts,

as "body",

"part of".

4 AXIOMS Axiom

RI:

element

(R,<)

is a w e a k l y

distributive

lattice

O and R ~ {O}

(see def.

(2244)

and compare

with prop.

(2246)).

with

least

This any

means two

v ^ w,

The

that

regions

ordered

v,

exist

w

6 R there

which

may

well

supremum

and

infimum

a i and R the

group

of

(312)

Axiom

R2:

(313)

Axiom

R3:

all

T

V v 6 R o -3 w V T 6 T, "Each

region

which

still

In

this

that

the

be

the

of

"empty

case

w

order

is

we

call

underlying

in w h i c h

R has

V T 6 T,

either

w

the

w,

relation v v w

and

and

for

infimum

O.

regions of

will

R

be

"non-empty"

automorphisms

of A u t

<,

(compare

of

denoted

by

regions; R.

prop.

(2332)).

that

TW < V. and

each

of

its

displacements

subregion".

w a kernel space

atoms

subset

preserving

a subregion

intersects

many

w < v such

A TW % 0 ~

the

region"

a subgroup

6 Ro,

contains

the

by

supremum

finitely

a. resp. ; R is i o

i=I AUt

R is p a r t i a l l y

fig.

(314)

of

(re T).

v

is

infinitely

thus

fulfilling

^ Tw = 0 or w <

~w

Axiom

divisible.

holds,

(see

R3

does

There

not

exist

mean, models

(318), (3110), (336)).

63

AS

a consequence

a topological

(315)

axiom

should

(316)

= def

{~6TITaAa~O},

= def

{T(a) [a 6 R o } .

not

Proof:

By

Let

be

is

confused

FI(U)

any

c~

a

A

Ta

=

{TalT6T}.

the

structure

of

set

of

This

holds

6 FI(U)

since

such

a 6 Ro,

of

a topology

t on

following finite

b kernel

need

intersections

V U 6 U,

that

4 points

Id R

in

to U,

6 U.

V 2 c U. of

a,

c kernel

of

b,

c T(b). ~ 6 V

~ o,

it

follows:

T b A b

% O

~ b < a y b, a #

a,

= U. U. 6 U such

T a A a ¥

Y ~ a

o

V2 c

V U

6 U

Let

U = T(a)

% o ~

X 6 T and

3 V

that T

-1

V ' I c U.

a ^ a % o,

6 U such

choose

V

T.

group.

the

V U 6 U 3 V Since

sub-base

denote

T a 6 T(a) Hence:

open

the

a c < b, Y u c <

an

1.2

T,

c ^ c

is

III,

U = T(a),

= T(c)

For

(d)

with

§

V U 6 U 3 V Let

(c)

with

a 6 R

a topological

[BGT]

~ FI(U).

1:

endowed

Proposition:

(T,t)

V

be

m

{~U]~6T,U6U}

(b)

can

Definition:

U

(a)

R3, T

group.

T(a)

T(a)

of

that

= T(x-Ia).

even

U

X V

X

-I -I

= U holds. c

U.

be then

shown:

64

For

any

fore:

~ 6 V,

X V

Finally, "V U Let

×

-I

X 9 9 -I

V

of

X 6 U

U = T(a)

and

open

B V

6

X 6 U,

= def

For

T(b),

any

~

where

6 X V

b it

= def

sets

U such that -I

V

9(x-la)

^

X -I

a

% o,

there-

= U.

U consists

6 U

a ^ a % o ~

X

follows

if

the

that

is

c

is

true:

X V c U".

:

X a m a % o.

def-1 c = a ^ X by

following

a.

definition

of

Clearly

Choose b,

c <

a.

V

d

= %0 b A X b # o def d < X b = c < a a n d d <

a ^ ~

6 T(a) Hence:

(317)

= U.

X V c U.

Axiom V

a % o

R4:

a E R V v

such "Each

that

E Ro, 3 n E ]q n a < V • v. i=i l

region

arbitrarily

can small

be

3 ~1,...,Tn

covered

by

finitely

order".

fig.

(31 8)

6 T

many

regions

of

65

Examples

of such tripels

be taken

from the various

be chosen Thus,

(319)

as the lattice

we may well

confine

Example:

(R,<)

(R,<,T),

fulfilling

geometries

in section

of all r e l a t i v e l y ourselves

4.2,

compact,

is taken as the lattice

closed

bounded,

(resp.

vals,

T as the group of lattice of the real

RI to R4,

where open

(R,<)

can may

subsets.

to a simple

real,

translations

the axioms

open,

resp.

of finite

joins

open-closed)

automorphisms

of

inter-

induced

by

line. /

The

independence

following g iven

(3110)

of the essential

counter-examples.

Counter-example (R,<)

The postulate,

which

clear by the does

not hold,

is

(weak distributivity) :

be the lattice

T the group

3112)

is made

in brackets.

Let

3111)

axioms

induced

Counter-example

sists

of all d i l a t a t i o n s

set of the n o n n e g a t i v e x ~

• x,

reals,

T con-

~ > o.

(axiom R4) :

is the lattice

functions

of ~ 2 ,

(axiom R3):

is the o r d e r e d

(R,<)

subspaces

by rotations.

(R,<)

Counter-example

of all linear

of continuous,

nonnegative,

real

with

f < g

~ V x 6 ~, f(x) _< g(x). def T is the g r o u p of a u t o m o r p h i s m s T , induced •

3113)

(f) (x)

= f(x-~), def

Counter-example (R,<)

by translations:

~ 6 Iq.

(axiom R3):

is the lattice

of bounded,

open

subsets

of ~ n + 1 ,

T is

68

induced

by the P o i n c a r ~ - g r o u p

The c o u n t e r - e x a m p l e s

(3111)

loosened without menacing Instead

of

"T(w)

w < v"

and

it w o u l d

N T(W i) w < v" w h i c h i=I...n 3 kernel" (w 1,...,w n) . A t h e o r y

worked

3.2 The

(3112)

of R3 to a l l o w

show,

the c o n t i n u i t y

"

version

of ~ n + 1 .

for

in T.

to p o s t u l a t e

to the c o n c e p t

of a "multi-

(R,<,T) , a d o p t i n g

for s p a c e - t i m e - g e o m e t r y

out by D. M a y r

a x i o m R3 may be

of m u l t i p l i c a t i o n

suffice

leads

that

this w e a k e n e d

, has r e c e n t l y

been

[MAY].

POINTS intuitive

precise

idea of points

through

representation

our c o n s t r u c t i o n of regions

the requirement, lattice

of

This

like

example

(329)

below.

uniform

structure

process

of c o m p l e t i o n

(321)

Definition: A subset

regions"

can be m a d e

of points.

It m a k e s

possible

as subsets

subsets,

of points.

We have

a,space

the c o n s t r u c t i o n

to fulfil

of points

and a

should p e r m i t one

a purely

I §6, Ex. It seems

present

lattice-theoretical

17-18,

necessary

to use the s u r r o g a t e

in the g r o u p T in o r d e r

of a u n i f o r m

con-

as is shown by the counter-

to imitate

of a the

space.

F c R satisfying

(i)

0 ~ F, and

(ii)

V a, b £ F H c £ F such that c < a ^ b, a prefilter

(iii)

a

space.

excludes [BGT]

small

if one starts w i t h

the g i v e n

condition

struction

that,

"resonable"

to r e c a p t u r e

as "very

basis.

If additionally,

V a £ F V b £ R, b > a ~ b 6 F prefilter. A prefilter

is said to be

F satisfying

holds,

F is c a l l e d

a

87

(iv)

¥ a 6 R O B Y £ T such Cauchy

F c G

say,

(322)

prove

G is f i n e r

than

¥ F,

F prefilter r such that U c

the

F a n d F is c o a r s e r

U satisfying

and a prefilter

V a, b ~ R,

in t h e c a s e w i t h

easily

that

G. A p r e f i l t e r

holds,

As

inclusion)

than

maximal, (vii)

a

the prefilter

(set-th.

we will

(vi)

~ a 6 F is c a l l e d

prefilter.

In the e v e n t (v)

that

F ~ U = F, is c a l l e d

P for w h i c h

a v b 6 P ~ a 6 P or b 6 P

is c a l l e d

filters,

prime

whose

prefilter.

elements

are

subsets,

one

can

following

Proposition:

Let

M # ~ be a family

of prefilters.

(i)

M

= D F is a p r e f i l t e r , def F£M

(ii)

M

= U F is a p r e f i l t e r , def F6M

M is l i n e a r l y (iii)

For

ordered.

any prefilter

maximal

if

prefilter

F there U, w h i c h

exists

a

is f i n e r

t h a n F.

(323)

Lemma: (i)

Let

If a 6 U, b 6 Ro, such

(ii)

U be a maximal

b < a, t h e n

either

b £ U or

t h a t b ^ c = o.

U is a p r i m e see

prefilter.

[BGT]

prefilter.

I §6,

Ex.

18b.)

(If R is d i s t r i b u t i v e ,

B c 6 U

68

Proof: (i)

Assume base

V c 6 U, and

Assume

^ c

generates

maximal, (ii)

b

hence

U

U

exist

neither

bl,

a I nor

(311),

O

=

Corollary

e.

but

A

Let

by

be

b I ^ b 2,

B i 6

b

{1,...,n}

finer

a 2 { U.

is

a prefilter

than

U.

According

aI ^ bI = O and,

by is

6 U.

property.

such

maximal

6 R

many

Ti b

Cauchy

{b}

But

U

is

6 U.

(blAb2) , w h i c h

Each

finitely

a <

b

a I ~ U and that

is

U

weak

to

(i)

= a 2 ^ b 2.

Thus

distributivity

impossible

because

U

is

:

a 6 U,

n y=

i.

such

a 2 meet

Proposition:

Proof:

= U,

U

[]

n V a. 6 U ~ i= I 1

(325)

case

{b}

a prefilter.

(324)

this

which

b2 6 U

( a l v a 2)

In

a prefilter

a I v a 2 6 U,

there

# o.

o

~i

prefilter

. According

b,

By

that

Ti

to

a. 6 U. 1

U

is

(317)

a Cauchy

a can

be

prefilter.

covered

6 T:

(324)

some

~i b

6 U,

which

is

just

the

[]

v

Proposition:

(326)

The

ductively

ordered

subfamily

~4 h a s

family

C of

downwards an

all

(i.e.,

Cauchy any

prefilters

linearly

is

in-

ordered

infimum).

v

Proof:

Take

= n F. W e h a v e to s h o w G 6 C. C l e a r l y G is d e f FEA~ (322) (i). Let be U any maximal prefilter U = ~ , a E R°

filter a kernel we

have

such

of

a

~ b

that

T

G

(see

def.

6 U with F

b

6

(314)).

some

F for

all

Since

~ 6 T. F 6 M.

U is

Likewise We

a Cauchy there

conclude:

prefilter

exists

some

a pre-

and

b

(325), TF 6 T

6g

U~M~F T

F

'r

F -1 o"

b

6

b

A O" b T

F

U,

b

a

b #

6

U

0

A b

4:0

-1

a

TF b < a b < a a

~F

a a 6 F. This

holds

for

From

(326)

it

all

F E M,

follows

F there

exists

(327)

Definition:

a minimal

each

unique

Cauchy

In

y c

minimal this

be

F.

x,

We

a 6 R°

Then

the

following

3 a,

~ E T

~b,

TbEF

such

~ b A T b % 0

T

-1

a b A b # O

ab,

~b, ~aEx,

b
a b < Y a y.

case

y

and

we

b

of

show

as

a kernel

of

6 x,

E y

a b

which

set

x

E P,

F + x

which

x = y.

~ b

each

of

prefilter

write:

to

for

Cauchy is

all

prefilter

coarser

minimal

than

F.

Cauchy

points.

prefilter

points,

o

G,

the

Cauchy

holds:

that

that

called

will

6 P two

have

any

-1

elements

For

Take

T

prefilter

Proposition:

Let

: x,

Cauchy

be

Proof: F

lemma

will

F.

P,

a a 6 G.

Zorn's

The

prefilters,

(328)

by

hence:

a.

are

F there which

or

lim

coarser

is

exists coarser

F = x.

than

a than

70

Now we

choose

a. F i n i t e l y cause

they

The

a n d y.

many

are

T a generates tion.

a T 6 T for e a c h

contained

x = H = y.

It w o u l d

H, w h i c h

assertion

these were

Ti a i have

consider

a nonempty

in t h e p r e f i l t e r

a prefilter

above

But

regions

a~6 R o a n d

x.

assumed

by

that

to b e m i n i m a l ,

family

of all

intersection,

Hence

is C a u c h y

~ a 6 x, y s h o w s

the

the

family

its v e r y

beof a l l

construc-

H is c o a r s e r

than

x

hence:

[]

be

convenient

at t h i s

point

candidates

for p o i n t s

to m a k e

some

comments

not used

in s e q u e l . Alternative subset

of maximal

however

shows

prefilters.

that

this

setting.

(329)

Counter-example: of real

I2v =

]a,b[U]b,c[

]a,b[v]b,c[=]a,c[ prime

The

following

Let be intervals

the p r i m e

would

(R,<)

or the

(counter-)

example

be unnatural

in a

the

of t h e

prefilters

lattice

form

of

U ~=O...n

finite

I2~ w i t h

]a2~,a2~+1[or[a2~,a2~+1[or]a2~,a2v+1]or[a2~,a2~+1],

satisfying

Since

The

construction

physical

unions

are

a I < al+1,

does and

I = O,...2n+I.

not occur

in R, w e h a v e

{]x-~,x+~[l~>O}

does

not

form the base

of a

prefilter.

only

prime

{]x,x+e[l~>O}

prefilter and

prefilters.)

filters

~ is i s o m o r p h i c

distributive,

this but

a r e of t h e

{]x-e,x[l~>O},

maximal

By the way,

bases

Hence

the

to ~

is a n e x a m p l e

where

space

form

x 6 ~ . (These

of p r i m e

(resp.

even

generate

maximal)

x {1,2} . of

not distributive.

a lattice, The

which

latter

is w e a k l y

follows

from

pre-

71

]a,b]

A

(]a,b[v]b,c[)

(]a,b]A]a,b[) If w e

v

restrict

becomes

=

]a,b]

and

(]a,b]A]b,c[) the

example

distributive

=

to o p e n

without

Thus,

generated

it b e c o m e s

depends reason

clear,

this

concept

AS POINT

For

(i)

this

of the b o u n d a r y

be ruled out

=

replace

prime

Now,

] a 2 v , a 2 v + 1 [ , (R,<)

(but n o t m a x i m a l )

~ ~ ~

concept

x {1,2,3}

.

of a " p o i n t "

of r e g i o n s .

for a p h y s i c a l

For this

geometry.

SETS

each

= def Hence : def

(332)

a 6 R we define

{x6Pla6x } . x 6 a ~ a 6 x. {~la6R}.

Let be a 6 R

(ii)

N

= a def

N

= def

o

.

{(x,y)6pxPIB~6T

such that x,y6~a}.

{Nala£Ro}.

Proposition: V a 6 R

o

3 x 6 P s u c h t h a t x 6 a.

"Each nonempty

Proof-

Let be b 6 R

U a maximal

x Cauchy

o

region

b 6 U

that

contains

a kernel

prefilter

prefflter

H ~ 6 T such T b,

If w e

are a d d i t i o n a l

sensitive

I2v

Definition:

(331)

{b},

how

must

~.

{]x-~,x+e[IE>O}.

o n the a r r a n g e m e n t

REGIONS

3.3

by

intervals

altering

a I < a l + I b y al ~ al+ I, t h e r e prefilters,

]a,b[.

T b 6 x

with

of a,

at l e a s t o n e p o i n t " .

F the o r e f i l t e r

U D F and x = lim

generated

U. It f o l l o w s :

by

72

~ b A b % O T b < a a6x

x6a.

(333)

Proposition: uniformity

Proof:

(BI )

We in

[BGT]

Nd c b d

have

to

P

6 x.

Set

prove

a ^ T b such

<0 a A

6 R

Choose

x

6

a

6 y,

a 6 R

(332)

(Uii)

(U~I I)

and

(v,z)

o

such

-c 6 T

6 N d.

that

such

It

follows:

z

(y,z)

6 Nb

6 P,

z

(x,x)

Cauchy,

£ N

there

"

exists

N B N b 6 N such

This

holds

: "V

a 6 Ro

trivially

b as

3 b

that

since

6 R°

a kernel

such of

a

~ 6 T

that

T b -I

u 6 T h ~ b T b

T b <

such % O,

A b

~ a,

that since

# 0 ~

~

-I

~ b < ~ a

< b

that

y

is

6 x,

Nb any

"

o N b c Na". (x,y),

y

and

a prefilter

T b < a

T a 6 x.

N b c Nat"

N a = N - Ia

a and

such

(y,z)

follows : 3 ~,

that

a

x is

Choose

of

§ 1.1:

V x

: "V N a 6

entourages

z

O

Since

of

1.1).

B d 6 R

o

any

~ d 6 y,

6 y,

6 N a n N b.

"V

take

f~ T b

(y,z)

II

and

that

§

system

4 items: b

6 N a and

(BGT)

II

a,

(y,z)

in

[BGT]

"V

# O

z and

a fundamental

(see

I § 6.3: fl N b " .

H <9 6 T

(u i )

on

Na

= def

N is

~ b

6 y,

z

6 N b.

It

a

73

o a 6 x, (x,z)

6

z N

a.

In s e q u e l

we will

structure

given

always

by

see

[BGT]

Let

Proof:

be

The

P is a u n i f o r m

II,

(x,y)

V a 6 R

family

because

(335)

F

=

6

Na,

D a6R

Hausdorff

space

(hence

regular,

which

means

o that

~ a has

a prefilter

0 a 6 x, y. the

y.

finite

intersection

of the prefilter

if, w h i c h

t h a n x a n d y.

is C a u c h y

By minimality

x.

property, Hence

they

by construction of x, y w e

infer

[]

Let be a 6 R = ~a def

(ii)

is a f u n d a m e n t a l

the D a6R

o

{Nala6Ro}.

~ V a 6 R O, def

system

II § I, Ex. equivalence Na(M)

.

L e t be M, N c p. M ~ N

P P([BGT]

o

{ (M,N) 6 F p x P P IMC/~a (N) , N C N a (M) },

= def

=

the u n i f o r m

Definition:

(i)

case

with

3).

~ a are e l e m e n t s

and coarser =

Prop.

o f all

all

generate

X

§ 1.2,

H ~ 6 T such

o

P to be e n d o w e d

N.

Proposition:

(334)

assume

of entourages

5), w h i c h relation

w e have:

(M,N)

6 ~

. a

of a uniformity

is in g e n e r a l

~ is n o t

M ~ N ~ M = N.

the

on

non-Hausdorff.

identity.

Since

In t h i s

74

The assignement weak

a ~ a is a l a t t i c e

representation

in the

following

sense.

(336)

Proposition:

L e t be a, b 6 R.

(i)

a < b ~ a c ~,

(ii)

a = ~ ~ a = O,

(iii)

a ~

= a n ~,

(iv)

a ~

m ~ U ~,

(v)

a~'~b

~ a U ~,

Define

for A c P P : A

(vi)

C f i l t e r on P ~ ~ p r e f i l t e r ,

(vii)

C Cauchy

(viii)

F prefilter

(ix)

F Cauchy prefilter

(x)

C minimal

Cauchy

f i l t e r on P

minimal

Cauchy

prefilter

see = def

(335)(ii), { a E R l a EA}.

f i l t e r on P ~ ~ C a u c h y

prefilter,

~ F f i l t e r b a s e on P, ~ ~ Cauchy

f i l t e r b a s e on P,

(i.e.

point).

Proof:

(i)

xE~ a 6 x b 6 x, s i n c e x is a p r e f i l t e r x6~.

(ii)

a % 0 ~ a ~ ~ by x 60

(iii)

(332).

~ 0 6 x in c o n t r a d i c t i o n

a^b
(i)

to "x is a p r e f i l t e r " .

75

x6~n~ a,

b

6 x

a A b

(iv)

6 x,

x

6 a ^ b.

a

v b > a,

a ~

(v)

since

x

is

b

m

a,

~ by

(i)

In v i e w

of

(iv)

it

arbitrary

v

Let

v b and

x 6 a

a v b

6 x,

a 6 u or

suffices

to p r o v e

a v b ~ ~v(aUb)

for

6 R o-

u and

b 6 u.

a v A a # O or y

a prefilter

£ ~ v A a

u be

a maximal

"u p r i m e Since

a v

o v A b

with

prefilter"

(323) (ii)

6 x,

u for

some

(332)

there

# O.

U ~ v A b c ~

prefflter

By

U ~,

u m x. we

~ 6 T,

From

infer:

conclude

we

exists

a

Nv(X) , hence

x c ~ v (~u~). (vi)

Follows

(vii)

The

from

Cauchy

V a £ R°

(i),

(ii)

property B C

and

of

(iii).

C means:

6

C such

that

V a

6 Ro B C 6

C ¥ x,

y 6 C B T 6 T such

Now

consider

any

b

6 R o.

We

C

x C c Na,

have

to

that

is:

that

show:

x,

y

H ~ 6 T

ff T a . such

that

obey. To

this

with £ T

end,

the

property

such

exists

consider

that

some

x

~a

£ oa,

^

a A ~

Ta

-1

Ta

#

0

T a

stated

# O

such

a 6 R

above.

~ a 6 x,

T £ T

conclude:

a kernel

that

that

x,

o

Choose is y

x

of

and

a point

6 ~a.

6 ~a

b

For

(see

a subset x

any

above).

6 C y

C

6 C

and

6 C there Now

we

76

-I

T a < b

T a < ~ b Ta c ~b y E ab. Since

y 6 C was

C c ~

and,

(viii)

Follows

(ix)

Let

(x)

By

C being

from

be a E R

set C

= def

(vii)

chosen

(i), o

shown:

E C. E q u i v a l e n t l y ,

ob 6 ~.

(iii).

exists

required

~ is a C a u c h y

filter

base

is c o a r s e r

a

~

we have

a ~ 6 T such property

that

a a 6 F. If w e

C x C c N

follows. a

prefilter.

prefilter.

According

on P and hence than

G = <.

Let be

(ii),

the

Cauchy

whence

a filter,

. There

~a,

arbitrarily,

to

(viii)

generates

C. B y m i n i m a l i t y

It r e m a i n s

a £ G.

Assume

to

show

of

F c ~ is a c o a r s e r

~ c C is a C a u c h y a Cauchy

filter

C we conclude

G, w h i c h

G = C,

~ = F.

It f o l l o w s :

E G

3 b 6 ~ such

that ~ ~ a

3 b 6 F such

that b < a

a 6 F. Let be

It is c l e a r ,

(337)

that

the

Let C be

According

to

Cauchy

filter

filter

o f x.

a Cauchy

to p r o v e

base

we

a

infer

a E G.

~ a need

not be

faithful.

P is c o m p l e t e .

(336) (vii)

In o r d e r

a E ~ c G

representation

Proposition:

Proof:

c C.

a 6 F. F r o m

x

Consider

filter

o n P.

~ is C a u c h y lim

prefilter.

C = x,

(336) (ix)

it s u f f i c e s

is f i n e r

a neighbourhood

Put

than

Na(X)

the and

x = l i m C, h e n c e

to s h o w t h a t

the

neighbourhood find

a T 6 T such

?7

that N

a

T a 6 x,

~a 6 x.

(x) is c o n t a i n e d

(338)

in t h e

Proposition:

Cauchy =

is an o p e n b a s e

x generates

the p r o o f

of

(337).

(336) (vi£)

filter generated

Na(X)

thus

~ ~a a n d

b y x.

L e t be x 6 P.

: {aia6x}

Proof:

S i n c e y 6 ~a ~ y 6 N a ( X ) ,

a filter,

of the neighbourhood

which

Consequently

and x minimal,

is f i n e r

x is f i n e r we have

than

than

U(x),

~(x).

x = U(x)

filter

U(x).

as s h o w n

Since

in

U(x)

is

and therefore

U (x) .

This

proves

x to b e a b a s e

sider

a n y y 6 a. W e

a 6

c U(y).

hence

open.

(339)

Thus

infer

of

U(x).

In o r d e r

to s h o w

con-

as above:

a is a n e i g h b o u r h o o d

of all p o i n t s ]

y 6 a and

u

Proposition:

P is l o c a l l y

compact.

Let be x 6 P and a 6 x a neighbourhood

Proof:

"a o p e n "

(338).

For

any b 6 R O

we have a < By

a covering

V T i b, i=I . . .n (336) (i),

(iv)

T i 6 T, b y a x i o m and

a c

U ~ i=]...n i

According

to

(v) :

U i=I...n

i=I...n =

"a p r e c o m p a c t " .

T . b,

hence

U ~r~--~. T h e i:1...n 1

[BGT]

R4.

II § 4.2

this

By c o m p l e t e n e s s

sets

rib are N - s m a l l .

covering of P,

property

a will

is e q u i v a l e n t

be a compact

to

neighbour-

h o o d of x.

(33310)

(3311)

Definition:

Corollary:

R

= def

R c R.

{AcPIA

is o p e n

and relatively

compact}.

78

(3312)

Proposition: induced

R is a d e n s e

by ~

(see

subset

of R w.r.

to t h e t o p o l o g y

(335)).

Proof :

I.

Let be

a 6 R a n d a 6 R o. A is c o m p a c t

finitely ~.

many

It f o l l o w s

Ac

U u=l...n

2.1

by

"r a u

It s u f f i c e s

2.

~ a,

B

~

to p r o v e

B'

assume

by

~ a n A % ~ for all

= def

,)=l...n

T a ,0

E R.

B 6 Na(A).

since

A c B.

B c N a ( A ) . L e t b e x 6 B,

2.2.

We may

covered

(336) (v): = def

A c Na(B ) holds

.

~=1...n.

and can be

that

is x 6 T a for

some

~=1...n.

It

follows:

a D A % ¢

(see above)

H y 6 A such

that

x 6 Na(A).

x,

approach

would

in d e f i n i n g

R by entourages

(3313)

(a,b)

6

~va

DJ

An equivalent consist

y

to t h e a

representation

(in g e n e r a l

NR(V) , v 6 R o, of t h e

£ N I (v)

~ def

of R b y p o i n t

non-Hausdorff)

sets

uniformity

on

form:

V w 6 R such

that w

^ a ~ 0

H ~ 6 T such

that w A ~ V $ 0

a n d ~ v ^ b % O, (a,b)

The

points

o f R.

6 NR(V)

could

be

defined

R is i s o m o r p h i c

verification

of t h i s

~ def

(a,b)

6 N I (v) a n d

(b,a)

6 N I (v).

as a t o m s

to t h e

approach

in t h e

lattice

Hausdorff

of compact

is l e f t to t h e

completion

subsets

reader.

o f P. T h e

79

3.4

CONGRUENT

In an o b v i o u s ~:

~PPINGS

manner

P ~ P, w h i c h

will

some properties

(341)

transport be named

of c o n g r u e n t

mappings

..congruent m a p p i n g s " . mappings

A bijection (of P)

shall

a n d of the a s s i g n e m e n t

f : P ÷ P will

compile ~ ~ ~.

(ii)

L e t b e x 6 P, x

(342)

= def

Proof:

Because

Cauchy

prefilters In o r d e r

equivalent

: (x,y)

T 6 T,

~ : P + P. M o r e o v e r ,

an a u t o m o r p h i s m

6 N a-

T is a l a t t i c e on minimal

to s h o w

automorphism

Cauchy

6 T

such

that

~ a 6 x, y

B ~'

6 T

such

that

T

B ~'

6 T

such

that

~'

-I

~' a 6 x, y a 6 T X, []

o f R,

prefilters.

"~ is an i s o m e t r y "

B ~

T is an i s o m e t r y

of the uniformity

assertions:

6 N a.

(fx,fy)

x 6 P.

6 Na

(TX,~y)

6 Na ~

T 6 T.

L e t be

T is a m a p p i n g

and hence

an i s o m e t r y

{Tala6x}.

Proposition: Then

be called

iff

V a 6 R ° V x, y 6 P

(x,y)

We

of p o i n t s

Definition: (i)

tive.

~ induce mappings

T y,

N o n P.

~ maps

Clearly

consider

the

minimal

T is b i j e c following

80

(343)

(344)

Definition:

: def

{~IT6T}. i

Proposition: (i)

Let

(ii)

~

be

• £ T,

: < ~

T is

isometries (iii)

Let

be

of

braic

on

T(a)

kernel T,

of

= T(a)

of

which

= from

sets

= def

the

is

~a. T

into

the

of of

homomorphism

transferred

T ~ T/K.

a subbase

consisting

~

group

of

P.

isomorphism

possesses

Then

a homomorphism

K the

topology

a 6 R.

Then

its the

~ is

~ and

to

T by

~ the

the

Hausdorff

Id- neighbourhood P

algeand

filter

form

{~6TI~ [a]Na%~} ,

a 6 R

. o

Proof:

(i) = {?xla x} =

x

= (ii)

Too

ra6~x}

{ylraey}

=

X =

{~oala6x}

=

T{eala6x}

Ya.

= T O ~ X .

(iii)

T/K

is

Hausdorff

Let

us

assume:

o ~ Id B x

iff

K is

3 ~ 6 K but

closed ~ ~ K.

([BGT] It

III§

follows

2.6

Prop.

18

a).

that:

P

6 P such

that

o x

# x %

3 a

6 R°

space.

such

that

Now c h o o s e

Na(OX)

n Na(X)

= ¢,

since

P is

a

Hausdorff

8~

b 6 R o such there

that N~ c N a and ~ 6 T

exists

~ b

A

•

~

a

T 6 a T(~b)

b * O.

~ x 6 ~ ~ b.

such

n K ~ ¢,

From ~ b 6 x we

t h a t ~ b 6 x.

which infer

Since

~ 6 K,

implies: a ~ b 6 ~ x and

Analogously:

X 6 T~ - ~ ,

further:

B z 6 ~ ~ b N • ~ b ~ ¢

(see a b o v e

That means: (ax,z)

6 N b and

(z,~x)

x = x and Na(~X)

Now T ~ T/K T(a),

follows

N Na(X)

is m a d e

a 6 Ro ,

6 N b,

form

into

(~x,~x)

6 N a in c o n t r a d i c t i o n

to

= ¢.

a topological

a subbase

from standard

hence

for

theorems

the

group,

such

that

Idp-neighbourhood

([BGT]

III§

2.6 Prop.

the

sets

filter,

as

17).

It r e m a i n s

to s h o w T(a)

= T(a) :

T(a)

= {rlTaAa#O}

= {~I~aNa%¢} { r l T [ a ] N a ~ ¢}

(345)

Proposition:

Proof: given

We have

U = ~ T(~b) We want z 6 ~

to p r o v e

x 6 M[~],

N

continuously

the c o n t i n u i t y a

(~x)

T 6 T such

'

a 6 Ro

that

~ y 6 Na(~gx).

n }[~b]

(<0,x)~-~M x. L e t b e

of ~ x. C h o o s e

'

of

(~,x).

Assume

It f o l l o w s :

%

~ y 6 %[Tb]

6 N b and

of

o n P.

(z,~y)

since ~

b 6 R

o

such

T b 6 x. H e n c e

× Tb is a n e i g h b o u r h o o d

to show: ~b

[]

T operates

any neighbourhood

that N b _ c N a and

(~?x,z)

= T(a).

x, y 6 Tb ~

2

6 N b ~ ( ~ x , ~ y ) 6 N b c N a.

[]

(~,y)

6 U.

82

(346)

Proposition: means:

Proof:

T operates

"almost t r a n s i t i v e l y "

on P, that

for any x 6 P, ~ x is dense in P.

Let be y 6 P and a £ R such that y 6 a any n e i g h b o u r h o o d of

y. it follows: x Cauchy prefilter H x 6 T such that ~ a 6 x x 6 ~ [a] T

X 6 a

~xn~,~. (347)

u

Proposition: (i)

The t o p o l o g y ~ on ~ c o i n c i d e s w i t h the t o p o l o g y of compact c o n v e r g e n c e

(ii)

~.

T is u n i f o r m l y e q u i c o n t i n u o u s Def.

(see [BGT] X § 2.2

2).

(iii) The u n i f o r m i t y of compact c o n v e r g e n c e

cu c o i n c i d e s on

w i t h the u n i f o r m i t y of p o i n t w i s e c o n v e r g e n c e ~u (see [BGT] X § 1.3).

Proof : (i)

The u n i f o r m i t y of c o m p a c t c o n v e r g e n c e is g e n e r a t e d by the f u n d a m e n t a l s y s t e m of e n t o u r a g e s of the form T ( K , N b) b6R.

= {(~,~)6T×TlYx6K,(~x,~x)£Nb}, def

K c P compact and

O

finer than 6:

C o n s i d e r any n e i g h b o u r h o o d ~(K,Nb) (~) of

U T i a as a finite c o v e r i n g of the compact set K, i=1...n T_ £ T. Take U = n T(~.~) as an Id - n e i g h b o u r h o o d w.r. 1 i=1...n ~ p to ~. We will show:

83

£0 o U c T ( K , N b ) ( ~ ) .

L e t be ~ 6 %0 o U.

It follows: ¥ x 6 K B j 6 {1...n}

<0[Tja] n x,

~[zja]

such that x 6 z.a and 3

# ¢. N o w ~ [ Y j a]is a k e r n e l

~ x 6 <0T ~ " . b a n d 3

of [p[~j b], h e n c e

(%0x,~x) 6 N b-

T h i s means: V x 6 K,

(%ex,~x) 6 N b

(£0,~) 6 ~ ( K , N b) 6 T ( K , N b) (~). coarser

than c:

It is s u f f i c i e n t

£0-neighbourhood w°r.

to c o n s i d e r

to ~ of the f o rm 9 o T(a),

some x 6 a and b 6 R ° such that Nb(X) is open.

Now consider

the c o m p a c t

~x,

because

set ~ and the c o r r e s p o n d i n g

= T ( a , N b) (~P) w.r. def ~ 6 M(M) implies:

V y 6 ~ H r 6 T such that ~y,

a 6 R o. C h o o s e

c a. This w o r k s

a given

to c. We w i l l

show:

9y 6 T b and further:

~y 6 ~[a]

x, ~ -1 ~ x 6 qo-1 ",, • [b] -I

~ x 6 Nb(X)

-I x, £0 ~ x

6 a

Cx c ~[~], ¢[a]

£ £0 o T(a). (ii)

T consists

(iii) F o l l o w s

3.5

CHAINS

of i s o m e t r i e s .

from

(ii) by

[BGT] X § 2.4 Th.

I.

I

The use of " c h a i n s "

to m e a s u r e

been suggested

by s e v e r a l

and W. B ~ c h e l

(in [BOE]).

the d i s t a n c e

authors.

between

We o n l y m e n t i o n

two p o i n t s G. L u d w i g

has

[LUD 2]

84

A chain

joins

overlapping

two

points

and

consists

of a s e q u e n c e

regions.

fig.

The

precise

(352)

of c o n g r u e n t ,

definition

reads

as

(351)

follows.

Definition: (i)

L e t be x, y, ~I'''"

Yn'

x', '

TI'"

y'

.. ~' n'

~ (x'y'v'TI''" .~n ) = X = X',

A chain length

E

v'

T.

and

~. v = ~' v' l l

9 is an e q u i v a l e n c e between

n 6 ~

(n+3)-tuples

6 Ro;

( x ' , ~v ' , v ' , ~ , . . . ~ ' n' )

y = y' , n = n'

Clearly, (ii)

6 P; v,

def for

relation.

x 6 P a n d y 6 P of o r d e r

is d e f i n e d of the

i=1...n.

v 6 R ° of

as an ~ - e q u i v a l e n c e

form

(x,y,v,T1,...~n)

class

such

of

that:

x 6 Xl v, y 6 ~n v and ~i v ^ Ti+ I v % O for

all

The

will

equivalence

class

i=1...n-1. be d e n o t e d

by

[x,y,v,~1-..Tn]-

If

[ x , y , v , r l . . . ~ n]

is a chain,

x i 6 Tiv n ~i+i v, N -chain V

in t e r m s

i=1...n-1. of

(X i, X i + I) 6 N v for

[BGT]

II

i=O,...n-1.

there Hence § 4.4,

exist (Xo=X, Def.

points x I .... Xn_ I, Xn=Y) 3, t h a t

is:

is an

85

The

existence

connected, Therefore

of c h a i n s

we need

into

assume

to a s s u m e

plausible,

even T should

it w o u l d

account

a further

that,

axiom

the

up to now, to a s s u m e

a "pre-connectedness". T, ~ E T w o u l d w.r.

N = {(<0,~)6TxTI~-1~6

instance,

P is n o t

x and y of order

of c o n n e c t e d n e s s .

space

P but

in s o m e

n E I~,

T is n o t

necessarily

connectedness

That would mean,

be connectable

by an

to the l e f t u n i f o r m i t y

sense

but

any

complete,

rather

two

"N-chain",

we

transport where

t s o n T:

D T(ai)} i=I . . .n

a. 6 R for i = 1 . . . n . 1 o

Clearly

it s u f f i c e s

to t a k e

(353)

Axiom

V ~ 6 T

R5:

V a I,... ~I'''" TI = IdR' V i=I

•

section

2)

considered

~ = Id R.

V n 6 I~

a n E R ° 9 m E l~ ~m E T

such

that

~m = ~ a n d

..n V j = 1 . . . m ,

"Every transport

this

v.

It is

{ (<0,~)ETxTlVi6{1...n},~aiA~ai#O},

where

From

for

be connected.

some entourage

=

If,

at all b e t w e e n

that not only

be unreasonable

mappings

insured.

t h e r e m a y b e no c h a i n s

physically

Taking

is n o t

can b e

Tj+ I a i A T3.

composed

from

(and f r o m the c o n s t r u c t i o n it is c l e a r as t r a n s p o r t

that,

small

i

~

"

O

transports".

of t r a n s p o r t

for i n s t a n c e ,

mappings,

a

although

mappings

reflections they

are

in

are not

isometries.

N is

88

(354)

Proposition: Let

Axiom

be U any

that

is,

R5

is e q u i v a l e n t

IdR-neighbourhood

T is g e n e r a t e d

in T,

to the

following:

then T =

by a r b i t r a r i l y

small

U U n holds; n6~ IdR-neighbour-

hoods.

Proof: I. A s s u m e

R5 h o l d s .

T2 6 V,

-I T3

O = T2O

~3 6 V,...

(T21~3)

2. C o n v e r s e l y , U =

T(ai).

o = o 2 o 3 ...

T ( a i) and and,

of T =

o i 6 U for

~ 6 V m-1.

~ 6 T be given. U n6~

U n,

(355)

~j+1

= Tj Oj+I

¥ j=1...m-1,

Proposition: chains

Proof:

Tj+I

i=2...m.

for j = 1 . . . m - 1 .

Since

x 6 ~v, -I

x and

p. T h i s

yields

v ^ Tj V % 0 a n d

Definition:

m]

apply

axiom

exist

R5

there

exist

T, p £ T such

for n = I, a I = v a n d such

that

~m = a. H e n c e

desired

length

there

v.

j=1...m,

TI = IdR' is the

order

prefilters

T. £ T, 3

The

[3

y 6 P, v 6 R o. T h e n

y of

x, y are C a u c h y

y 6 pv. N o w

and

=Tm

Tj+ I a i ^ ~j a.x * O.

L e t be x,

between

[x,y,v,TTI,~T2,...T<

(356)

as

follows:

V i=1...n

o = ~

Define

~ can be w r i t t e n

o = q 2 o 3 ... ~m = (T~IT2) (~21~3)''" (~m~ITm) --I ~j T j + 1 = o j + 1 6 U, t h a t is:

that

It f o l l o w s ~

since

(~ ~11a _ ):

a n 6 R ° and

~ 6 T.

define

~I = I d R a n d It

O...

Because

om where

N i=]...n

-I Tm_ I ~ £ V,

( ~ I T 4)

0

let al,..,

D i=1...n

Now

L e t be V =

chain.

of c h a i n s

of o r d e r

D

v between

x and

87

y is b o u n d e d

from below

which

the minimal

call

(357)

attain

them minimal

Lemma:

Proof:

v < w ~

Let be

[ x , y , w , ~ 1 . . . T n]

Lemma:

(358)

Then

Proof:

length

length

Hence

n m i n d e= f

l(x,y,v)

is a chain,

exists

hence

l(x,y,v).

We

shall

~ l(x,y,w)

chain.

It f o l l o w s ,

~ n.

[]

~(x,y,w)

that

t h a t x % y.

a v 6 R o such that

P is H a u s d o r f f , there

are chains

chains.

[ x , y , v , % 1 . . . % n] a m i n i m a l

there

there

there

l(x,y,v)

exists

is no c h a i n b e t w e e n

~ 2.

a v 6 R o such

x and y of order

that v and

I.

Proposition:

(359)

L e t b e x, y 6 P a n d x % y.

{ X ( x , y , v ) Iv6R o}

Proof: further

Assume regions

l(x,y,v)

S n. C o n s i d e r

a I 6 R ° such

ak E R ° such According

to

(355)

chain we will

there

set o f

arbitrary

t h a t N (2k) c . . . c N and ak ao

exists

construct

some

integers

a ° 6 R° and

that N 2 c N , al ~o t h a t N ~ 2 c Nal c N a o , etc . . . .

a chain

[x,y,ak,T

an a k _ 1 - c h a i n

L e t b e x 2 6 T 2 a k N T 3 a k % ~, h e n c e

in t h e

x2 6 N

6 T s u c h t h a t x, x 2 6 T

ak_ I,

until

2k a n.

.... •

]. F r o m o following manner:

(x) c N k

H T k-l)

The

is u n b o u n d e d .

a 2 6 R o such

this

I. T h e r e f o r e

L e t b e x, y 6 P s u c h

Because

y ~ Nv(X).

by

(x) ak- I

88

fig.

etc.,

until

we

obtain

[ x , y , a k _ I T~k-1)

([...] From

denotes

last

the

Hence

l ( x , y , a O)

(3511)

"r,0v, " r

Let

v £ R

V y 6 ~,

[TIT]

6 T,

11 ~ 2 k-1. ak_2,

We p r o c e e d

ak_3,..,

= I in c o n t r a d i c t i o n

o

p.

choose

to

recursively,

until has

the

two

ak_ k is r e a c h e d .

length

(358).

sets

of

i k g 2k - k

= I.

[]

be K c p a p r e c o m p a c t

. Then

subset,

integers

and

{l(z,y,v) Iz,y6K}

(from

function).

(k-k)] [ x , y , a k _ k , ~ I (k-k) ,.. . ik

{l(x,y,v) ly6K}

Proof

integer

of o r d e r

Proposition: x 6 P,

11 = [1°+I]

[-~--]

greatest

chains

chain

that

• 11

1 ° _< n _< 2 k we o b t a i n

constructing The

a chain (k-l) ] such

. . . . .

(3510)

196,

are b o u n d e d .

3.4):

points

Cover

K with

finitely

many

y~) 6 -c,~v a n d n o t i c e :

X(x,y,v)

-< X(x,y~,v)

V z 6

V y 6 •~'~ v,

i (z,y,v)

(3512)

Proposition:

+ I and <

x(y~ ,y~,v)

+ 2.

[]

L e t be K c p p r e c o m p a c t .

H v 6 Ro V a 6 Ro,

a < v ~ V n 6 I~ ,

Nn(K) a

is p r e c o m p a c t .

89

Proof

(see

compact.

[FRE]

1.3,

L e t b e u 6 Ro,

a covering

o f K,

x 6 Ta a n d

some y 6 ~

We will

1.4):

v a kernel

a 6 R o,

show

first,

~v v'

of u a n d K c

U v=1...n t h a t is:

a < v and x 6 Na(K),

N K,

t h a t N a (K)

is p r e ~v 6 T

T 6 T.

It f o l l o w s : B v 6 {1...n}

such that

y 6

re c Tv y 6 ~v A r v # ¢ V

T

-I -I

T v A v # 0

T v < u

cv < Y U x 6 T u.

This

shows:

Na(K ) c

U

•

v:1...n

Now,

Na(K)

is a s u b s e t

u v

of a finite

union

of p r e c o m p a c t

sets,

hence

precompact.

The

precompactness

The

last propositions

compact" plete

for

spaces.

(3513)

Proof: n = max x 6 ~[L]

For

H

Choose

valid

since

the next

L e t b e K, = def

follows

remain

"precompact",

Lemma: Then

o f Nn(K) a

if o n e

we need the

"relatively

coincide

in c o m -

following

L c P precompact.

to

D K # ~ a n d y 6 ~[L].

~-Ix

is p r e c o m p a c t .

(3512)

(this e x i s t s

a v-chain

u

substitutes

the two c o n c e p t s

section

v 6 Ro according

with

induction,

U{M[L]I~6~,~[L]NK#~}

{l(x,y,v) Ix,y6L}

be connected

by

by

let b e

(3511).

and -ly

of l e n g t h

and

Consider

are points

k ~ n. T h e

latter

any

in L a n d c a n is a l s o

true

90

for x and y. This

shows y 6 N~(K) ~

subset of a p r e c o m p a c t

The

length of m i n i m a l

the d i s t a n c e accurate

This

a 6 R

To o v e r c o m e

this

more

and m o r e

depends

the chain q u o t i e n t

Ro/T,

[a] ~

(downwards) [a] C

This

[b]

directed

of a".

is scaled down,

difficulty,

as "unit

then

manner.

class

Note

of a,

as a real

fixed x , y , u , v

6 P.

proved

by using

(332).

limit of the

function

f in the sense of

l(x,y,a) l(u,v,a)

This

2), w h i c h will

leads

which

Hence

that

[a] 6 Ro/T,

a I < b I.

3, Ex.

the

a is chosen.

= 3 a I 6 [a], b I 6 [b] such that def

Def.

of

l(x,y,a)

and c o n s i d e r

ordering

I § 7.3,

measure

of the distance,

may be c o n s i d e r e d

set w.r.

but

length"

following

' for

D

let us fix a pair

the smaller

in the

As a

It can be m a d e more

to the partial

speak of the

lim a£R

H is precompact.

as a m e a s u r e

= l(x,y,a) X(u,v,a)

can be e a s i l y

[BGT]

o

H c N~(K).

is an a p p r o x i m a t e

only on the c o n g r u e n c e

h ence

f[a]

l(x,y,a)

accurate,

can be m a d e p r e c i s e

l(x,y,a)

(3512)),

points

l(x,y,a) l(u,v,a)

"chain quotient"

therefore,

x and y "in units

6 P × P of d i s t i n c t

becomes

(by

chains

if the r e g i o n

increases. (u,v)

between

set

and,

function

f on

Now."Ro/T is defined

by

it is m e a n i n g f u l

be d e n o t e d

a

to

simply by

to the

o

(3514)

Definition: the lim a6R

If for all x , y , u , v

6 P

(such that u % v)

limit ~(x,y,a) l(u,v,a)

o exists,

A

it is c a l l e d

chain distance we w i l l

= def

(x,y,u,v) the chain d i s t a n c e

quotient)

abbreviate

between

A (x,y,u,v)

(more precisely:

x and y. If u,v are fixed,

= d(x,y).

91

3.6

COMPLETION

In ord e r

to connect

have to show, (w.r.

OF THE GROUP our theory with

for instance,

transitivity

to c u) of the group of c o n g r u e n t

not n e c e s s a r i l y completion [FRE]

complete,

of T and of proving

for this

completion

formity

convergence

example, n

the group

Another

in

counter-example

transitivity

[BGT]

T is

of an a p p r o p r i a t e axioms

in

at all).

of ~ w.r.

is not p o s s i b l e

is c-dense

we w o u l d

~. Of course,

and the o t h e r

that a c o m p l e t i o n

GL(~,n)

(~) , as m e n t i o n e d

[FRE],

and c o m p l e t e n e s s

mappings

(if p o s s i b l e

emerges

of compact

in

and it is thus a q u e s t i o n

Now the d i f f i c u l t y

M

the axioms

to the uni-

in all cases.

For

in the set of all m a t r i c e s

X § 3.5.

is the d i s c r e t e

space

~ , endowed with

the

metric d(n,m)

= ~rl [0

if n # m if n = m

and the group of isometries t ogeth e r axioms

with

the

lattice

of ~

([BGT]

of finite

subsets

if we choose

another

of T,

some extra work will be n e c e s s a r y

[FRE]

under

c onsid e r

the m o d i f i e d

the t o p o l o g y

(resp.

of the two

right) c

z

= c

s

convergence,

conditions.

c (=s=~ by

uniformity

This

satisfies

cs

uniformity

group

the

to prove

(347)) (=au)

v c , the t w o - s i d e d

on

several

by F s

theorems

cd)

uniformity.

and the s u p r e m u m Further,

the u n i f o r m i t y

(P,P).

in

in this way and

and the c o r r e s p o n d i n g

(resp.

P ÷ P, endowed with

will be d e n o t e d

for the c o m p l e t i o n

Let us p r o c e e d

d

space of all m a p p i n g s wise

of ~

19c).

RI to R4.

On the o t h e r hand,

left

X § 3 Ex.

the

of point-

92

(361)

Proposition: space

Fs

Each c Z - c a u c h y - f i l t e r on T converges

(P,P). The set of limit points T c Fs(P,P)

uniformly equicontinuous

group of h o m e o m o r p h i s m s

in the is a

of P. T is

s Z - c o m p l e t e and T is S-dense in T.

Proof:

(362)

[BGT] X § 3 Ex.

Corollary:

19d)

[]

T is cZ-complete,

T is c-dense in T. Hence

may be i d e n t i f i e d w i t h the c Z - c o m p l e t i o n of T. The m a p p i n g s 6 T, w h i c h also w i l l be called a u t o m o r p h i s m s of the u n i f o r m i t y

Proof:

(363)

Follows

from

Definition: T(A,B)

~(x,B)

= def =

[BGT] X § 2.4 Th.

In a c c o r d a n c e w i t h

"congruent mappings",

are

N on P.

I

D

(344) (iii) we define

{T6TIADT[B]%¢} ,

~({x},B),

def T(A)

= T (h,h) . def

Analogously:

(364)

T(A,B),

T(x,B),

T(A).

Proposition: (i)

The tripel

(R,c,T)

corresponding isomorphic

satisfies the axioms RI to R5. The

space of points P is c a n o n i c a l l y

to P as a u n i f o r m space.

(The bar denotes in seguel a t r a n s f e r of a d e f i n i t i o n w i t h i n the (ii)

(R,<,T)-theory onto the "model"

(R,c,T),)

The sets T(A), A 6 R , form a subbase of the t o p o l o g y o on T, w h i c h c o i n c i d e s w i t h the t o p o l o g y of compact convergence

c.

93

Hence

to e a c h p r o p o s i t i o n

new proposition

p r o v e d w i t h RI to R5 there c o r r e s p o n d s

(the "bar" version)

Proof:

(i)

I.

is a d i s t r i b u t i v e

(R,c) empty,

which

lattice with

is v a l i d

for

l east e l e m e n t

a

(R,c,T).

~. R is not

s i n c e P # ~.

2. T c o n s i s t s

of h o m e o m o r p h i s m s

automorphisms

and thus

induces

(which we h a v e i d e n t i f i e d

3. L e t be A 6 t o. S i n c e A is open, that ~ c A. C h o o s e b E R

(R,c)-

w i t h T).

t h e r e exists

as a k e r n e l

a g r o u p of

some a 6 Ro such

of a k e r n e l

of a, h e n c e

O

N b3(~)

c a • We w i l l

of A. T h u s

assume

show,

that ~ 6 R c R is a k e r n e l

T 6 T such that T[b]

(w.r.

to T)

N ~ % ~, say

~y c ~[~] n ~. Further exists

let x 6 b be a r b i t r a r y . S i n c e T is e - d e n s e % a ~ E T N T(b,Nb) (~). This implies:

V z 6 5 B a 6 T such that

~z,

~z

in T, t h e r e

6 a[~]

•y 6 b • Y,

~Y 6 at[b]

~x 6 ~

[b]

~x,

E

Hence: Since

Tx

~

Tx E N

[~] ) c a c A.

~x 6 ~[~] was

arbitrary,

we h a v e p r o v e d

T[b] c A,

thereby

R3. 4. L e t be g i v e n A, V 6 R . T h e r e o can be c o v e r e d

by f i n i t e l y m a n y

this the c l a i m of R4 f o l l o w s Notice (347)

that part(ii) (in the

discussion.

is some a 6 R Tl[a],

such that a c V and

~I 6 T,

l=1...n.

From

immediately.

of the p r o p o s i t i o n

"bar" version)

o

follows

from p r o p o s i t i o n

and m a y be u s e d in the

following

94

5. Let be o 6 T and U some Id - n e i g h b o u r h o o d in T w.r. to c. T being P c-dense in T, there exists some ~' 6 T such that , - I ~ 6 U. Moreover, B e t w e e n Id

U' P

= U R T is a Id - n e i g h b o u r h o o d in T w.r. to c. def P and o' there exists a U ' - c h a i n by a x i o m R5:

(~I = I d p , T 2 , . . . T n = 0'). C o n s e q u e n t l y ,

(~1,...Tn,~)

is a U-chain.

6. P ~ P . 6.1. Let F be a filter on P. The a s s i q u e m e n t on P onto p r e f i l t e r s

in

F ~ F N R maps

filters

(R,c). We will show that the m i n i m a l

C a u c h y filters on P are b i j e c t i v e l y m a p p e d onto the m i n i m a l Cauchy prefilters

in

(R,c,T).

Let F be a C a u c h y

filter on P and B 6 Ro' ~ c B, a ~ £ Ro . Since

is a C a u c h y p r e f i l t e r

in

(R,<,T)

(see

(336) (vii)),

exists some T 6 T such that ~ a E [, that is:

there

T[a] 6 F. By t h e

filter p r o p e r t y of F we have x[B] 6 F n R, w h e r e ~ 6 ~ c T. This proves

F n R to be Cauchy.

Conversely,

let C be a C a u c h y p r e f i l t e r in

(R,c,T)

and F the

filter on P g e n e r a t e d by C, hence C = F N R. In view of (ix) it suffices

(336)

to show that [ = ~ is a C a u c h y p r e f i l t e r in

(R,<,T). Let a 6 R ° be given and choose some b 6 R ° such that Nb2(~) c ~. The C a u c h y p r o p e r t y of C yields a ~ 6 T such that T[~] 6 C. Since T is c-dense in T, we find some 6 T N T ( b , N b) (~). Like in 3. one can show T[b] c o [ a ] ,

hence ~[a]

6 C and

~a6C. T h e r e b y we have shown, which

that there exists a C a u c h y filter

is m a p p e d by F ~ F N R onto a

(R,c,T). Now we know, neighbourhood injective,

filters,

F on P,

given C a u c h y p r e f i l t e r

C in

that m i n i m a l C a u c h y filters on P are just since P is complete.

Hence F ~ F n R is

if r e s t r i c t e d to m i n i m a l C a u c h y filters.

map r e s p e c t s the i n c l u s i o n of

Since this

(pre)filters, minimal C a u c h y

95 filters

on P are b i j e c t i v e l y

filters

in

6.2. We w i l l

mapped onto minimal

(R,c,T).

identify

P and P as sets a c c o r d i n g

to 6.1.,

s i d e r the u n i f o r m i t y

N on P g i v e n by e n t o u r a g e s

Na = {(x'y)6P×PIH~6~

such that x , y 6 T [ a ] } ,

It r e m a i n s 6.2.1.

C a u c h y pre-

to s h o w that N and N c o i n c i d e

and con-

of the form

a 6 Ro"

as u n i f o r m

structures.

N is f i n e r t h a n N. L e t N a, a 6 R o, be g i v e n and c h o o s e b 6 R ° as a k e r n e l We will

s h o w N ~ c N a. Thus t a k e any

x, y 6 ~[b] Hence

property

x, y 6 T[a]

and

N is c o a r s e r open,

some ~ 6 T. By p a rt

there exists

kernel

6.2.2.

with

T a k e any

(x,y)

have

to p r o v e

(365)

in T.

# ~. By the

hence

s o m e ~ c a, ~ 6 R o. We w i l l

and,

since T c T,

(x,y)

S i n c e a is s h o w N b c Na"

of T w.r.

in

~ 6 T. We

to the u n i f o r m i t y postulates,

[FRE] u n d e r our w e a k e r

L e t K c P be compact.

some

6 Na .

C u = C s, as H. F r e u d e n t h a l

some t h e o r e m s

Proposition:

follows,

a 6 Ro' be given.

B e c a u s e we h a v e o n l y the c o m p l e t e n e s s the c o a r s e r

n T[b]

6 N b, that is: x, y 6 ~[b] w i t h

i n f e r x, y 6 ~[a]

C z instead

(ii), ~ is t - d e n s e

6 Na .

t h a n N. Let Na'

(x,y)

6 N~, w h i c h means:

a Y 6 T such that ~[~]

of b, a[~] c T[a]

it c o n t a i n s

(x,y)

of a.

T(K)

we

conditions.

is p r e c o m p a c t

w.r.

to ~z.

Proof: ~(K)

According

to

[BGT]

II § 4.2 Th.

can be c o v e r e d by f i n i t e l y m a n y

sets, w h i c h

I. L e t N = N s N N d be any ~ Z - e n t o u r a g e , the f o r m

3, it is to p r o v e that are ~ Z - s m a l l .

which may assumed

to be of

96

N s = {(~,~)6TxTIVi=I... =

Nd

{ ( ~ , ~ ) 6 T~ x T~I V i = I . . . 1 , $ ~

where

a i 6 R o,

suppose lemma U

Let e

Aai%0},

i=I...i.

without (3513)

loss

there

-I a i Aa.~O} z '

Because

of K' D K ~ ~(K')

of g e n e r a l i t y ,

exists

that

a compact

D T(K)

K contains

subset

we may

all

H c P such

ai"

By

that

[p[K] c H.

c. 6 R be k e r n e l s l O

= def

X i=I...i

W is c o m p a c t

~

~w

of a. and l

= def

m

as a c a r t e s i a n

on P x p x...x

p by m e a n s

~ ( X l , X 2 , . . , x I)

= def

The

open

W c

U ~=1...m

2. N o w w e

covering

product

W c

of c o m p a c t

sets

and T o p e r a t e s

of

U

T[c]

for

contains

T 6 T. a finite

covering

• [c].

to c o v e r

of c i c K w e h a v e

i=

~.

.....

(~x1,~x 2 .... Tx I)

are p r e p a r e d

2. I. B e c a u s e fore

~x~

.IX...l ¢[~i ] c W.

[c] of the

X i=I...i

X

above

Let

be

~ 6 T(K).

¢[c i] c H for all

i=I . .. 1 a n d t h e r e -

Hence

covering

X ~[~i ] m e e t s i=I...i and we c o n c l u d e :

some

~[ci] n ~ [c] ¢[~i] n

i=I ...i

X

T(K).

X

~[ci ]

i=I...i

~[~i] n

~ [~i ] , f u r t h e r

i=I... 1 H U 6 {1...m} 6

2.2.

U U=1...m

V i 6 {I...i},

T

0 [ ~

n i=I...i

9 6 T(K)

~ 4 -I

conclude

analogously:

m -I ¢ % @[c i] D T [Ci],~T U @ 6

T (C i) ]

6 T(K).

= def

We may

n T(ci)~ i=I . . . i

U T Z-

repeat

the

argument

of

2.1.

and

97

B v E {1...m}

6 T for v=1...m.

V

6

2.3.

U ~=1...m

Because

[

~ i] N ov[ci], ~ ~ 9-I [c

V i 6 {I...1},

If follows:

n i=I...i

9 E T(K)

where

~(~i ) 0 o

]

was a r b i t r a r y ,

= def

U S.

we h ave c o n s t r u c t e d

a finite

covering

~(x)

3.

~

u

It r e m a i n s (SvNT)

T

.

to show: (SvNT~)

x

Let be ~, = T

N

S

~ 6 S

c Ns

N d = N.

N

n T . As e l e m e n t s

0 ~ such that V i 6 {i...l},

such that V i = {i...l}, in

the

properties

of c i in o r d e r to c o n c l u d e :

(~,~)

~,

(366)

By

~ E S . V

The

proof

Analogously

"bar"

of

(316)b)

the

a[a i] N ~[a i] # ¢, w h i c h (~9,~)

version

of

E Ns

(365)

But T(K)

l a) and T is ~ z _ c o m p l e t e .

is

inferred

kernel

proves

from

reads:

is r - c l o s e d

This gives

(366) we h a v e p r o v e d e s s e n t i a l l y

(367)

uses

(K) is t - c o m p a c t .

is ~ z - p r e c o m p a c t .

Let us r e c a l l

one

[]

Corollary:

Proof:

Th.

6 Nd .

by

[BGT]

III § 4.5

the a s s e r t i o n .

that T o p e r a t e s

"properly"

the c o r r e s p o n d i n g

Definition

0 B

8[c i] N c i ~ ¢.

as

Hence

, they are of the form

e[ci ] N ci % ¢' ~ = T

Similarly,

c i c ~[ai] , B[ai].

T(K)

of T

(see

A topological

[BGT]

III§

g r o u p G, w h i c h

4.1 Def. operates

I): continuously

on a

on P.

98

topological

space X, is said to o p e r a t e properly,

iff the

mapping 8 : G × X ÷ X x X

(T,x) ~ is proper, mapping

(368)

that is, if for every t o p o l o g i c a l space Z the

8 x Id Z m a p s c l o s e d sets onto c l o s e d ones.

Theorem:

Proof:

(x,~x)

T o p e r a t e ~ p r o p e r l y on P.

Since P is locally compact,

the c o m p a c t n e s s of T(K) Th.

the a s s e r t i o n is e q u i v a l e n t to

for any c o m p a c t K c p

I.c). Hence it follows from

(366).

(see [BGT] I I I §

4.5

[]

F r o m the cited t h e o r e m in [BGT] we further infer:

(369)

Theorem:

T is locally compact.

(3610)

Theorem:

~ o p e r a t e s t r a n s i t i v e l y on P.

Proof:

Let be x £ P, By

But T x is closed, properly.

(3611)

Proof:

(346) T x is dense in P, hence also T x.

since T and {x} are closed and T o p e r a t e s

[]

Theorem:

T is connected.

T is g e n e r a t e d by any I d - n e i g h b o u r h o o d

locally compact.

By

[BGT] I I I §

4.6 prop.

connected.

(3612)

Definition:

Let be x 6 P.

(see

14, Cor.

(354)) and is

2, T is

The

group

subgroup point We

(3613)

Proof: T/Jx J

Jx by

x.

shall

simply

J for Jx'

Theorem:

P is a t o p o l o g i c a l

(see

III§

[BGT]

(3613)

means,

if x is k e p t

homogeneous

fixed.

space

of

2.5).

that

the c a n o n i c a l

map

~TX

is a h o m e o m o r p h i s m . since

T operates

This

Theorem:

Proof:

[BGT]

(36.15)

Proposition:

Let

and

from

III§

4.6

[BGT]

III§

transitively.

P is c o n n e c t e d

subgroup

Proof:

follows

properly

(3614)

Prop.

4.2 Prop.

4,

[]

and paracompact.

13.

J x does

not contain

any p r o p e r

invariant

N of T.

be v E N, y £ P.

It f o l l o w s

that

3 T 6 T such

that

T x.

v y = v T X

= def

V = I~.

With

write

÷ P

X

y=

= T({x}) = { y 6 T ] T x = x } , w h i c h is a c o m p a c t def (366), is c a l l e d the s t a b i l i t y s u b g r o u p of the

since

~ 6 N c Jx"

Hence:

[]

the

proved

T ~ X = T X = y,

exception

in o u r

of a x i o m

context.

(Z) all of the

axioms

in

[FRE]

are

now

100

3.7

CHAINS

II

For

later

and

(R,c,T)-chains.

really

Above

use,

new

chain

all,

a

Our

aim

is a d d e d

~(a)

that

to e a c h and v i c e with

L e t be a 6 Ro'

is T - d e n s e

between

(R,<,T)-chains

the c o n j e c t u r e

that

no

by c o m p l e t i o n .

(R,c,~)-chains

Lemma:

the r e l a t i o n

is to v e r i f y

(R,c,~)-ehain

compare

(371)

analyze

it is clear,

corresponds will

we w i l l

(R,<,T)-chain

there

versa.

first

In the

s t e p we

(R,c,T)-chains.

x 6 P.

in T ( a ) ,

likewise

~(x,a)

in T ( x , a ) .

T(x,a)

is

T-open.

Proof:

T(a)

6 T(a)

we

analogous hood,

exists

t o

such

[c]

x £ T o

(316).

Thus

o 6 ~ D T(a)

is valid.

that

c

: ~(a).

-I

For T(x,a)

T £ T(x,a),

~[a]

we may ~ a

in any n e i g h b o u r h o o d

L e t be

t h a t Nb(X)

of g e n e r a l i t y

a o 6 ~ such -I

by

some

argument

loss

~ o

-I

find

b 6 ~o'

Without

From

is o p e n

and

assume

an

U some

c a kernel

of a

Id-neighbour-

of b.

x 6 c c b c

T[a].

There

6 U N T(c).

n c % @ follows: -I

c c b c

~[a]

x 6 a[a] o

e

This The

~(x,a). means: same

reasoning

contained

For will

the

~(x,a)

also

in T ( x , a ) .

purpose

consider

~ (x,y,a)

is d e n s e

which

in T ( x , a ) .

shows

Hence

T(x,a)

of c o m p a r i s o n

a formulation refers

only

that

the

~-neighbourhood

is open.

of m i n i m a l

of the

chain

definition

to the g r o u p

~.

(T(c)) -I

[]

lengths of

we

T is

101

Definition:

(372)

~(x,y,a)

(~)

T(x,a)

(373)

Lemma:

Proof:

Let be x, y 6 P, x ~ y, a 6 R • o

= n will

T(a) n-1

be d e f i n e d

n T(y,a)

~(x,y,a)

satisfying

% ~.

such that i=2...n.

and ~i 6 T(a),

Yn = Y1 cP2
where

We conclude:

x 6 T1[a]

~I 6 T(x,a) ~2[a]

n a ~ ~, ~2

= ~ ~2 def I

T2[a] D ~1[a]

~

~3[a]

fl a # ~,

T3

= def

~2 ~3

T3[a] N ~2[a]

#

~n[a]

N a ~ ~, Tn

= def

Tn-1 ~n ~ Tn[a] N Tn_1[a]

~n 6 T(y,a)

%

y 6
Thus

each

element

and,

of course,

therefore

integer

= l(x,y,a).

Let be T n 6 T(y,a)

as the least

T n of

(372) (~) yields

vice versa.

Minimal

the same as m i n i m a l

chain

a chain

n such that

[ x , y , a , T I , . . . T n] (372) (~) holds

length.

is

[]

Analogously,

Lemma:

(374)

Each

T(x,a)

T(a) n-1

yields

a

Now we can c o m p a r e

element

of

N T(y,a)

(R,c,T)-chain

chains

and vice versa.

of d i f f e r e n t

type.

Each

(R,c,T)-chain

(R,c, ~) -chain. Hence

~

= def

I (R'c'~) (x,y,a)

~ ~

= def

i (R'c'T) (x,y,a).

is a

102

Now consider

some

(R,c,T)-chain

[x,y,a,~1,...Tn].

As in the proof of

(373) we have Tn = ~I ~2

"'" ~n'

TI 6 T(x,a),

T n £ T(y,a),

~i E T(a),

i=2...n.

There exists a T n - n e i g h b o u r h o o d

U n c T(y,a),

since T(y,a)

is open

(371). The m u l t i p l i c a t i o n neighbourhoods

V I of TI, V i of ~i'

T~ 6 Vl, ~i' 6 Vi, choose

in T is continuous,

i=2...n:

T{ 6 V I N T(x,a),

T'n E Un n T c T(y,a).

~

i=2...n,

~= T~ ~ ~n' def

a

such that for all !

... ~n 6 U n. Using

6 V i N T(a),

This yields

hence there exist

i=2...n,

(R,c,T)-chaln

(371) we

and conclude [x,y,a,~{...T'n ]

of the same length n. Hence ~ ~ ~ and we have proved the

Proposition:

(375)

The completion

length of minimal

chains.

In the next step we will compare

(376)

of ~ to T does not alter the

(R,c,T)-chains

with

Lemma:

Let U be an open I d - n e i g h b o u r h o o d

topology

t on T, K c p compact.

e E Po'

(R,c,T)-chains.

w.r.

to the

Then there exists

such that for each x £ K, U operates

an

transitively

on

Ne(X) •

Proof:

Let V be an open I d - n e i g h b o u r h o o d

satisfying VV -I c U.

Let us assume the negation of the above claim: V e 6 R

o

3 x(e)

Clearly x(e)

6 K 3 ~ 6 T such that x(e)

6 ~[e] but ~[e] ~ U x(e).

depends only on the congruence

class

[e] 6 [R o]

~ Ro/T. Notice that [Ro ] is d i r e c t e d downwards in a def natural way. Since K is compact, there exists a filter F on [Ro], which

is finer than the section

Vz is open

([BGT] I I I §

N2(z)= c Vz. x(F)

filter,

2.5 Prop.

÷ z 6 K yields

such that x(F)

15), hence

+ z 6 K.

some a E Ro fulfils

some f 6 Ro such that

103

x(f)

6 Na(Z),

where

the existence x(f)

f c a may be assumed.

of some y 6 6[f]

The above negation

such that y ~'U x(f).

implies

Since

6 u[f], we have

y 6 Nf(x(f))

c Na(X(f))

Together with x(f) contradiction

6 Vz this implies y 6 VV

to y ~ U x(f).

Lemma:

(377)

c N2(Z)a c Vz. -I

x(f) c U x(f)

[]

Consider the right uniformity ~ d

usual uniformity

N on P. Further

with the corresponding gies , likewise

in

on T and the

let FP and PT be equipped

canonical

uniformities,

resp.

topolo-

the subset Ro c Pp. Then the two mappings

o (i)

a ~ T(a),

(ii) a ~ T(x,a) are continuous.

Proof:

(i) Let M be any ~ d _ e n t o u r a g e

(~,~0) 6 M ~ ~ - I

6 U, where U is some open Id-neighbourh0od

t. C o n s i d e r the corresponding ~(T(a))

= {S6PTIscM(T(a))

The subset K = N2(a) yields

and T(a)cM(S) }.

is compact

w.r.

= {b6RolbCNd(a)

Let be b 6 ~d(a). prove T(b)

(3512),

the application Ne(X),

to the uniformity

(376)

on which U operates

~ on -Ro given by

let ~ 6 T(b)

N Nd(~[a])

6 ~(T(a)).

be given.

N T1[d],

In order to

From b 6 4)[b] #

# ~. Hence there exist x, y, z 6 P;

T I, T 2 6 T such that x 6 ~[a]

of

and aCNd(b) }.

We will show that T(b)

c M(T(a)),

follows Nd(a)

to

Let d be a kernel of e. We may assume d c a. Consider

the a-neighbourhood Nd(a)

w.r.

T(a)-neighbourhood

for each x 6 K some neighbourhood

transitively.

of the form

y 6 ~1[d]

D T2[d] , z 6 ~2[d]

D a.

104

N d (M[a])

N d (a)

fig.

We

conclude

such It

that

x

6 N2(a) a

~ x

=

c K

and

z 6 N

e

(378)

(x),

hence

there

exists

a ~ 6 U

z.

follows: ~[a] = def

n a # ~ M

6 T(a)

6 M(T(a)) Thereby

ii)

c M(T(a))

roles

a c Nd(b ) ~ This

of

T(a)

completes The

can

set

is

a and

6 M

proof of

proved.

b one

c M(T(b))

the

continuity

(.One e v e n

(~,~)

.

T(b)

Changing

and

a

d = e.)

shows

similarly,

. that

a ~ T(a)

T(x,a)

is

is

proven

continuous. with

the

same

technique.

105

Lemma:

(379)

L e t K c T be compact.

PK × PK ÷ PK 2, logies

Proof:

(A,B) ~

II § 4 . 1 T h .

mapping

P(KxK)

striction

of o r d e r T(x,a)

a 6 R

[BGT]

T(a) n-2

minimal

N T(y,a)

Lemma:

T(x,a)

Proof:

In o r d e r

to d e r i v e

From

[BGT]

T 6 T(x,a)

6 T(y,a),

T(a) n-2

T(y,a)

PK 2.

continuous

the c o r r e s p o n d i n g hence

its re-

between

x and y, x % y,

= @.

2, Cor.

we w i l l

of open

some T - n e i g h b o u r h o o d

in c o n t r a d i c t i o n

sets

V. B e c a u s e

is open, of

to

[]

for a ~ l(x,y,a).

L e t be x, y 6 P, a 6 Ro" T h e n t h e r e

a d 6 Ro such that for all b 6 ~d(a) i l ( x , y , a ) - l ( x , a , b ) I _< I.

that

I we infer

T(a) n-2. A p r o d u c t

= ~.

assume,

intersection.

a type of c o n t i n u i t y

Proposition:

PK resp.

(374):

in the a b o v e

Prop.

N T(y,a)

N o w we can p r o v e

(3711)

(R,c,T)-chains

a contradiction

T(a) n-2 c o n t a i n s V meets

on

[]

T(a) n-3 N T(y,a)

T(a) n-3 c T(x,a)

thus T(x,a)

T(x,a)

II § 1.2,

to the topo-

= ~.

(3710)

some T 6 T is c o n t a i n e d

6d)

equicontinuous,

PK × PK.

a n d l e n g t h n. By

o

uniformities

II § 2 Ex.

+ PK 2 is u n i f o r m l y

consider

w.r.

K × K ÷ K 2 is u n i f o r m l y

2). By

to the s u b s p a c e

N o w we w i l l

AB is c o n t i n u o u s

i n d u c e d by the c a n o n i c a l

The multiplication

([BGT]

T h e n the m a p p i n g

N Ro'

exists

5

Proof:

For each A c

fi

N(A) ([BGT] I1 § 1.2, Prop. 2, Cor. I), hence (3710)

n

=

the closure is given by

~ € 2 implies, that for some .entourage N E

zd,

~ ( ~ ( x , a ) ~ ( a )n~ ~(f?(~,a)) -~) = 6. For all b c Na(a) the subsets ? (x,b) and fixed compact K c

f!

T (b) are

contained in some

5

(this follows easily from [BGT] I11

4.5, Th. 1).

Thus we may apply (377) and (379) in order to show that ~ ( x , b ) ~ ( b ) " - ~ c ~ ( ~ ( x , a ) ~ and ( a ) T(y,b).c ~-~) N ( T ( ~ , ~ )hold )

, if b is

chosen "close to a", that is, if b c Nd (a) and a c Nd(b) for some %

appropriate d E Ro, d c a. Now the relation ?(x,b) T(b)n- 3

n ff(y,b)

=

6

shows (374), that X (x,y,b) 2 n

-

/ 1, hence h (x,y,b) 2 h (x,y,a) - 1.

Since the conditions on a and b are symmetric, we infer likewise

-

1, which completes the proof.

Proposition:

Let R' be a dense subset of

X(x,y,a) 2 X(x,a,b)

(3712)

X, y, u, V E P,

U

*

go

and

V.

Then the net converges iff

converges.

Proof:

For the non-trivial implication choose for each a E

go

some

ba E R' "near" the region a according to (3711). This yields a directed subset[R"]c[R1]. The convergence of the go-net now follows from

107

and

lim b a 6R'

l(u,v,b a) =

lim b a £R'

l(x,y,b a) =

(see

(359)), if x % y. In

the case x = y clearly both nets c o n v e r g e to O.

NOW from

(3312) and

(375) we can derive the m a i n result of this

Theorem:

The c o n v e r g e n c e of the chain q u o t i e n t in the

section:

(3713)

(R,c,~)-theory implies t h e c o n v e r g e n c e of the chain q u o t i e n t in the

(R,c,T)-theory.

The limits coincide.

THE H E L M H O L T Z - L I E

4.

In s e c t i o n

3.6 we have d e v e l o p e d

Tits/Freudenthal

solution

to study

the r e l a t i o n

quotient

in s e c t i o n

the d i f f e r e n t i a l with

argument more

between

mobility

geometry

of this

4.1.

using

the c l a s s i f i c a t i o n the d i m e n s i o n

results

of H i l b e r t s groups.

geometrical

some

J. v. Neumann, et al.

OF THE T H E O R E M

fifth problem,

After

In sequel we

decades

view

general

suppose

to be formulated)

follow

is r e f o r m u l a t e d

is made

of p h y s i c a l

possible

C. Chevalley,

solution

and use

analysis

was

D. M o n t g o m e r y ,

g i v e n by H. Y a m a b e

to fulfil

essentially

the axioms only

structure

6 such that

(411)

(and R6 proved

3.

a differentiable

(for d e f i n i t i o n

=

Zippin

[YAM].

the p r o p o s i t i o n s

equipped

(T,~}

of Lie

L.

RI to R5

a Lie group,

induced

to

B. K e r e k j a r t o ,

be c a l l e d

to

using

space

characterization

(G,T) will

topology

an

by the s o l u t i o n

group

the

[FRE]

T to be a Lie

A topological

Lie gr o u p

of

and f u r t h e r m o r e

of w o r k by L. Brouwer,

(R,<,T)

(:R,c,T) in section

with

of the c h a i n

OF Y A M A B E

the t o p o l o g i c a l

L. P o n t r j a g i n ,

the m o s t

prove

[FRE]

of orbits

from a pure t o p o l o g i c a l

the d i f f e r e n t i a l

of

In o r d e r

an e x p o s i t i o n

We shall

shall

the

[MSZ].

IMPLICATIONS

The p a s s a g e

problem.

to give

problem.

first we

to apply

and c o n v e r g e n c e

4.2 it is c o n v e n i e n t

concerning

recent

the p r e r e q u i s i t e s

of the H e l m h o l t z - L i e

two m i n o r m o d i f i c a t i o n s :

group w i t h o u t

for

PROBLEM

see e.g.

[B&C]

by ~ c o i n c i d e s

with

12.1)

and

~. This

if G can be (G,~)

if,

is a

in addition,

applies

especially

(T,c).

Theorem:

There

exists

a basis

of Id 6 T and an infinite

U of the n e i g h b o u r h o o d

net of s u b g r o u p s

filter

109

(Nu)u6U,

downwardly

directed,

For

e a c h U 6 U,

(i)

N U c U,

(ii)

N U is a c o m p a c t ,

(iii)

T / N U is a L i e

with

connected,

the

following

invariant

properties:

subgroup,

group.

Further, (iv)

if T i t s e l f

is n o t

a Lie group,

V U, V 6 U, U ~ V ~ N U ~ N v,

we may

assume

hence

V U E U, N U ~ {Id}, (v)

if ~ i t s e l f V U 6 U,

The point

Proof:

(v)

By

is o n l y

[YAM]

neighbourhood L i e group. "trivial"

Let

compact.

Let

bourhood

v[U]

since

case

v

-I

where

the

case

to b e

included.

v[U]

assiquement

c a n be a c h i e v e d

Because

compact

with

N U. N o w

an i n v a r i a n t

(open)

[YAM]

i.e.

T/N U

to a c o a r s e r

canonical

quotient

map.

The

the

subgroup

is d i s c r e t e ,

for any b a s i s U c U'.

hence Id-neigh-

subgroup

T / N U is a L i e

by its c o n n e c t e d

isomorphism:

we have

closed,

invariant

basis

is a

subgroup

in U, h e n c e

implies

an Id-

subgroup

Either

be an i n v a r i a n t

U ~ N U is p e r f o r m e d

by passing

invariant

any non-trivial

(M) w o u l d

= {Id},

equipped

Id-neighbourhood.

it is to be r e p l a c e d

of t h e

group

no n o n - t r i v i a l

contain

the maximal

connected

locally

a s s u m e N U is m a x i m a l

including

If the

of a r g u m e n t .

v : T + T / N U b e the

in t h i s

assume

for s a k e o f u n i f o r m i t y

(V) o r U c o n t a i n s

cannot

(trivially)

adduced

U be a compact

We may

group we may

N U = {Id}.

containing

case

N U % {Id}.

any

is a L i e

M,

in U group,

is u n d e r s t o o d

U',

(iv)

always

If N U is n o t

Id-component

o N U-

110

NU Nu/N~ (see [BGT] I I I §

2.7, Prop.

2.2, Cor.),

o Nu/N U, since N U is compact, -

the two t o p o l o g i c a l groups T/N U and

o

T/N u are locally isomorphic. Lie groups.

and the finiteness of

Thus both T/N U and also T/N~ are D

In sequel we c o n s i d e r some fixed N = NU, x 6 P and J = Jx' the s t a b i l i t y subgroup.

(412)

Definition: w.r.

We shall d e n o t e by P/N the space of orbits

to the o p e r a t i o n of the group N on P, endowed w i t h the

q u o t i e n t topology.

(413)

Proposition:

T h e r e exist the following c a n o n i c a l

homeomorphisms: P N

_~ T / J (i) N

~ __T ~ T/N (ii) JN (iii) JN/N

Further, (iv) P/N is a H a u s d o r f f space.

Proof: (i)

P ~ T/J w.r.

to the map Tx ~ ~ Jx' see

(3613).

(iii) Since the s u b g r o u p JN o p e r a t e s c o n t i n u o u s l y on T on the right,

it follows by

[BGT] I I I §

2.7. Prop.

c a n o n i c a l map T JN ~ {TjNIj6J} = {~NjNIjN6JN/N} , is a h o m e o m o r p h i s m

JN

JN/N

T 6 T,

22, that the

111

(ii)

T operates on P/N by means of Ny ~ invariant. III§

TNy = NTy,

This o p e r a t i o n is transitive,

2.4 Prop.

since N is

continuous by [BGT]

11 and proper, w h i c h can easily be inferred

from [BGT] I I I §

4.2 Prop.

5i). The "point" Nx 6 P/N has w.r.

to this o p e r a t i o n the stability subgroup JN = NJ. Hence

[BGT] I I I §

4.2 Prop.

4c) shows the canonical map

T JN ~ NTX to be a h o m e o m o r p h i s m T/JN ~ P/N T/JN ~ (iv)

and by

(i),

(T/J)/N.

Being a product of two compact subsets, closed s u b g r o u p space

(iv) and

group of T/N. P/N ~ JN/N

2.5 Prop.

(iii)

13).

o

shows that JN/N is a closed sub-

thus is a q u o t i e n t of a Lie group and a

closed s u b g r o u p and can be endowed with a C ~ - d i f f e r e n t i a l c o m p a t i b l e with its topology d i s c r e t e case).

(414)

hence

. T h e r e f o r e T/JN is a h o m o g e n e o u s H a u s d o r f f

(see [BGT] I I I §

The above proof of

JN is a compact,

(see [B&C] Prop.

12.9.4,

structure

including the

In brief:

Corollary:

P/N is a C~-manifold.

For later purpose we need the following general propositions.

(415)

Definition: A mapping

Let any group G operate on spaces X and Y.

f : X ÷ Y is called e q u i v a r i a n t

all x 6 X and g 6 G, f(g.x)

(416)

Lemma:

(re G) iff,

for

= g • f(x) holds.

Let any group G operate on C ~ - m a n i f o l d s MI, M 2 by

means of C - d i f f e o m o r p h i s m s Li,g Let T Li,g

: M i ÷ Mi,

i=I,2,

g 6 G.

: T M 1 ÷ T M.1 be the induced o p e r a t i o n of G on

112

the

tangent

the

derivation

operation

The

functorial T(L2,g)

of

and

-I

: TM I ÷ TM 2,

tangent

equivariance

property o Tf

of T

0 TLI,g

Corollary: f(x I) (w.r. Tx

f : M I ÷ M 2 an e q u i v a r i a n t

f, Tf

of G on the

Proof:

(417)

bundle

x 2. T h e n to the

Mi,

is e q u i v a r i a n t

(w.r.

Then

to the

bundle).

of

f means

L -I 2,g

(see

[B&C]

4.3 or

= Tf.

Let

C -mapping.

o f 0 L1,g [ABR]

Th.

= f. By the 5.7)

w e have:

D

x i 6 M i be

fixed

Txlf

M I + Tx2 M 2 is e q u i v a r i a n t

induced

: Txl

operation

points

of G and a s s u m e

of G on the

tangent

spaces

i=I,2).

1

Now

let us

basis

return

to

situation

of I d - n e i g h b o u r h o o d s

of

(411)

to

of T a c c o r d i n g

(414), to

where

(411).

We

U is the shall

write

e = Id 6 T.

(418)

Proposition: Further

L e t U, V 6 U such

consider

the

vU

: P ÷ P/N U

and

VV

: P + P/Nv"

Then

(i.e.

a surjection ÷ P/Nv

canonical

there with

: P/Nu

(i)

~U,V

o

(ii)

VU,V

is e q u i v a r i a n t

(iii)

let y 6 P/N V,

vU

JNv/JN U -

U c V,

quotient

exists

maximal

~U,V

that

N U c N V-

maps

a canonical rank,

hence

see

submersion

[B&C]

6.1)

satisfying

= v V,

then

w.r. VU~V

to the o p e r a t i o n (~)

is h o m e o m o r p h i c

of T, to

113

Proof:

Consider

the diagram

T.

P___~

t

defined

~

~

JNu

by

T

TJNu ~

~U,V ~I

~U,V is thereby

uniquely

T

_~P

"-JNv

~JN V

;V

VU,V

where

(413)

is used.

If the diagramm

is commutative,

v U and ~V are submersions.

Considering

for instance

determined.

~U' this follows

%

from the commutative

~/jN u

=

diagram

p N U

I

and the conclusion "if ~ and ~U are submersions, Prop.

12.9.4,

clearly

become

Composition

then ~U is a submersion"

6.1.2 and 6.1.3).

The homeomorphisms

C -diffeomorphisms,

of diagrams

hence

now yields

(see [B&C]

denoted

by

~U is a submersion.

~U,V o ~U = ~V' hence

~U,V is

also a submersion. In order

to prove the equivariance

diffeomorphism

P/N ~ T/JN,

we use the formula

namely NTX ~

(413) (ii)) and take N = NU, resp.

~ TJN

for the C ~-

(see proof of

N = N V to get the diagram:

114

x ;

NU

m~ JN U

~U,V

x ;

NV

This

I~U,V

shows

~

JN v

the equivariance

of ~U,V:

L NuY:

>N U ~ y

L NvY ,•

~-

NV

~

y,

where y £ P and ~ 6 T is arbitrarily In order to show property -I (y) of points ~U,V transitively

chosen.

(iii) we notice

y 6 P/N V are homeomorphic

on P/N V and ~U,V is equivariant.

= NvX~-C---~eJN V in order to calculate

:

that all inverse

~

~

because

T operates

Therefore

the inverse

images

we may take

image:

Ivc e JNv

= nU[~vI[eJNv ]] = ~u[JNv] = JNv/JN u.

Now consider notation

[]

an arbitrarily

by writing

chosen U £ U. We shall simplify

our

N U = N, Nx = x, T/N = T, P/N = P, JxN/N = J.

115

Let T be the Lie algebra of T and d the subalgebra subgroup

J.

Consider

any 3 6 J. The C~-mapping

onto e, its derivation

belongin~

to the

t ~ J t 5 -I , t 6 T, maps e 6

at the point e thus being a linear

endomor-

phism of TaT ~ T, which will be denoted by J

:

T-+

(419)

T

Lemma:

Let t 6 T and I 6 ~ , then

exp(It)

Proof:

~-I : exp(l~(t)).

The left hand side defines

thus is of the form exp(It'), the possible

definitions

transformation

j ~

is a restriction

9.6).

It is faithful,

(j~x=~jx=~x=j=e). definite

represented invariant

complement, Consider

inner product,

[]

of the group J, of T

no central~elements T may be equipped

(see [F&V] except with a

denoted by <,>, such that J is

endomorphisms

to all 9, hence this holds

(see [F&V]

35.1).

d is

also for its <,>-orthogonal

which will be denoted by K.

the following

K C-~T exp~ ~

(4110)

representation

Since J is compact,

by one of

5.6).

a linear representation

since J contains

of ~,

~, namely by the induced

(see [ABR] Def.

of the adjoint

by <,>-orthogonal

w.r.

= t' follows

of the derivation

forms

which

positively

t' 6 T. ~(t)

of smooth curves

The assignement

a l-parameter-subgroup

~ ~T/J

C=-map

f:

~ P, that is:

Definition:

Its derivation

composed

at O, f'

f(X)

= def

(expX)

x for all X 6 K.

= Tof, decomposes def

in the following

way:

116

K~-~T

To,~

~ r& '~

~ r3.; (,~/~) ~ r::: ~'

Td

T = J • K

where

z denotes

Therefore,

the c a n o n i c a l

j

= K

quotient

up to i s o m o r p h i s m s ,

local C~-diffeomorphism

J~K

~

map

f' is e q u a l

a n d for a s u i t a b l e

O • K + O • K/J. to Id K. H e n c e

f is a

x-neighbourhood

U the

restriction f-11U

: U ÷ K ~ ~P

is a C ' - c h a r t canonical

of P a r o u n d

chart. (see

[K&N]

(4111)

Proposition operates

coincides f-1

(compare

[FRE ]

on P by l o c a l l y

= ~(k)

exp and ~ are l o c a l l y

2.7.): orthogonal

transformations

via

chart.

~IK is o r t h o g o n a l

w i t h 3~ w h e n

3 f(k)

since

to as the

I, I 4.2.).

the canonical

Since

x. It w i l l be r e f e r r e d

It is e v e n a n a l y t i c a l ,

analytic

Proof:

the p o i n t

computed

for all

it s u f f i c e s

to show that

in the c a n o n i c a l

chart.

3 locallY

T h i s means:

3 6 J and k 6 K.

It follows: f(k)

=~ 5 ( ~ o e x p ) ( k ) = 3 (exp k) : 3 (exp k) 5 -I

J.

f-1 ~ f(k) = l o g ( j ( e x p =

(4112)

3 (k),

Lemma:

k)) ~-1 by

There

(41 9 ) .

exists

[]

a local C ~ - i n j e c t i o n

T : U ÷ T such

117

that

~(y) (x) = y for all y in some n e i g h b o u r h o o d

of x.

oo

Being

Proof:

a submersion,

v o T = Id U

(see

[B&C]

Identifying

T/J = P, we may

v has a local

Prop.

6.1.4.). express

v T(y)

= ~ J where

a 6 T such that ox = y.

Hence

T(y)

6 J

(4113)

Theorem:

and T(y)(x)

such that T o p e r a t e s

Proof

(see

[F&V]

inner p r o d u c t

= aj(x)

P may be e q u i p p e d

6.4.3.1,

= y in the form

= a(x)

with

isometrically

[K&N]

T,that means:

Let y 6 U.

T(y)J

= aj,j

C -section

2, X 3.):

= y.

a Riemannian

metric

g,

on P.

The p o s i t i v e

<,> on K ~ T ^ P has to be t r a n s f e r r e d

definite

to the o t h e r

X

tangent

spaces

T~ P, z 6 P. This m a y be done by the linear

I^ f : T^ P ÷ T ^ P, w h e r e x x z The c o n s t r u c t i o n satisfying

by

on the choice

of f. Let g 6 T also

f-] gx = x, j

Y~ g = T~ f 0 Ti j, w h e r e T ^ P invariant

f 6 T, fx = z.

does not d e p e n d

gx = z, then

isomorphism

= flg £ ~, def leaves the inner p r o d u c t

Ti j

in

(4110).

X

In this way an inner p r o d u c t transporting

maps

differentiable Consider with

Hence

-I

= T^z g

o T~ g

-I

-I

to d e p e n d

(4112),

l o c a l l y.

hence

The

on z^ in a

C ~ -

z ~ g~ is C ~ too.

fixed

and T~(hog) , the p r o d u c t

= T~ h is an isometry. isometrically

g is u n i q u e l y

the claim only

way by use of

T operates

Whereas

can be chosen

globally.

any z 6 P and h 6 T. Let g 6 T such that gx = z. T o g e t h e r

(T~g)

T~(h0g)

g~ on T~ P is d e f i n e d

on

determined

that T C o n s i s t s up to p o s i t i v e

(P,g).

[]

by the inner p r o d u c t

of isometrics, definite

<,>

on T P and

the inner p r o d u c t

linear maps

commuting

<,> is

with

the

118

r e p r e s e n t a t i o n of J on K. At any case, remains

4.2.

a p o s i t i v e scaling factor

facultative.

M O B I L I T Y AND D I S T A N C E M E A S U R E D BY CHAINS

We now turn to the last and d e c i s i v e a x i o m of Freudenthal. developed

It is

from earlier p o s t u l a t e s c l a i m i n g for i n s t a n c e that t w o

sets of points, w h o s e internal d i s t a n c e s are p a i r w i s e equal, may be m a p p e d onto each other by a

(global)

using the notion of a metric, Freudenthal

(421)

isometry

"mobility"

(and s i m i l a r l y by Tits)

(Birkhoff). W i t h o u t

is now f o r m u l a t e d by

in the following way:

There exist two points x, y E P such that the orbit Jx y dissects

the space P,

w h e r e "dissection" means:

(422)

Definition:

Let B be a t o p o l o g i c a l

space, A c B. A is

said to d i s s e c t B, iff B~A is not connected.

The p o s t u l a t e

(421)

m e a s u r i n g distances. (3514))

is i n t i m a t e l y c o n n e c t e d w i t h the p o s s i b i l i t y of The c o n v e r g e n c e of the chain q u o t i e n t

(see

cannot be p r o v e d on the basis of the axioms RI to R5, as is

Shown by the f o l l o w i n g

(423)

C o u n t e r example:

R is the lattice of b o u n d e d open subsets

of 2 2 , T is induced by translations.

Since the chain links do not freely rotate, l(x,y,a) h l(u,v,a)/a £ R

the limit of

o

depends on the shape of the links as is shown in the following

119

figure:

] I j i

]

I I

I....

I

fig.

Clearly,

t

(.424)

the counter example does not satisfy the r e q u i r e m e n t

(421).

This is of course not accidental as c o n v e r g e n c e of the chain q u o t i e n t and m o b i l i t y are e s s e n t i a l l y equivalent,

(425)

Theorem:

as we will show.

In a system in which axioms RI to R5 hold,

the

following 3 assertions are equivaleDt: (i)

There exist two points x, y 6 P such that the orbit Jx y dissects the space P

(ii)

(Freudenthal).

There exists a point x 6 P and a n e i g h b o u r h o o d V of x such that each orbit Jx y' x # y 6 V, dissects the space P

(Tits). Further,

(iii) The chain q u o t i e n t w.r. on regions

T is a Lie group. to

(R,<,T)

converges u n i f o r m l y

(in the sense defined below),

or P is homeo-

m o r p h i c to the real line.

Explicitly,

the m e a n i n g of

(iii)

is c o n t a i n e d in the following

120

Definition:

(426)

uniformly u % v,

Here

and each

region

for

is the

is the

space

the

uniformity

of c o m p a c t

Cauchy-convergence

the

sense

section

of

(427)

[BGT]

filter

detailed

defined

x,

u, v £ P,

the m a p p i n g

by

set of c o n g r u e n c e

of real

functions

convergence

on

I § 7.3, [R o]

that

is a C a u c h y of

(426)

H x 6 P V u, v 6 P such V b 6 R

is,

classes

on P~f

(equivalently:

such

o

l ( x , a , a 2)

l-~v~a

l ( u , v , a 2)

that

[BGT]

X § I. Th.

Fc(P~f,~)

I, a n d

is c o m p l e t e

Proposition: converges

the

of r e g i o n s

endowed

with

uniform

conver-

If

(427)

to a f u n c t i o n

Theorem:

Under

A(x,-,u,v)

6 Cc(P~f,~),

on P~f.

the

reads

u % v

that

be u n d e r s t o o d

under

base

on

as

F

Q(x,-,-) (P~f,~)

o

such

of the .

that

V e > 0

a I < a ° and

a 2 < a ° V y 6 b:

c

conclude

of the

is f u l f i l l e d ,

~,

the

space

following

the

chain

quotient

A(x,y,u,v).

assumptions the

of

space

in

follows:

V f 6 R

completeness

and we m a y

will

image

b ~l f = ~

< I)

the

thus

that

l(x,y,a I )

Q(x,-,-)

filter

H a ° 6 R ° V a I, a 2 6 R ° s u c h

(429)

each

is C a u c h y - c o n v e r g e n t .

of the m a p p i n g

formulation

x 6 f

(428)

x 6 P and

to c o n v e r g e

on r e g i o n s ) .

The

By

is said

f 6 S ° containing

directed

Fc(P~f,~)

quotient

some

(y~ l(x,y,a)/l(u,v,a))

and

The

iff

: [R o] ÷ F c ( P ~ f , ~ )

[R o] = R o / T

gence

(R,<,T)-chain

on r e g i o n s

Q(X,-,-) [a] ~

The

(428), of c o n t i n u o u s

functions

121

Proof:

P~f is locally compact and by

C(P~f,~) mate

is closed in Fc(P\f,~).

l(x,y,a)/l(u,v,a)

converges Consider

to O w.r.

by a continuous

the 2 closed sets NaX

continuous

It suffices

to the section

(3614), hence normal

[BGT] X, § 1.6, Th.

([BGT]

function

filter on

and

therefore

2, Cor.

3,

to approxi-

such that the error

[Ro].

C N2a x. P being paracompact

IX § 4.4, Prop.

4), there exists a

function ~ : P + [1,3] which is equal to I at every point

of NaX and equal to 3 at every point of

C N2a x. Hence

sup{ l~(x,y,a)-~(y) I Jy6N~x} ~ I. In this way, we inductively : P ÷ ~

construct

a continuous

function

with the property

sup{ 11(x,y,a)-~(y) I Iy6P} ~ I.

Hence sup

{ ~ (x,y,a) l(u,v,a)

~(y) - l(u,v,a)

y6P\f

}

I ~ l(u,v,a)

and the approximation

of the chain quotient by a continuous

is arbitrarily

close.

The approximation

the uniformity

of uniform convergence.

It is not known whether convergence

(425) (iii)

of the chain quotient.

(see

(4214)).

is even performed w.r.

can be weakened

is not

P, that is P ~ S I or P ~

This is due to the existence of infinite

in the case P ~ ~

to

to pointwise

Note that convergence

given in the case of one-dimensional

function

fractions:

for instance one can easily compute

2 ~(O,~,a n) lim l(O,1,an ) - ~, where

a

= n def

]-n-22-n,n-22-n[

and the representation

U ]2-n(1-n-2),2-n(1+n-2) [ 2 ] = 0.101070...

This is the reason for including whereas

in binary digits

the case P ~ ~

the case P ~ S I is ruled out by

(425) (i).

in

is used.

(425) (iii),

~22

The

remainder

theorem

of s e c t i o n

PROOF

We

use

(4211)

OF

"(i)

the b a s i s U c Uo, orbit

use

this

Proof: Jy

loss

A and B b e i n g

subsets

In o r d e r

N

= Nz 6 ~[A] not Jy,

In o r d e r means

to show

that

that

assume

4.1.

U° 6 U

for e a c h the

(U is

U 6 U,

corresponding

V U 6 U, U c U ° and we w i l l

disjoint

subset. f o r m P~J y = A U B, The quotient

where

~[A]

and

~[B]

the n o n - c o n n e c t e d n e s s

[BGT]

Since

map

I § 11.1

are n o n - e m p t y

Def.

are of

2) that

and

= ~.

latter,

Nz c

P.

Hence

U ~[B],

(see

(P~[J~)N~[B]

N

Jxy= J y dissect

subsets.

and open.

to i n s u r e

to s h o w

the

orbit

it is of the

= ~[A]

In o r d e r

N ~[B]

to

that

non-connected

that

non-empty,

(P~[Jy])

hence

contradiction

we m a y

(425) (i) some

c ~[p~Jy]

to s h o w

such

a y 6 P such

is s u r j e c t i v e

and

an I d - n e i g h b o u r h o o d

(411))

is an open,

if s u f f i c e s

0 ~[B]

to

in s e c t i o n

P.

I § 11.1)

of P.

(P~v[Jy]N~[A]

but

to

open,

P~J y = P~[Jy:]

~[A]

of

henceforth.

P~Jy

: P ÷ P = P~N

exists

exists

of g e n e r a l i t y

([BGT]

x 6 P, J d e f i n e d

There

there

According

P~[Jy]

T,

according

convention

means

open

to the p r o o f

(ii)"

J y dissects

is closed,

This

=

the n o t a t i o n s

Proposition:

Without

is d e d i c a t e d

(425).

4.2.1. shall

4.2

assume

(P~[Jy]). P~Jy

It f o l l o w s

= A ~ B and Nz

t h a t Nz m e e t s

A and B

is n o n - c o n n e c t e d

in

(411) (ii). (P~[Jy])

any a 6 A lies

N ~[A] on some

% ~ assume orbit

~[A]

N j y,

c ~[Jy].

j 6 J.

But

This there

123

exists

a neighbourhood

chosen

sufficiently

We w i l l the

use

sense

small.

the n o t i o n

of the

defined

restriction

in

(see

[B&C]

For

topological

with

the u s u a l

Cot.

I).

From

[H&W]

(4212)

D

(in symbols: where

"dimension"

of the b o u n d a r y

metric

spaces

d i m A)

of A.

is not

paracompact

too

in

is

The limited

Riemannian

3.4).

notion

the n o t i o n

of d i m e n s i o n

IV 4. Cor.

Proposition:

I we

of d i m e n s i o n

in

of the m a n i f o l d

infer

[H&W]

([H&W]

coincides Th.

IV 3.

immediately:

Let be d i m P

= def

^ ^

then

[H&W],

P is a c o n n e c t e d ,

manifolds

Th.

of a set A

and Wallmann

to s e p a r a b l e

since

f r o m Jy and N a c V if N is

is a c o n t r a d i c t i o n .

by the d i m e n s i o n

[H&W]

for o u r p u r p o s e s ,

This

of d i m e n s i o n

Hurewicz

inductively

manifold

V of a, d i s j o i n t

p

and Jy d i s s e c t i n g

P,

d i m Jy ~ p - I holds.

We can n o w a p p l y

the r e s u l t s

groups

Let

on o r b i t s

of c o m p a c t

transformation ^

in

[MSZ].

us a s s u m e

that

for

some

neighbourhood

V of x,

^ ^

d i m Jy S p - 2 for all y 6 V. T h i s

means

that

the

set

^ ^

{y6Pldim each

Jy~p-2}

orbit

(4211)

and

(4213)

of J has (4212).

Lemma: that

We

shall

without

(4214)

contains

an i n t e r i o r

dimension This

Each

proves

at m o s t the

neighbourhood

point p - 2,

and by

[MSZ],

Th.

in c o n t r a d i c t i o n

I Cor., to

following

of x c o n t a i n s

some

point

y such

d i m Jy a p - I holds.

discuss assuming

the c a s e

p = I separately.

The

following

holds

425) (i).

Proposition:

If d i m P = I, P is i s o m e t r i c a l l y

isomorphic

124

either to the real line ~

Proof:

or the circle S 1.

The group of isometries of a connected,

p-dimensional

I R i e m a n n i a n m a n i f o l d is of d i m e n s i o n at m o s t ~ p(p+1) Th.

3.3,

(see [K&N] VI.

3.4).

In the case p = I it follows that dim T = I b e c a u s e T o p e r a t e s t r a n s i t i v e l y on P. Thus the m a x i m a l d i m e n s i o n of T is a t t a i n e d and in this case to ~I

[K&N], N o t e

10 Th.

I, states that P is isometric either

or S I, since the h y p e r b o l i c and elliptic

c o i n c i d e w i t h these.

(4215)

Corollary: that P ~ ~

spaces of d i m e n s i o n I

[]

If T is a Lie group and dim P = I, it follows and hence

(425) (ii) is fulfilled.

N o w we c o n s i d e r the case p > I.

We shall i d e n t i f y K and ~ P

(see

(4110))

such that the inner p r o d u c t

<,> on K c o i n c i d e s w i t h the usual inner product on ~ P . W e recall that w.r. linear,

to the c a n o n i c a l chart f-1

orthogonal,

: U + ~ P , f,1 ~ f consists of

locally d e f i n e d m a p p i n g s

and dim Jy ~ p - I a c c o r d i n g to

~P

÷ ~P.

(4213). We define r =

Let be y E U lif-lyll and

S(O,r)

= {~6~PILI~II=r} . We may choose y 6 P such that f-1[U] def c o n t a i n s S(O,r) and hence the image of the orbit y

= f-1 ~ c S(O,r). Y is a c o m p a c t s u b m a n i f o l d of S(O,r) of def d i m e n s i o n p - I, hence open in S(O,r). By v i r t u e of p > I, S(O,r) c o n n e c t e d and we infer Y = S(O,r). £ ~P

~

linearly.

l~,

proves:

The c o n t r a c t i o n s

O < I < I map J - o r b i t s onto J-orbit,

Hence in some n e i g h b o u r h o o d

a sphere w.r.

is

since J operates

f-1[V] c f-1[U]

each orbit is

to the c a n o n i c a l chart and thus d i s s e c t s P. This

125

(_4216)

Proposition"

In a certain n e i g h b o u r h o o d V of x each

o r b i t Jy, ~ # 9 6 V, dissects _D, if dim P > I.

Now we are ready if T can be shown to be a Lie group, = T and

(425)(ii)

follow

either by

(4215)

since then

for p = I or by

(4216)

for p > I.

(4217)

Lemma:

For all U, V 6 U the following holds:

dim P/N V = O or dim P/N v = dim P/N U.

Proof:

Let us assume U c V and p = dim P / N u > dim P/N V = q > O.

The s u b m e r s i o n VU, V : P / N u ÷ P/N V is e q u i v a r i a n t w.r. o p e r a t i o n of J J-orbits.

By

(see

(417)

(418) (ii))

to the

and hence it maps J - o r b i t s onto

the same holds for its d e r i v a t i o n

TNuX ~U,V : TNux(P~Nu)

÷ TNvx(P/Nv)'

which can be identified w i t h a

n o n - t r i v i a l linear p r o j e c t i o n ~ : ~ P

÷ ~q.

J - o r b i t s are the spheres in ~ P

~q.

resp.

As m e n t i o n e d before,

But a sphere S p-1

the

is not

p r o j e c t a b l e onto a sphere S q-l, 0 < q < p, because for instance -I (O) D S p-1 tion.

# ~ implies O 6 ~[S p-I] = sq-1 , w h i c h is a c o n t r a d i c []

(4218)

Theorem:

T is a Lie group.

Proof:

Let us assume the c o n t r a r y and consider a c o u n t a b l e infinite

d i r e c t e d s u b - f a m i l y of U according to ding i n v a r i a n t subgroups Ni,

(411)

i 6 ~ . By

and call the correspon-

(4217)

the sub-family may

be chosen in such a way that the spaces P/N i have the same dimension. Hence the submersions images

vi+1,i

(see [B&C] Prop.

: P/Ni+I

6.2.1.).

÷ P/Ni have discrete inverse

126

By

(418) (ii),

open

(see

compact spaces

[BGT]

space and

subgroups only

for

all k ~ k By

J N i / J N i + I is d i s c r e t e .

(413)

III§

2.5 Prop.

is the

topological

consequently of the

contains

f o r m JNj.

finitely

14)

many

Further, subgroup

only

i E ~ . Thus

such we

two that

that

we may

we have

P/N

different z ~ W.

~ T/JN,

hence

points

y,

subfamily

we obtain

PROOF

(4215)

that

OF

P/N k = P / N k + I ,

we may

~ : P x P+

We

consider

confine

~

of

JN k = J N k + I for

let a £ R ° w i t h

means

N k % {Id},

a neighbourhood

(see

U in such

(313))

a way,

there W of y

containing

that

because

that

N k c T(V)

z.

If

for

N k z is c o n t a i n e d 0

(iii)" ourselves

connected The

to the

case

Riemannian

dim P >

manifold

corresponding

metric

I. We

recall

on w h i c h

shall

be d e n o t e d

.

two p o i n t s

be a s s u m e d

consider

~

isometrically.

by

later

of d i f f e r e n t

which

Because

of W

a contradiction,

"(ii)

P is a c o m p l e t e ,

operates

of

J N i + I ~ JN i is v a l i d

in W and y E N k z is i m p o s s i b l e .

By

number

assume

z 6 N k y and

L e t V be a k e r n e l

the

k Z ko,

4.2.2.

number

.

O

construct

some

a finite

a finite

We conclude

and

of JN i. But JN i as a

s u m of o n l y

for all k Z K ° and y £ P, N k y = N k + I y. exist

J N i + I is a c o m p a c t

to be

x, y 6 P and

a region

sufficiently

small

a < ao,

(R,c,T)-chains

be a r b i t r a r i l y instead

of

a O £ R O which

(depending

chosen.

By

(R,<,T)-chains.

on

(3713) We

~).

will Further

we may

shall

write

a for ~ etc. One

defines

It is f i n i t e a minimal

(4221)

A(a)

= sup{~(r,s)

since

chain.

Lemma:

Ir,s£a}

a is r e l a t i v e l y

The

triangle

~(x,y)

as the

compact.

inequality

< l(x,y,a)

diameter Let

implies

A(a)

of the r e g i o n

[ x , y , a , T 1 . . . T n] be the

following

a.

127

In o r d e r

to f i n d an u p p e r b o u n d

chain between

x and y w h i c h

by u s i n g a g e o d e s i c exists

a geodesic

d = 6(x,y). dissecting

x

P

consists

(see

geodesic [K&N]).

(4216))

char t

of l e n g t h

~(r)

Lemma:

Proof:

Assume

([K&N]

be done

4.2.

= x, y(d)

there

= y and

the J x - o r b i t s

0 < r < ro, w.r.

to

We c h o o s e

of the o r b i t

is a c o n t i n u o u s

r o > 0 so small,

is an i s o m e t r i c m a p

because

there 3.6)

a l w ays

f[S(O,r)] map,

t h a t each

("minimizing"

exists

in

"convex"

a n d P is a h o m o g e n e o u s

space.

r I < r 2 ~ 6 (r I) < 6 (r 2) .

r I < r2,

such that ~(0)

f[S(O,rl)]

at a p o i n t

z 2 6 f[S(O,r2)]

and ~ : [O,~(r2)]

= x, ~(~(r2) ) = z 2. It m e e t s

÷ P being

the o r b i t

z I # z 2. H e n c e

< $(X,Z 2) = 8(r2).

The c o n v e r s e

holds

Thus

the f o l l o w i n g

locally

S(O,r),

to x. r ~ ~(r)

I, IV Th.

a geodesic

= ~(X,Zl)

I, IV Th.

of x, in w h i c h

h e n c e all p o i n t s

1 S ~(r o)

is p o s s i b l e

(4222)

~(rl)

are s p h e r e s

This w i l l

a

f-1.

f is c o n t i n u o u s .

neighbourhoods

[K&N]

y : [O,d] ÷ P such that ¥(0)

of ~ - i s o m e t r i e s ,

This

we h a v e to c o n s t r u c t

minimal.

j o i n i n g x and y. By

h a v e the same d i s t a n c e because

is "almost"

L e t V be the n e i g h b o u r h o o d

the c a n o n i c a l J

for X(x,y,a)

for any s t r i c t l y m o n o t o n e

notions

real

[]

function.

are s y n o n y m o u s :

O r b i t Jx y' I m a g e w.r. Equidistant

to f of the s p h e r e sphere

S(O,r)

{ z 6 P I 6 ( x , z ) = 6 ( r ) }.

N o w a s s u m e a ° 6 R ° is s u f f i c i e n t l y Further These

let Xo, Yo 6 a be p o i n t s

exist because

c ~P ,

a is compact.

small that

A(a)

I

S A(a o) < ~ ~(ro).

of m a x i m a l

distance

6 ( X o , Y O) = A(a).

Consider

a congruent

mapping

128

6 T such implies

that

Tx O = x.

If Yl

= ~Yo' 6(x'Yl) def 0 < r < r o.

J x Yl = f [ S ( O , r ) ] ,

Define

K(O,r)

Either

the

f[S(O,r)]

=

= ~(Xo'Yo)

< ~(r°)

{ ~ E ~ P l II~II< r}.

geodesic

y[O,d]

at a p o i n t

fs c o n t a i n e d

x I, s i n c e

in f [ K ( O , r o ) ]

f[S(O,r)]

= Jx Yl

or

dissects

it m e e t s P.

Yl

....

....../ y

x2 xI

Yo o

fig.

In the

first

In the

second

case we put m = O and proceed

c a s e we h a v e

there

exists

chain

to be c o n s t r u c t e d .

Otherwise

we

some

j 6 Jx

repeat

the

yl[A(a),d]

for y etc.

Evidently,

after

a such

that

(4223)

m steps

6 (Xm,Y)

x I = y(t),

such

that

x I = j YI"

construction

< A(a)

and

indicated

0 < t S d.

If y = x I, w e

we obtain

as

are

Because

We put

x I 6 Jx Yl

T I = jT for the

finished.

substituting

a chain

below.

between

x I for x,

x and x m of o r d e r

129

(4224)

The

m = [ A(a)

latter

geodesic ~(x,y)

~ "

follows,

and

because

are p o i n t s

~ ( x , x I) = ~ ( X l , X 2) = ... g ( X m _ 1 , x m)

on an

= g(a),

(isometric) hence

= ~ ( x , x I) + g ( X l , X 2) + ... 6 ( X m _ 1 , x m) + ~(Xm,y).

N o w we h a v e to c o n s t r u c t chain,

X,Xl,...Xm,Y

the last two c o n g r u e n t

mappings

of the

Tm+ I, Tm+ 2. L e t be ~, ~ 6

Xm+ I

xm

Y

fig. s u c h that ~ x ° = x m,

(4225)

~ x O = y and put x' = ~ Yo' y' = ~ Yo" We n e e d

the f o l l o w i n g

(4226)

Lemma:

Proof: defined

J

x

x' D J m

The geodesic

y

y'

# ~.

yI[d(X,Xm),d]

can be e x t e n d e d

for all t 6 ~ , s i n c e P is c o m p l e t e

Consider obtaining

extensions a geodesic

([K&N]

to a g e o d e s i c I, IV Th.

at b o t h ends Xm, y by a l e n g t h of A(a),

4.1.). thus

130

y'

: [~(X,Xm)-A(a),d+A(a)]

2A(a)

+ 6(Xm,Y)

< 3A(a)

Hence

the p o i n t s

÷ ~

with

< ~(ro).

= y' (~ (X,Xm)-A (a)) def

and

= y' (~(X,Xm)+A(a)) def

satisfy

~(Xm,X)

= ~(Xm,~)

= A(a)

length

By the choice

and t h e r e f o r e

of ro,

y' is i s o m e t r i ~

lie on the orbit

Jx

x'. m

Further: 6(x,y)

= ~(x,x m) + 6(Xm,Y)

> A(a)

(~,y)

= 6 (~,x m) - ~ (Xm,Y)

< A(a)

Hence

x lies

A(a),

which is

and Jx

x' m assertion.

(4226)

in the i n t e r i o r

is

J

Y connected

y',

o

of the sphere

x lies

in the

and contains

x,

around

exterior,

y with J

Y ~. T h i s p r o v e s

y'

radius dissects

the

D

proves

Xm+ 1 6 J x

just

and

the e x i s t e n c e

of a

point

x' n Jy y'. we put Xm+ I = ~ x' = ~ y',

and Tm+ I = j ~,

Tm+ 2 = ~ ~. This

yields

the chain

~ 6 Jxm'

~ E Jy,

[x,y,a,T I .... T~]

where m

=

if x

= y,

m

m + 2 if x m # y. A minimal hence

chain

k(x,y,a)

(4227)

between S m, and,

l(x,y,a)

x and y m u s t by

_< [ A(a)

Let d £ R ° be such that ll(x,y,a)-l(x,y,a')

] + 2.

a' = Nd(a)

I -< I (see

~(x y,a)

< [~(x-t-~!)] + 3 _< ~!x,y) [

A(a)

J

satisfies

(3711)).

> l(x,y,a'),

-

(not strictly),

(4224):

k(x,y,a)

'

be s h o r t e r

hence

A(a)

+ 3 "

a c Nd(a)

implies

P

131 Together

(4228)

with

(4221)

follows:

6(x,y) A(a)

< l(x y,a) '

< 6(x,y) - A(a)

6(U,V)

<

~ 6(u,v)

+ 3

and

{ X(u,v,a)

A(a------7---

A(a)

if u, v 6 P is a n o t h e r We will

assume

X(x,y,a)

= XI

X (u,v,a)

= X2

A(a)

= A

6 (x,y)

= 6

6 (u,v)

= 62 .

A A

= def

of p o i n t s . L e t us w r i t e :

1 We obtain:

(6i+3A)

11 < -12

(62+3A)

61 -> 62

62

61 11 - 3 ~ ~--

From

pair

x % y and u % v in sequel.

61

61 / 1 ' ~ A : ~22 { ~ /

+ 3,

A -

A

62

1 1+__3_3___ X2-3

follows

B,

def

62

I A ~ 11_3

, hence:

361 61 + XI-3 B

61 /I +

3

<

82

This

62

proves:

(4229)

--

61 3 XI 62 12 - 12

_ _

- -

By proposition a 6 R

o

decreases

<

(359),

61 61 3 62 - 62 11-3 <

11 a n d

12 i n c r e a s e

and we obtain:

indefinitely,

as

132

(42210)

lim aERo

X(x,y,a) X(u,v,a)

In the case x = (constant Hence

= 6(x,y) 6(u,v)

y, which was hitherto

and)

equal

the limit

to I, whereas

(42210)

exists

In order to prove the uniform

excluded,

X(u,v,a)

and is equal

convergence

from above by,

say,

is

tends toward

infinity.

to zero.

on regions

assume b, c E R, 5 N c = ~, x £ c, y C b. NOW bounded

X(x,y,a)

in

(425) (iii),

{6(x,y) lyCb}

is

6 > O.

36 Let e > O be given and chose some K E ~ , K Z 6(u,v------[+ 3, and a O 6 R O so small that 36

~(u,v,a o) ~ ~

The latter

36 an d V y E b, ~(x,y,a O) ~ 6(u,v)

for instance

holds

'

if in addition

N Ka

+ 3.

(x) c c. For any o

a E RO,

a ~ ao,

it

follows

by

(4229)

that:

1~2 - 6~21 < ~2 61 min {~2 , ~13-~3} < mln ' { 3X~ , (11_3)62 36 } < min{e,e}

4.2.3.

PROOF OF "(iii)

A homeomorphism in this case the

to

(i)"

can be replaced

(425) (i) clearly

case where

According

P ~ ~

~

holds.

the chain quotient

(3514),

(3713)

and

by an isometry

(4214)) ourselves

converges. (428) we will write

otherwise. (4231) Theorem: d : P × P + ~

(see

Hence we may confine

lim l(x,y,a) = A(x,y,u,v) = d(x,y). a6~o l(u,v,a) The pair u,v £ P, u ~ v, is kept fixed in sequel,

(i)

= e.

is a metric,

if not mentioned

and to

133

operates

(ii) (iii)

d-isometrically

if a n o t h e r d'

pair

coincides

u',

with

v'

on P,

6 P is chosen,

the c o r r e s p o n d i n g

d up to a c o n s t a n t

factor

metric

~ > O.

Proof:

(i)

I. d(x,y)

< -

by

(425)(iii).

2. d(x,x)

= 0

because

3. d(x,y)

= O ~ x = y. O t h e r w i s e ,

l(x,x,a)

= I and

l(u,v,a)

in c a s e

lim l(u,v,a) = a£R l(x,y,a) o = d(y,x) b e c a u s e a c h a i n

÷ ~.

of x % y, w e w o u l d

conclude 4. d(x,y) between 5. d(x,z)

S d(x,y)

This

S l(x,y,a) by

mappings

follows

l(x,y,a) l(u',v',a)

Definition:

(i)

L e t M c P, x 6 P,

(ii)

infimum

The

infimum

and

d(x,-)

between

The

chains

12

x and

be o n l y

assertion

(both of z of

shorter, now

hence

follows

limits. onto minimal

chains.

l(u,v,a) l(u',v',a) D

a 6 R O,

since

L e t M c p be compact, = def

z of l e n g t h

could

and t a k i n g

map minimal

inf{l(x,y,a)

is a t t a i n e d ,

d(x,M)

chain

x a n d y of

from

(4232)

The

to a c h a i n

limits.

= def

between

y and

+ l(y,z,a).

l(u,v,a)

= l(x,y,a) l(u,v,a)

by t a k i n g

l(x,M,a)

between

11 + 12 . A m i n i m a l

by d i v i s i o n

(iii)

A chain

a) c a n be c o m b i n e d

l(x,z,a)

Transport

+ d(y,z).

11 a n d a c h a i n

length

(ii)

x a n d y is a c h a i n

y and x.

length order

between

ly6M}. l(x,y,a)

has

only

values

in ~ .

x 6 P,

i n f { d ( x , y ) Iy6M}.

is a t t a i n e d , is c o n t i n u o u s

since

either

on P~f by

x 6 M or M c P~f,

(429).

f 6 R o, x £ f,

134

(4233)

Proposition:

(i)

d(x,M)

(ii)

Let x { M, M =

=

Let x 6 P and M c P be compact.

lim l(x,M,a) a6R l(u,v,a) o W Mj, w h e r e j6J

all Mj are compact.

Then

I (x,Mj ,a) I (u,v,a) a6R ° > d (x,Mj) uniformly

Before

proving

(4234)

Lemma:

w.r.

this

proposition

Let

functions

to j 6 J.

we c o n s i d e r

I be a d i r e c t e d

li : X + ~ <

the f o l l o w i n g

set and

converging

(li)i61

uniformly

a family

of

to a f u n c t i o n

I : X ÷ ~ + . Let X =

~ Xj, J some index set, and a s s u m e that j6J its i n f i m u m on Xj at the p o i n t xij 6 Xj and I at the

ii attains p o i n t xj Proof:

Let

6 Xj.

Then

6 > 0 be given.

lim !i(xij) i6I 3 i

= l(xj)

6 I ¥ i > i

o

o

uniformly V x £ X

w.r.

to j £ J.

the f o l l o w i n g

holds: lli (x) -i (x) I < 8, I i(x)

- l(x)

li(xij) l(xj)

that

is

< ~, I (x) - I i(x)

- l(xj)

< li(xj)

- l(xj)

- li(xij)

< l(xij)

- li(xij)

l l i ( x i j ) - l ( x j) I < 6 Proof (i)

of

We may

lim a6R

assume

< ~. Hence:

x ~ M and u n i f o r m

convergence

of the

( y ~ l(x,y,a))x(u,v,a) a6~o on M. Let the i n f i m u m

!(x,y,a) l(u,v,a)

infimum

= d(x'Ym)

d(x,M)

at Ym"

By

system

by a n a l o g o u s

as in s e c t i o n

4.1,

(4234),

= d(x,M).

invariant

reasoning.

we c o n s i d e r

subgroups

arbitrarily

N c T,~such

small,

of

l(x,M,a)

o

follows

connected,

and

[]

at Ya £ M and the

Now,

< {

(4233):

functions

(ii)

< ~. In p a r t i c u l a r :

compact,

that T/N w i l l be a Lie

be

135

group.

Let us recall

= 9/N,

~ = JN/N,

further

the a b b r e v i a t i o n s

~ = P/N,

we set

= Ro def

{&l&={Nxlx6a}cP,a6Ro}.

(4235)

Theorem:

The c h a i n q u o t i e n t

uniformly Proof:

on regions

Let a, f E R

w.r.

to

(R,c,T)

(in the sense of

(426)).

converges

and x 6 f such that x 6 f. We will

o

^

form c o n v e r g e n c e ^

of the c h a i n q u o t i e n t

on the subset

show uni-

^

a~f.

Since

^

a~f = {Nyly6A}, W M j6J 3

where

= a~Nf, def (4233) (ii).

and a p p l y

The u n i f o r m

A

convergence

l(x,y,a)

= l(x,Ny,a).

(R,c,T).

This m e a n s

now follows,

Let

identify

M = NA =

if we can show that

[x,y,a,T1,..~l]

= {NTI

Further,

be a m i n i m a l

xlx6a}'

thus

N m 2 ~2[a],

that

shorter

a-chain

to N, and this (x,y,&) (4236)

to

chain between

between

It is m i n i m a l

x and some ~ 6 Ny w o u l d

x and y if one c o n s i d e r s

is c o n t r a d i c t i o u s

H m I, m 2 6 N

hence

-I -I n I m I x I E n I ~1[a] N n I m I m 2 ~2[a]. N o w put -I n2 T2 = nl ml m2 ~2 and c o n t i n u e until a c h a i n def [x,xl,a,n1~1,...nlxl], w h e r e x n 6 Ny, is obtained. a shorter

c h a i n w.r.

3 n I 6 N such that

Xl 6 TI ~ N T2 ~ % @ m e a n s

such that x I 6 m I T1[a]

because

W Ny = y6A

in p a r t i c u l a r :

6 ~i ~ = {~I N x l x 6 a } x 6 n I ~1[a].

we may

yield

the q u o t i e n t

to I = l(x,y,a).

This

a w.r.

shows

= l(x,Ny,a) .

Theorem:

Let d : P × P ÷ ~ +

according

to

(i)

d(x,-)

(ii)

Let us define Then E>O.

(4235)

: &~{~}

÷ ~+ K~(x,e)

denote

(u,v6P are kept is c o n t i n u o u s =

each n e i g h b o u r h o o d

the c h a i n

distance

fixed). for all & 6 R • o

{y6PIa(~,9)~} U of x contains

(analogously: some ball

K<,K=).

Ks(x,e),

136

(iii)

The uniformity uniformity

induced by d coincides

with the given

on P.

Proof:

(i)

follows

(ii)

We may assume

analogously

to

(429).

that U has a compact boundary

P. Let V m U be a compact uniformly

on V~U

~U which dissects

set. The chain quotient

. Now assume

for each ~ >~0 there exists Let a 6 Ro be a sufficiently connected)

the converse

of

converges

(ii), namely

some y¢ ( U such that d(x,yE) small connected

that P

is locally

and consider

between

x and y . Its intersection

with

region

a minimal

~U contains

that ~ ~.

(notice a-chain a points

such that lal ( ~ 'n~ ' a )d

d(&,§~) select

ld(x'~ ( £ )I~ S ,E " since $ , ~(~'~ a E' ) a) -~ I(~ 'Y~ ^ 'a)

~ ~ we have d(x,z E) S 3e. By compactness

of ~U we

e i ÷ 0 such that z e. + z 6 ~U, and, by l of d(x,-), d(x,z) = O. Since x # z this is a

a sequence

continuity

contradiction. (iii)

By

(i) and

(ii) the two topologies

corresponding

uniformity

coincide.

is generated

In each case the

from the topology

by the

group T. (4237)

Corollary:

r I ~ d(~,~) point

S r 2. Then B b £ R o ¥

z on a minimal

d(x,z) Proof:

Let O < E, r I < r 2 and y 6 P such that

sup{d(r,s) Ir,s6b}

r I ~ d(x,w)

a-chain between

x and y satisfies:

< r 2 + e.

Choose b 6 R

approximated

a £ Ro such that a c b each

o

such that

< ~/3 and the distance

up to E/3 by each a-chain S r2 + ~

(according

to

d(x,w)

quotient,

(4235)).

can be uniformly a c b, if

Assume

d(x,z)

> r2

137

SSnce a has a diameter

less than ¢/3, there exists a point w on the

a-chain between x and y such that r 2 + ~2 ~ < fi(.x,w)

minimal

~ r 2 + E.

Now

fi(~,~)

s ~(.~,w,a)

(&,~,a)

~ ^

^

+ E/3

+ ~/3

^

~(u,v,a) 2 2 Sr2÷~c, which is a contradiction. NOW consider the canonical

f-l:u÷

D chart around x £ P (see section 4.1)

K

and the dilatations Dt : K ÷ K Z ~ t-Iz,

t C ]0,1].

Each congruent mapping

~ £ T induces a local congruent m a p p i n g

Dt d -1 • f Dt 1, w h i c h o p e r a t e s Intuitively canonical

speaking,

in the O-neighbourhood

f-l[u]

one looks at the congruent mappings

chart with a m a g n i f y i n g

glass. One then expects

o f K.

in the the space

to look almost Euclidean. Indeed,

by

(4111)

f-1 ~ f consists of local,

commuting with dilations.

It remains

orthogonal

to examine transformations

occuring

in exp K. We set for t E ]O,1] and X £ K

(4238)

~(t,X)

= Dt f-1 exp(tX) f Dt I def and notice that the local transformations generally be non-linear.

However,

a p p r o x i m a t e d by translations (4239) Lemma:

are defined,

~(t,X) will

fort t.T_Q1__thgy may be

of K, as we will show.

Let X,Y,K 6 K and e,B £ ~

expression

transformations

be such that the following

and put ~(t,X,Y)

= T(t,X)Y. def

138 ^

(i)

K = T(t,X,Y)

(ii)

T(t,X,O)

(iii)

T(t,~X,BX) ~(t,X)

(iv)

~ exp(tK)x

= exp(tX)

exp(tY)x,

= X,

-I

=

(~+B)X,

= ~(t,-X).

Proof:

(i)

By

(4110),

f(X)

K = ~(t,X,Y) hence (ii),(iii) locally

(iv)

are

chosen

(4239) (i) y i e l d s

and,

f(tY)

exp(K1)x

a method

f-1[U]

of

T(t,X),

= exp(tX)

exp(tY)x,

following

fact:

exp(tY)x.

consequences

from

by d e f i n i t i o n

= exp(tX)

= exp(tX)

diffeomorphic,

K I and K 2 are

(expX)x

~ f(tK)

exp(tK)x

and

=

of the

= exp(K2)x c

implies

Since

f is

K I = K 2, p r o v i d e d

K.

[]

of c o m p u t i n g

K = T(t,X,Y).

Find

a Z 6 T

satisfying

(42310)

exp

Z = exp(tX)exp(tY). !

This

can be a c c o m p l i s h e d

(see

[VAR]

written

(42311)

2.15.).

Z(X,Y)

as an a b s o l u t e l y

Z =

[ haO

where for c

o

by

the

Baker-Campbell-Hausdorff

is a n a l y t i c convergent

w.r.

to

(X,Y)

t n Cn(X:Y) ,

the c n are p o l y n o m i a l

mappings

Y × I + I of d e g r e e

instance: = O

c 2 = l[x,y]

Further

= 7!~[x, Ix,Y] ] - 7~[Y, Ix,Y] ]

we o b t a i n

a n d c a n be

series

cI = X + Y

c3

formula

a locally

unique

decomposition

n,

139

(42312)

exp Z = exp K' exp J', K' Z ~ K', chart

(42313)

K'

=

Z ~ J' are l o c a l l y

and exp:

tK,

J'

a n a l y tic,

T ÷T are analytic.

since

the a s s i g n e m e n t s the c a n o n i c a l

If we set

tJ

=

we h a v e K = ~ (t,X,Y)

(42314)

6 K, J' 6 J w h e r e

and

K =

[ t n Kn(X,Y), J = [ t n J n ( X , Y ) , from w h i c h K n can be n~O n~O c o m p u t e d for any p o w e r of t in the f o l l o w i n g m a n n e r . Inserting exp(tX)

into

(42310)

exp(tY)

and u s i n g

= exp(tK)

exp(tJ)

(42311)

we o b t a i n

and

tn Cn(X:y ) = [ tn Cn(K:J ) n~O

n~O

tn Cn(

= n~O

~ tVKv(X,Y) : ~ t ~ J ~ (X,Y)) vaO u~O

The p o w e r

series

coincide.

Since c

are i d e n t i c a l o

iff the t m - c o e f f i c i e n t s

= O, no t ° - t e r m s

occur.

For the t l - t e r m s

we o b t a i n

t 1 Cl(X:Y)

= t 1 Cl(K:J)

Thus o n l y the t ° - p o r t i o n

modulo tm-terms, of c1(K:J)

m > 1.

counts,

which

is K o + Jo"

Hence: t(X+Y) and,

(42315)

= t(Ko+Jo) ,

s i n c e X + Y 6 K,

K O = X + Y, Jo = O.

T h e n e x t terms I KI = ~ [ X ' Y ] K '

are I X Y] J1 = 2[ ' J _

K 2 = I[X-Y,[X,y]]K

I

~ [ X + Y , [ X , Y ] O]

J2 = I ~ [ X - Y ' [ X ' Y ] ] J ' where

the s u b s c r i p t

corresponding

subspace.

c o u l d be e x t e n d e d homogeneous

K

(resp.

Further,

to o b t a i n

spaces.

j) d e n o t e s

the p r o j e c t i o n

[K,d] c K

a B.C.H.

formula

is used. for

onto the

The c a l c u l a t i o n s

(reductive)

140 Now we consider K-K ____90 = t

the norm of

[ tn-1 K = L(t,X,Y), n n>1 def

Since L is analytic

in

(t,X,Y),

t 6 ]O,1]. it has a b o u n d e d derivative

at

(0,0,0),

hence IIK(t,X,Y)-Ko(X,Y) II = tlIL(t,X,Y) II S t(llL(O) ii+itlliXiiliY1iliL I It), where ilL(O) II = llK1il = II½[X,Y]KII ~ CliXliliYli and t,X,Y are sufficiently We recall K(t,X,Y) (42316)

Lemma:

small.

= T(t,X,Y)

Theregxists

and Ko(X,Y)

= X + Y and obtain:

a connected O - n e i g h b o u r h o o d

W c f-1[U] c K such that ¥ ~ > 0 B to > O V t 6 ]O,to[

¥ X, Y £ ~,

II~(t,X,Y)-(X+Y) II < ~IIXII I[YII. Let us summarize: local mappings (t)

on K by means of the class of

T (t)

= Dt f-1 def

T (t) = ~(t,X)

The group T operates

= Dt f-1 ~ f Dtl " Each local t r a n s f o r m a t i o n def T f D -I t £ T(t) can be w r i t t e n as a product

j such that X = T(t)o and j £ f-1 ~ f

£ T (t) has a domain of d e f i n i t i o n

containing

= J'. Each def -I D(~) = ~ [W] n W. def

If we Consider the class of regions R' a6R,acU

and O£a'}

= {a'la'=f-1[a], where def in K, it is appropriate to speak of (R',c,T(t)) -

chains between points the chain quotient? O and Y £ W, say minimal

in W. What can be said about the convergence

Notice

that a minimal

[ O , Y , a ' , ~ t) .... T~t)],

chain between

is just the image of a

(R,c,T) -chain

[x=foD -1 (O) ,foDtl (Y) .foDt1[a'],~ . . . I. . provided

(R' ,c,T(t))-

of

that a' ~ D(~i(t)))

But this can be achieved by necessarily

independent

~11

for i=I . . . 1. (4237),

if a' is sufficiently

of t). Let us make the following

small

(not

141

(42317)

Definition:

S&(X,~)

{YEKI iJX-YiJ~e} , a n a l o g o u s l y :

= def

S< and S=. T h e n w e can state (42318)

the

Proposition:

B r > O such that Ss(O,r)

V V,Y 6 S=(O,r) lim O6R' Proof:

V t 6 ]0,1[

l(t) (O'Y'a) l(t) (O,V,a)

Let

c,r 2 in

exists

(4237)

r > 0 satisfying

f[Ss(O,r)]

This

by

is p o s s i b l e

c W and

and will

be d e n o t e d

be such that Ks(x,r2+E)

by d(t) (O,Y).

c W and choose

c Ks(x,r2).

(4236) (iii).

= d(x,f0DtIY) < r 2. def Thus each p o i n t w on a m i n i m a l

Now apply

(4237)

for

rI

= f 0 D~ I Y lies this m i n i m a l This

proves

(42319)

a-chain,

in K s ( x , r 2 + E ) ,

hence

c h a i n onto a m i n i m a l

provided

a c b, b e t w e e n

in W. T h e r e f o r e

(R',c,T(t))-chain

the assertion.

x and

D t o f-1 maps

between

O and Y.

D

Proposition:

Let V,Y 6 K satisfy

V c £ ]O,1[H

t o 6 ]O,1[V t 6 ]O,to[

liVii =

iIYiJ = r.

the f o l l o w i n g

2 assertions

hold: (i)

V a £ R' V T 6 T (t)

(ii)

If an def = S<(O'~n

,

lIT(O) II < 4r ~ ~(x[a]) --

(I+~)),

then

< (I+~) --

(t) (O,y, an) llim (t) - I n÷~ I _.(O,V,a n)

~(a)

.

< E.

Proof:

(i)

Apply

(42316)

for e _< c/8r in order

to find t

6 ]O,1[.

Take

O

any X,Z

£ a and w r i t e

iI~X-~ZIi <

~ = ~ o j, w h e r e

IJajX-(U+jX) lJ +

(by (42316))

< ~HUBJJljXJJ

(.since O£a)

< A(a)

(2~HUJJ + 1)

< ~(a)

( 2 . ~ . 4 r + I)

< (I+~)

A(a).

~ = T (t,U)

il~jz-(u+jz) ii +

+ ~JJUJJHjZIJ

+ HX-ZJl

and U = r (0).

ll(U+jX)-(U+jZ) Ji

142

(ii)

Let e 6 ]O,1[ be given and chose t o 6 ]0,1[ as in part

(i).

It is easily shown that [O,V,an,~(t,2~ because

-

V) ,~(t,~n V) ,...

ni V = T(t, 2 1

V)

(t,_~n__2n-1v)]

is a chain

V = T(t,--~--

v,

-

(4237) (iii). Let

[x,foDt I (V) 'f°D~(an)' ' ~ I ' ' ' ' ~ ]

Clearly

(R,c,T)-chain.

I -< n. (t)

[O,V, an,-r 1

Let

be a minimal

it) ,...T 1 ] be the c o r r e s p o n d i n g

(R',c,Tit))-chain.

We infer: l(t) i O , V , a n ) 1

>_

The same argument

I _>

r r i t ) -> 1 n sup{A(~ [an] ) Ij=I...X } n(1+~)r(1+n)

>

n+1

'

=

n(1+e

applied to Y yields:

l(t) (O'Y' an) n+1 _ > ~ n n(1+~)

Combining

the inequalities

l(t) (O,Y,an) n+_____!__1~ (1+e)n l(t) (O,V,an) Now the assertion

dissects

if n ÷ ~.

to show that for some V 6 S=(O,r)

the O - n e i g h b o u r h o o d

portrayed (42320)

S n(1+~) n+1

follows

We are now p r e p a r e d

we obtain

in the introduction

Theorem:

the orbit J'V

W. The idea of the proof has been (see fig.

(131)).

There exists a V 6 K such that W~J'V is not connected.

JLVII = r and

(We recall

that W was

assumed to be connected.) Proof: I. Let V 6 K and

llVLi = r. J'V is a closed subset of S=(O,r),

latter dissecting assume

W. In order to derive a c o n t r a d i c t i o n

the

let us

143

(I) J ' V ~ S = ( O , r ) . Let <,> V X,Y

denote

the

6 S=(O,r),

<X,Y>

(2)

H Y 6 S=(O,r)~J'V

2.

Let We

inner

= r

2

chains

K. S i n c e

V X 6 J'V,

> O be

n

on

~ X = Y, w e m a y

H 6 6 ]0,1[

n 6 ~ , t 6 ]O,1[,s consider

product

J'V

and

conclude:

<X,Y>

f o r the m o m e n t

[O,V, a n , T o . . . T n _ 1 ] ,

is c l o s e d

where

~ 6r 2. arbitrarily

chosen.

a n = b n U c n,

(3) b n c S < ( O , e n ) I V,e n) , (4) c n c S< (~ and

~i = T ( t , ~ V )

The

chain

minimal

for

property

chain

i=O...n-1.

is e a s i l y

could

only

be s h o r t e r ,

(5)

I (t) (O,V,a n)

S n.

3.

NOW

consider

6 > 0 according

(6)

e <

< ~.

We

set

may

e = e/8r and choose

also

use

we choose

(7)

~

(8)

en <

n

<

e

4r ~

r 2n

n

o

(42319) (ii).

hence

(2) a n d c h o o s e

> O according

t % to be

to

fixed.

to For

~ > O such

(42316). each

that

Thus,

we

integer

n 6

(42318)

and

en > O s a t i s f y i n g

r 8r ~ ~n < ~

Let

t

to

analogously

(> 0 s i n c e

3n > I + s),

hence:

"

is c h o s e n

so t h a t

I

< n

some

3r 2(I+E)

Further, (9)

(42319).

proved

4n2 (i+c) r

(IIO)

£n <

2

and

2(I+c)

(11)

1

(> O s i n c e

(1+2a) (I+E)

Consider

an = b n U cn according

(42319) (ii),

-

]

e n < 2-n

to

I E < ~).

(3) a n d

(4).

By

A

144

l(t) (O,Y,a n) lim n÷~

= d(t) (O,Y)

l(t) (O,Y,an)

(12)

B n

E ~ V n ~ no o

< l(t) (O,V,an) (I+2e)

and by

(5):

I (t) (O,Y,a n) < n(1+2e). From

(2) it f o l l o w s

that,

I V) , (13) ¥ X E J' (--

4.

< I + e, h e n c e

It(o,V,an)

Now consider We w i l l

This

a minimal

always

(14) n 2 ~ 2+e 16r

< ~r2/n.-

assume

chain

[O,Y,an,T1...~l].

n Z no,

further

"

implies:

8__~r > 1 2+~ - 2n 2

> En (9)

I 4r ~ ~n + 2 e e n 4r(I-~

(15)

en/8r)

4r > I-E where

~n/~

prove

¥ i=I...I (16) < Y , W >

is s t r i c t l y

the f o l l o w i n g

iIWll -< 3r.

4.2.

"i=I" = def

positive

auxiliary

by v i r t u e

assertion

of

(8).

by i n d u c t i o n

V W E ~i[an],

< ir[r(~+e)/n+4Cn(1+~)]

(17)

If U I

'

the d e n o m i n a t o r

4. I. We w i l l

(18)

Z en, hence:

and

T 1 (O) we h a v e

~I = T(t'UI)

0 J1'

Jl £ J'

We m a y w e l l

confine

W E ~l[Cn]"

Let W = T I W o, W O E c n, and W I

ourselves

to the case O E T 1[b n] and = Jl Wo" def

Hence

on i:

145

W = T (t,U 1)w I and r llW III = IIWoll < ~ + E n-

(19)

From

(13) we conclude

(20) < 6r2/n, because

(21)

Jl is a

further

V-Woll

< e n since W O 6 c n and,

II. II-isometry,

I l J l ( HI V)_Wll I < E n .

4.2.1.

We d e f i n e

W2

= U I + W 1 and will def E

(22)

IIW1-W211 = IlUlll

Proof:

n

We conclude:

IIUIII =

IIU1-~IZ111S (42316)

IIZIII < e n

From this

(22)

IIJI(ZI)II

IIUI-(UI+JIZI) S IIJIZ111

+

II +

< en

II(UI+JIZI)-T(t,UI)JIZ

I

11UIII " liJIZ111 " e/Sr

en + IlUlll

4.2.2.

show:

< 1-e ¢/8r " n

O £ T1[b n] -I Z1 = T (O) 6 b n and def I

~n ~/Sr.

follows.

N o w we infer ilW-W2il

=

ilT(t,Ul)W1-(U1+Wl)

and by v i r t u e

(23)

II

lIW2-Wii

E < lle n e/Sr n

We c o m b i n e 1

of

(21),

(22) and

(r/n+en)

(22)

and

il S llUlll lIWlil c/8r (42316)

(19):

E " 8-~

(23) concluding:

1

1191( ~ V)-WII < I l J l ( ~ V)-Wllt en < en + 1-e n-C/St [ 8r = en 1 + 8r_e n e

+ IIW1-W211 + IIW2-WII [ I+

re + n e n £] 8nr _

8nr+re+n 8nr

en e]

r(16n+e) = en

n(8r-E

n

~)

146

r (I 6n+e) < E = 4 ~ £ n (8) n n ( 8 r - r ( 4 - ~-~)) Together

with

(20),

this

I < + < r[r(6+e)/n+4

which 4.2.3.

(24)

is

(16)

Combining

IITI(O) II =

I IIYI} IIW-j I (~ V) II -< 6r2/n + 4r ~n en(1+e) ]

for i = I.

(22)

and

(15) we infer

IIUIII < 4r

and may a p p l y <

A(T1[an])

shows

(42319)(i).

(I+~)

Hence

A(a n) <

(I+~)

(r+ 2 e n )

and thus JlW-OII

<

This p r o v e s

4.3.

r r 3r r (~+ 2 ~n ) < (I+~) (n+~-~e-n) (9)

iI+~)

(17)

= 3r.

for i = I.

"i ~ i+I". Assume : W 6 ~i+1[Cn]' Ui+1

= ~i+I (O) , W i 6 Ti[c n] N ~i+l[bn]

Ti+ I = T ( t , U i + 1 ) J i + 1 W = ri+1

where

(25)

following

H W I II =

HWoll

Ui+1 •

holds: r < n + ~n'

=
• I Wo> -< +

< ~r2/n + r en' (13) (26) < Y , W I >

6 J',

W o, W o 6 c n,

WI = 3i+I W o , W2 = W The

Ji+1

% ~'

< r(r6/n+En).

hence

IIYII IIWo-l_n Vtl

147 ^

4.3.1.

Let W i = ~i+I Vi' Vi 6 bn, and ~ = a

--I

and ~i+I

= d o i, w h e r e

d = T ( t , W i)

^

0 Ti+ I. F r o m

~ V i = 0 we infer 0 6 ~[b n] and thus ^

we may apply particular,

the results

of 4.2.

concerning

II~(0) II < 4r by v i r t u e

of

Iio(0) II = llWill < 3r by the i n d u c t i o n

~I to T. In

(24),

and

hypothesis

(17). Thus we

^

may a p p l y

(42319)(i)

A(ri+1[bn])

= A(oo~[bn]) 2

-< (I+¢) llUi+111 (27)

for both

<

llWiJl +

~ and d and o b t a i n

< (1+s)

2 a . n

A(~[bn])

< (1+e) 2 A(b n)

Hence

llUi+1-Will

< 3r + 2(I+¢)

2

a

< 4r, n(IO)

llUi+ I II < 4r. Therefore, j=1...i, IIWII <

(42319) (i) applies and the t r i a n g l e

(i+I)(1+e)

to Ti+ I, as well

inequality

as to Tj,

yields:

A(a n)

< 1(I+¢) (r+ 2 a n) r < n(I+2¢) (I+¢) (n + 2 e n) (12) < 3r. (11)

This proves 4.3.2.

From

(28) By (29)

(17) .

(16)

as the i n d u c t i o n

hypothesis,

it follows

< ir[r(6+s)/n+4an(1+s)].

(42319)(i)

IIWi- Ui+1 II =

applied

to ~i+I we o b t a i n

II~i+I Vi-Ti+1Oil

-< (1+e)

sn-

Further: IJW2-W I [I =

IIW-Ui+I-WI II =

< I I U i + 1 II IIW 1 II a / 8 r (42316) r _< 4 r - ( ~ + a n) • s / 8 r , (25),(27)

IJT(t,Ui+I)WI-(Ui+I+WI ) II

hence,

that,

148

(30)

r

IlWl-W211

< ~'(~+ en).

We combine

these results

= <

to compute

llWi-Ui+111

-< ir[~6+e)/n+4£n(1+e)] e

+ IIYII

IIW1-W211

(by (29))

n

(by (26))

r + r ~ ( ~ + Sn)

(by (30))

(i+I) r[r(6+s)/n+4en(1+~)].

This completes

the proof of

4.4. The chain property by

+

(by (28))

+ r(r6/n+~ n)

<

for .

+

+ IIYII

+ r(1+s)

an upper bound

of

(16).

[O,Y,an,tl,...t ~] implies

Y 6 tl[a n] and

(16):

(31) = r2
(5),

11

= x(t) (O,V,an) def

i-_>! 11 n

> r (31) r(6+E) + 4n En(1+e)

Taking

the limit n ÷ ~ the left hand side tends

and the right towards (32) d (t) (O,Y) 5.

~ n, hence

By

I because ~-~-~

1

4n ~n (1+s) < nI bv

< I + e, thus

(32)

+ ~

1 < (I+E) (6+s) 3+26-6

< (1 + . ! ~ . . ) ( 6 + 12.---~-6) (6) 2 4<3+26-6 1 < 6(2-6) This

d(t) (O,Y)

(9) " Hence,

I > "~+s

(42319) (ii) we have dt(O,Y)

--
towards

= 1 -

(1-6)

2

is a contradiction.

< 1. []

implies:

149

The existence of dissecting completes

orbits Jy in P = P/N U for each U 6 U

the proof of "(iii) ~

used in the section

"(i) ~

(i)" because only this property was

(ii)" to show that T is a Lie group and

thus P = P.

4.3.

TITS/FREUDENTHAL

CLASSIFICATION

If the preceding axioms RI to R6 hold, (P,T)

can be exhaustively

but considering

enumerated.

the resulting geometries We will adopt them from

only connected groups T.

The proof of this classification

is much involved and it is no place

here to give an account of it. Before enumerating possible

[FRE],

the list of

geometries we need some further definitions.

As usual, we denote by the field of real numbers, ¢

the field of complex numbers, the

(scew)

field of quaternions,

the alternative

algebra of Cayley numbers

(or octaves);

or ¢ or ~. S n is the n-dimensional pn the n-dimensional

real sphere and

projective

real space,

i.e.

We will use the notation of the classical groups, to physicists, [GIL]).

as for example SO(n)

Further we will consider

exceptional

or U(n,~)

v(G 2) denotes automorphism

to the real

by the signature of the C a r t a n - K i l l i n g

and F4(_20)

group usually represented

(see for instance

G 2 and F 4. The different

metric which will be indicated in brackets. the real groups F4(_52)

which is familiar

some groups corresponding

simple complex Lie algebras

forms may be characterized

pn = sn/{±l}.

We only need to consider

and B4(_36),

which is the abstract

as SO(9).

the 7-dimensional

real representation

group of non-real Cayley numbers,

of G 2 as the

which operates

150

transitively dimensional

on S 6, h e n c e on F 6. F u r t h e r , real

spin-representation

(more p r e c i s e l y :

of its u n i v e r s a l

c a s e 1 = 3 its L i e a l g e b r a suitable

basis

of ~ ,

let ~ SO(21+I)

of the s p e c i a l

covering

representation

group

be the

orthogonal

Spin(21+1)).

21group

In the

m a y be g i v e n by u s i n g

a

(e i) i=O...7:

7 (aij) ~ Hence

(x6~i,!=1

~ SO(7)

aijei(ejx))"

operates

For all d e t a i l s

transitively

on S 7 and p7.

see

[FRE],[FRE3]

and

We are now p r e p a r e d

to f o r m u l a t e

the list of p o s s i b l e

w i l l be i n d i c a t e d P + T

either

IF&V].

geometries.

They

in the f o r m

or

P ÷o J if P is an a b e l i a n

subgroup

and T = P ~

J

(semi-direct

product),

or TI j~,

where

J1

is a g r o u p w h i c h

into the g r o u p T I in TI a (more or less) o b v i o u s m a n n e r . In the t h i r d case, P and T is J1 e q u a l to T I m o d u l o the k e r n e l of the n a t u r a l o p e r a t i o n of T I on P. certain

m a y be e m b e d d e d

We w i l l

invent

(431)

Theorem:

If RI

precisely:

[TF-isomorphic,

following (i)

~n

denotations

for the n o n - c l a s s i c a l

to R6 h o l d

, (P,T)

is i s o m o r p h i c

see s e c t i o n

5.3)

geometries. (more

to one of the

geometries:

÷o V ( n ) ,

where

V(n)

is e q u a l

to

= ~

: SO(n)

or

= ¢

: U(n,¢)

or SU(n,~)

or

= @

: U(n,~)

x K,

K is the g r o u p of r i g h t

where

multiplications

in Qn w i t h

b E Q,

Ibl = I

or

b 6 ¢,

Ibl = I

or

numbers

of the f o r m

b = 1. A geometry

of this k i n d

is c a l l e d an " a f f i n e

~-Hilbert

space

151

(or parabolic) (ih)

142 ÷o v SO(9), "parabolic

(is)

octavian

spin line".

I~7 + o v (G 2 ) , "imaginary

(ii)

(iih)

plane".

I~ ÷o v SO(7), "octavian

(io)

geometry".

octavian

U(n,1;]F) U ( n , ~ ) x U(I,~F)

line".

'

hyperbolic

geometry".

F4 (-20) ,

"hyperbolic

octavian

plane".

B4 (-36) (iii)

U(n+1 ;~) U(n,]F) xU(;I,IF) "IF - e l l i p t i c

(iiih

,

geometry" .

"elliptic

octavian

plane".

spin

line".

F 6 + v(G2), "elliptic

imaginary

(iv)

SO (n+1) SO(n)

(ivs)

S 7 + v SO(7),

= sn'

"spherical (ivo)

= IR)

F 7 + v SO(7), "elliptic

(iiio)

(n_>2 for ~

F4 (-52) B4 (-36)

(iiis

'

spin

line".

n > 2,

"spherical

line".

S 6 + v(G2) , "spherical

imaginary

line".

geometry".

5.

CHARACTERIZATION

The be

classification singled

OF EUCLIDEAN

of section

out by two

Vanishing

curvature

(ii)

dimension

3.

It w o u l d (R,<,T) been

be desirable in o r d e r

achieved

Finally

we will

obtained

5.1

geometry

could

postulates:

to formulate

respect

study

that Euclidean

and

to s t r e s s

with

4.3 s h o w s

additional

(i)

GEOMETRY

the

their

these

empirical

to t h e class

postulates

axiomof

in t e r m s

meaning.

of

But this has only

dimension.

of Euclidean

representations

in t h i s way.

DIMENSION

It w i l l (511)

be natural

to e m p l o y

Definition:

a notion

of dimension

based

on coverings.

L e t b 6 R O, V c R.

M(b,V)

~ ~ u 6 R ¥ v £ V B ~ £ T def o t h a t u < ~b a n d Tb ^ V ¢ O.

such

(V is b - a p p r o x i m a t e l y (512)

Lemma:

Proof:

M(b,V)

~

A N b ~ ¢ 9. v6V

"~" L e t x 6 u, t h e n x 6 N b ~ f o r a l l v £ V. "~"

The

overlapping.)

If V is f i n i t e ,

following

D N b ~ is n o n - e m p t y v£V

a n d open,

u£~.

m

o

definition

(513)

Definition:

(i)

Dim

only

presupposes

Let N 6 ~

hence

contains

the axioms

RI

a region

a n d R2.

U {-1,0}.

R S N def

V a £ R V v £ Ro 3 n 6 ~ V i=1...n V V c (ii)

B a 6 T such

H Vo...v n 6 R

t h a t v i < av,

{Vl...Vn} , ~(Vo,V ) ~

Dim R = N

~ Dim def

R S N,

a <

satisfying: (vivv2v...Vn)

IVI S N + I.

but

n o t D i m R ~ N - I.

and

153

Thus

it is r e q u i r e d

number

of a r b i t r a r y

"overlapping" (514)

of

satisfying

V i=O...n

3 o 6 T

U

(514)

from

Consider

v,

being

U i=1...n

a metric.)

the c o m p a c t

U i=I..,

than

Theorem

V8 Cor.,

there).

Since

dimension

manifolds

finite,

that

show

R7:

U i=I...n

NV

that

open

~ has

(see

o

~.1

But

(513)(i)

we

of d e v i a t i o n .

(335)

One

and the

coverings

intersect.

open

[H&W], single

out

mesh Then

to

sense

IV 3, Cor.

if w e p o s t u l a t e :

the

that

[H&W],

of P, n a m e l y

just

(514)

S e, such

(in the

subset

Theorem

of

to 4.2 P is e n d o w e d

with

S N

[]

diameter

D i m R S N. A c c o r d i n g

dimension

list,

implies

for any v ° £ R o.

according

assume

that we w i l l

D i m R = 3.

of

form:

that ~ c b c ~

a nonempty,

P in F r e u d e n t h a l ' s

two p o i n t s

~ i=IU...n ~i"

of the c o v e r i n g

shows

IVl < N + I.

1

(Recall

~ N as a m a n i f o l d

Axiom

Vo

we w i l l

~ contains

arguments

at the

~ > O such

~/2.

set ~ has

N + I members

holds:

~ v..

N

=

Further

at m o s t

following

a ~-version

is of the

a c i=1.--\/n vi..

a, b E Ro,

smaller

are

~. c o 1

completely

~ c

V Vl c N V i=I...n o

the

n N ~. % ¢ ~ v ;6V Vo 1 1

the o t h e r

(336)(i), (v),

then

that

an i m p l i c a t i o n

1

N ÷ 1 ones

and

(512), v

i=1...n

at m o s t

1

is a l m o s t

to p r o v e

V

a <

v. Vo

{~I .... Vn},

Since

follows

N

. . . n

such

that

by a f i n i t e

(511)).

6 R

need

(515)

sense

a 6 R can be c o v e r e d

v i such

B ~o'''Vn

Proof:

These

regions

If D i m R < N,

V V c

with

region

Proposition:

i=1

By

each

small

(in the

a c

only

that

defined ~, P has

I).

3-dimensional

154

5.2

CURVATURE

We will

prefer

of E u c l i d Recall

expressing

the flatness

but by a local p r o p e r t y

that A(x,y,u,v)

d(x,y) / d(u,v),

(521) A x i o m There

which

denotes

of P not by the p a r a l l e l

of triangles,

the q u o t i e n t

is u n i q u e l y

definable

following

H. Busemann.

of d i s t a n c e s (as o p p o s e d

to d(x,y)).

R8: exists

u,v,x,y,z

a neighbourhood

6 U satisfy:

U in P such

that all points

If

A(x,u,u,z)

= A(x,v,v,y)

= I

and

A(u,z,x,z)

= A(v,y,x,y)

I = ~ ,

then

A(u,v,z,y)

I = ~.

z

x

v

y

fig. From

[BUS]

identically.

§ 41 it follows,

axiom

(522)

that the c u r v a t u r e

tensor v a n i s h e s

155

5.3

EUCLIDEAN

In o r d e r

REPRESENTATION

to s t a t e

the m a i n

to d e f i n e

more

structure

"Euclidean

(531)

precisely

what

space"

principle

(ii)

an a u x i l i a r y

base

(iii)

a structural

term

(iv)

and a x i o m s sional,

additive E

an

It is w e l l - k n o w n phic,

i.e.

that

set

species

of

by

~,

p(~xVxV)

x

P(VxVx~)

x

P(VxExE),

expressing linear

space

(V,~,+,-,) equipped

bilinear)

(V,+)

We w i l l

that

but

set e =

this

any two

[3 is c a t e g o r i c a l .

inner

operates

We will

with

an

(positively

product

transitively

,

and

of this

gather

in our kind

freely

its on

component

context.

are

some w i d e l y

and

.

as an a d d i t i o n a l

is not n e c e s s a r y

structures

is a 3- d i m e n -

(E,V,~;+,.,,o)

on E is c o n s i d e r e d

term,

appropriate

E, V,

x

group

"orientation"

structural

sets

it seems

by the

It is g i v e n

symmetric,

(via o).

book

6 P(v×v×v)

real,

definite,

of the

[3"

base

of this

is u n d e r s t o o d

(i)

(+,',,o)

Often

theorem

[3-isomorused

definitions. (532)

Definition: kind

Let

(E,V,~,+,.,,o)

be a s t r u c t u r e

of the

[3"

A mapping

Tv

translation.

: E ÷ E of the A linear

= <xlx>

rotation some

e =

of V and D

group

mapping

D

v 6 V,

is c a l l e d

: E + E, d e f i n e d

is c a l l e d

generated

a

: V ÷ V satisfying

for all x 6 V and det D > 0 is c a l l e d

f i x e d A 6 E,

A. T h e

form A ~ v 0 A,

a proper

by t r a n s l a t i o n s

by D(v0A) rotation

=

(Dv)

a proper 0 A, f o r

of E w i t h

and p r o p e r

center

rotations

of

156

E will The

be d e n o t e d

linear

mapping

reflection; fixed

S

A 6 E,

: E ÷ E,

the

L 1 : V ÷ V, L l v

T(~)

of E, R(~)

Now

let

[TF

(S;~,G)

S : V + V,

will

as the

Iv;

that

(S,T)

LI,

is c a l l e d =

(Sv)

of E w i t h

I > O,

be d e f i n e d subclass

as the

of open,

be the

0 A

center

= def

class

Recall

(at l e a s t

that

structure A glance (533)

the

N t°p

slightly

set of p o i n t s

P

(327)

with

the

at F r e u d e n t h a l ' s

list

shows

Theorem:

Assume

the

Then

exists

a structure

there

such

Consequently,

situation

that

axioms

(P,Nt°P,T)

(R,c,T)

is e n d o w e d

RI

N

with

a of the k i n d

represented

(p,Nt°P,T)

(~,=,¥)

(E(~) ,T (~) ,T(~) )

fig.

[3

is [ T F - i s o m o r p h i c

(R,<,T)

/

(534)

subsets

given

by

of are

in this

book).

a topological

(333).

to

for

(R,<,T).

("Euclidean to

(E,~(~),T(a)).

(R(~),c , T(~)).

in the

Z3

I°

[FRE]

used

to R8 h o l d i n g

Z2

(R(a),c,T(a))

o A.

at once:

is [ 2 - i s o m o r p h i c

can be g r a p h i c a l l y

A.

and G a g r o u p

version

uniformity

some

subsets.

of s t r u c t u r e

space

weaker

associated

space")

The

in the

for

(Llv)

homeomorphisms and the f u r t h e r a x i o m s of F r e u n d e n t h a l satisfied

(point)

of all o p e n

bounded

species

is a t o p o l o g i c a l

a

are d e f i n e d :

L 1 : E + E, Ll(v0A)

(Tits/Freudenthal)

such

= -v, def by S(voA)

a reflection

dilatations = def

Sv

defined

is c a l l e d

Analogously,

Finally,

by T(e).

following

manner:

157

The

representation

since

[TF

F of all with

the

than A u t ~ 3 ( ~ ) is o n l y 2 3

(R,c,T)

is c a t e g o r i c a l .

the g r o u p equipped

of

F will

we m a y

(535)

group

The

symmetry

it w i l l

[TF-automorphisms structure

~(~), to the

physically

be the c l a s s

Therefore

(R(~),c,T(~))

However,

corresponding

partially

by

is e s s e n t i a l l y

be

instructive

of a g i v e n

to c o m p u t e

Euclidean

space

T(~).

This

group

fact

that

the E u c l i d e a n

relevant.

If E(~)

of a d m i s s i b l e

will

be

is c h o s e n

coordinate

unique,

E(~)

larger structure

as the

space

transformations

define: F

= Aut 7 (E(~),T(~),T(~)) def LTF g r o u p of p h y s i c a l g e o m e t r y .

(5136) T h e o r e m :

F is the g r o u p

rotations,

reflections

It W i l l

be a d e q u a t e

result,

not

using

to g i v e

generated

the m a c h i n e r y

of Lie

be c a l l e d

by t r a n s l a t i o n s ,

and d i l a t a t i o n s

an e l e m e n t a r y

will

the

proper

of E(~).

proof

algebra

for this theory

elementary

or p r o j e c t i v e

geometry.

Proof:

Let

satisfying

y 6 F. T h a t

(~) y T(~)

¥

is,

-I

y : E(~)

÷ E(~)

will

be a h o m e o m o r D h i s m

= T(~).

If A 6 E and Hence

~ JA

rotations

yA = v 0 A = T A, t h e n ~ = T y maps v def -v ~ = J A if J A is d e f i n e d to be the g r o u p -I

with

center

More

generally,

Each

rotation

angle

~(D)

6

~ JB

~ ( ~ D a -I)

= ~(D).

Proof:

Consider

a(D)

= const.

subgroups

We

cyclic

Under

of the

-I

= JC'

if ~B = C.

is a r o t a t i o n

[O,2~[.

around

an axis

~(D)

6 V with

some

claim:

subgroups

D ÷ ~ D -I

same

of p r o p e r

A.

e

D E JA

A o n t o A.

order,

of JA of o r d e r

they

hence

are m a p p e d

their

angle

n,

defined

I : I onto of r o t a t i o n

by D n = i,

cyclic is

158

conserved:

~(D)

= 2~k = ~ ( ~ D

-I).

Now,

the u n i o n

of all

finite

cyclic

n

subgroups

is d e n s e

continuity

Further, fixed

in J A , ~

= def

{D6JAIa(D)=~}

and the

claim

follows

by

of ~.

~ throws

points

of

the

~ D ~

set of -I

fixed

points

of any

in an 1 : I fashion,

D6J B onto

a n d thus

maps

the

set of

lines

onto

lines. Especially, lines The

if w e d e n o t e

through

latter

there

follows

a linear

f r o m the

lift

the

a

(linear)

of the

equation

lines

~ permutes

between

the

them.

equivalence t h a t ~(R)

rotation

through

between

of ~ to V,

angles

= e ~ 3 R 6 J A such

exists

permutation

~ the

O 6 V conserving

a(~(D1),a(D2))

Now

by

unit

= e a n d D I = R D 2 R -I

8 of V w h i c h

O as ~, b e c a u s e

induces the

the

same

coefficients

of

vectors

e 3 = 11 e I + 12 e 2 may

be e x p r e s s e d

X(~l , e÷ 3 ) This

functions

of the

angles

a(~1,e2),

~(e2,e3),

.

result

rotations maps

as

may

be w r i t t e n

~ in V.

orbits

Or,

6

= def

o f the g r o u p

as

~ ~ ~-I

= ~ ~ ~-I

8 -I

e commutes

~A onto

for all p r o p e r V

V

orbits,

with

that

all

is,

s u c h D.

spheres

Hence

onto

V

spheres, lines

and

leaves

through

O.

Consider

the

by

Clearly,

them.

y be m a p p e d If

invariant

Further,

z 6 g D 6g,

O ~ g a n d the

6 leaves ~g.

then

L e t us p r o v e 6 -I

invariance

z and

of the

properties

of

will

e,

sets

6 maps

6 V and Let

of D, lines

the

the

line

"g p a r a l l e l

~g."

line

(by the

6 statet

point

that onto

plane

is,

on the

above

intersect

through

the

and

O and

P spanned

g through

circle

~-I

around

z under O through

its c o n t i n u i t y ) .

circle

at a s e c o n d

the

lines.

z w o u l d lie on g in c o n t r a d i c t i o n

~ is c o n s t a n t

lines

O,x,y

P invariant.

If ~z = z,

of t h e s e

with

points

~z % z.

one

fixed

together

3 noncollinear

onto

the

x and

to ~,

if

z

If g # ~g, point

w.

159

Then from ~z = z and 6w = w it follows that ~g = g. Hence g is parallel Now,

to ~g at any case.

it follows by the familiar

proportional segments

on transversals,

product of a dilatation

~

fact that parallel

lines intercept

that ~ is a dilatation

or a

and a reflection.

x

5y dilatation

g x

~y

\

~x

I

~g

fig.

The converse, satisfy

that translations,

(~), is immediate.

6 dilatation times reflection

(537)

rotations,

dilatation []

and reflections

6.

REFERENCES

[ABR] R. Abraham:

Foundations Benjamin,

[B&C] F. Brickeil, R.S. Clark:

of Mechanics.

New York,

Differentiabie Van N o s t r a n d

[BGT] N. Bourbake:

Elements

Manifolds.

Reinhold Comp., New York,

Elements

Frankfurt

a.M.,

Geometrie

Paris,

1968

of Geodesics.

Press,

New York,

1955

und Erfahrung.

(27.1.1921) Ullstein, [FRE] H. Freudenthal:

1976

of Mathematics.

The Geometrie Academic

[EIN] A. Einstein:

1966

in: Erkenntnis

Theory of Sets, Hermann, [BUS] H. Busemann:

Paris,

Protophysik. Suhrkamp,

[BTS] N. Bourbaki:

Herman,

Geometry by Ropes and Rods. To be p u b l i s h e d

[BOH] G. B~hme:

1970

of Mathematics.

General Topology. [B~K] W. Balzer, A. Kamlah:

1967

In: Mein Weltbild,

Berlin,

Neuere F a s s u n g e n

1955 des Riemann-Helmholtz-

L i e s c h e n Raumproblems. Math. [FRE 2] H. Freudenthal:

Z. 63,

(1955/56),374-405

Das H e l m h o l t z - L i e s c h e

R a u m p r o b l e m bei

indefiniter Metrik. Math. Annalen [FRE

3] H. Freudenthal:

[F&V] H. Freudenthal, H. de Vries:

156

(1964),

263-312

Lie groups

in the Foundations

Adv. Math.

I (1964),

145-19o

Linear Lie Groups. Academic

Press, New York,

1969

of Geometry.

161

[GIL] R. Gilmore:

Lie Groups, Lie Algebras, Applications.

[HELl H.v. Helmholtz:

and Some of Their

Wiley, New York,

Uber die Thatsachen,

1974

die der Geometrie zum

Grunde liegen. Nachr.

Ges. Wiss. G~ttingen

]868,

Reprinted in: uber Geometrie, gesellschaft, [H&W] W. Hurewicz, H. Wallmann:

Darmstadt,

193-221.

Wiss. Buch-

1968.

Dimension Theory. Revised Edition 1948, Princeton University Press

[K&N] S. Kobayashi, K. Nomizu:

Foundations of Differential Geometry. Wiley, New York,

[LUD I] G. Ludwig:

1969

Deutung des Begriffs

"physikalische

Theorie" und axiomatische Grundlegung der Hilbertraumstruktur der Quantenmechanik durch Haupts~tze des Messens. Lecture Notes in Physics 4, Springer Berlin,

[LUD 2] G. Ludwig:

1970

Einf~hrung in die Grundlagen der Theoretischen Physik. Band I: Raum, Zeit, Mechanik. Bertelsmann,

[LUD 3] G. Ludwig:

DUsseldorf,

1974

Grundstrukturen einer physikalischen Theorie. Springer,

[MAY] D. Mayr:

Berlin,

1978

Zur konstruktiv-axiomatischen

Charakteri-

sierung der Riemann-Helmholtz-Lieschen Raumgeometrien und der Poincar~-Einstein-

162

Minkowskischen-Raumzeitgeometrien

durch

das Prinzip der Reproduzierbarkeit. Dissertation, [MSZ] D.Montgomery, H. Samelson, L. Zippin:

MOnchen,

Singular points of a compact transformation group. Ann. of Math.

[REI] H. Reichenbach:

1979.

(I) 63

(1959),

I-9

Axiomatik der relativistischen

Raum-Zeit-

Lehre. In: A. Kamlah, M. Reichenbach "Gesammelte Wiesbaden, [SCH I] H.J. Schmidt:

Werke",

(eds.)

Band 3, Vieweg,

1979.

Zur Charakterisierung

des Euklidischen

Raumes durch Gebiete und Transporte. In: W. Balzer, A. Kamlah

(eds.)"Aspekte

der physikalischen Begriffsbildung", Vieweg, Wiesbaden, [SCH 2] H.J. Schmidt:

1979.

Entwurf einer Laborkinematik. Lecture held at the Conference

"Grund-

strukturen einer physikalischen Theorie: Raum,

Zeit und Mechanik"

in MOnchen,

May 1979. To be published in the proceedings. [TIT] J. Tits:

Sur certaines classes d'espaces homog~nes de groupes de Lie. Acad.

Roy. Belg.

CI. Sci. Mem. Coll. 29

(1955), fasc. 268 pp. [VAR] V.S.Varadarajan:

Lie groups,

and their Representations.

Prentice-Hall, [YAM] H. Yamabe:

1974.

A generalization of a theorem of Gleason. Ann. of Math.

(2) 58 (1953), 351-365

163

7.

NOTATIONS Most

notations

perhaps

with

FA A = B ~ C

in t h i s b o o k

the e x e p ~ i o n

the

are c o m m o n l y

of the

s e t of a l l

the n u m b e r

-1

the

¢

the n e g a t i o n

fig [x] [x] ~

,

, of

the

of

the m a p

finite

s e t M,

~,

of the r e l a t i o n

r-,

f: A ~ B to t h e

the e q u i v a l e n c e c l a s s of x 6 A w.r. to t h e e q u i v a l e n c e r e l a t i o n ~ o n the s e t A,

[x]

if x is a r e a l n u n ~ e r : less o r e q u a l x,

[a,b]

the

]a,b[

open,

closed,

[a,b[

fdifferent

]a,b]

t half-open

a/b+c

of A

of the r e l a t i o n

the r e s t r i c t i o n s u b s e t M c A,

or simply

subsets

of e l e m e n t s

inverse

in set t h e o r y

following:

A = B U C and B N C = #

IMJ

used

a

~ + c.

intervals,

the g r e a t e s t

integer

etc.,

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