Formalizing Multiple Location

Non-Classical Logics, Model Theory and Computability, A.I. Arruda, N.C.A. da Costa and R. Chuaqui (eds.) 0 North-Holland Publishing Company, 1977

FORMALIZINGMULTIPLE by

F.

G.

LOCATION

ASENJO

0, PURPOSE, Whitehead c r i t i c i z e d t h e concept o f s i m p l e l o c a t i o n as

insufficiently

descriptive o f the spacial relationships o f actual e n t i t i e s i n the physical w o r l d (see Whitehead 1967,chapters I I I a n d I V , s p e c i a l l y p a g e 6 5 ) . T h e concept o f f i e l d o f f o r c e s p r o v i d e s a good example o f t h e k i n d o f f o r m a l u b i q u i t y which r e a l i t y e x h i b i t s and which s i m p l e l o c a t i o n cannot convey. E n t i t i e s i n a f i e l d e x e r t t h e i r dynamic i n f l u e n c e t h r o u g h o u t t h a t f i e l d and a r e i n turn i n f l u e n c e d by a l l t h e f i e l d ' s o t h e r e n t i t i e s as w e l l as by i t s general d i s t r i b u t i o n o f f o r c e s . Because space i s t h o u g h t t o b e a s i n g l e a n d u n i f o r m d e p o s i t o r y o f p h y s i c a l e n t i t i e s , t h i s p l u r a l i t y o f dynamic e f f e c t s i s u s u a l l y d e s c r i b e d as b e i n g m e r e l y p a r t o f t h e general phenomenon o f a c t i o n a t a

d i s t a n c e . A c t u a l l y , e n t i t i e s a r e n o t p l a c e d i n a n i n d i f f e r e n t space; i s a changing p r o p e r t y o f t h e dynamic c o n d i t i o n s o f a f i e l d , which i s

space spe-

c i a l l y e v i d e n t a t t h e m i c r o p h y s i c a l l e v e l . Every r e g i o n i n a f i e l d has what elsewhere we have c a l l e d mcLetipLe LocutLon (see Asenjo 1962), i.e.,

a real

and e f f i c i e n t e x t e n s i o n o f each r e g i o n i n t o and t h r o u g h o u t o t h e r r e g i o n s o f the field.

Such an

idea

o f m u l t i p l e l o c a t i o n provides us w i t h a

more c o n c r e t e conceptual approach t o t h e r e a l i t i e s o f t h e p h y s i c a l w o r l d .

No l o n g e r m u s t o n e t a k e r e f u g e i n t h e m y s t e r i o u s a c t i o n a t a d i s tance;

instead,

o n e c a n t h i n k o f e n t i t i e s as a c t i n g upon o n e a n o t h e r b y

v i r t u e o f t h e i r mutual c o e x t e n s i o n a l i t y . T h i s i d e a i s what

we s h a l l f o r -

m a l i z e here. The problems i n r e a l i z i n g t h i s p r o j e c t a r e a s f o l l o w s . Class i c a l s e t theory lends i t s e l f too n a t u r a l l y t o i n t e r p r e t a t i o n i n t e r m s o f s i m p l e l o c a t i o n . Elements i n a s e t a r e c l e a r l y d i s t i n g u i s h e d f r o m o n e a n -

25

26

F. G . ASENJO

o t h e r and a l l a r e r e l a t e d e x t e r n a l l y t o t h e s e t t h a t g a t h e r s them

together

t h r o u g h membership-a r e l a t i o n s h i p t h a t u n a v o i d a b l y performs a r a d i c a l s e l e c t i o n f r o m t h e e l e m e n t s ' many p r o p e r t i e s . I t i s a p p r o p r i a t e t o c o l l e c t b e r s i n t o a s e t , say, b u t one cannot c o l l e c t t h e e n t i t i e s o f

a

num-

physical

f i e l d i n t o a s e t without doing violence t o the wealth o f concrete r e l a t i o n s h i p s t h a t t h o s e e n t i t i e s have between one a n o t h e r . C u r r e n t p o i n t s e t

to-

p o l o g y i s o f no h e l p , e i t h e r , because i t i s a f o r m o f a p p l i e d s e t t h e o r y . W e need t o be a b l e t o f o r m a l l y p l a c e one s i m p l e l o c a t i o n i n t o a n o t h e r ; t h a t i s ,

U 2 as

we want t o f o r m a l i z e t h e f a c t t h a t s i m p l e l o c a t i o n V l has l o c a t i o n

w e l l , t h a t a p o i n t l o c a t e d a t V l a l s o moves i n t o t h e l o c a t i o n V 2 - a l t h o u g h not necessarily vice versa-just

as f o r c e s may be e i t h e r e x c l u s i v e l y o u t ~ O W L C M and

going o r exclusively ingoing i n a given r e l i o n ( t h e so-called

bi~bn)W . e s h a l l use t h e concept o f d i r e c t e d graph f o r t h i s purpose, v e r t i -

ces r e p r e s e n t i n g prima f a c i e s i m p l e l o c a t i o n s , and d i r e c t e d edges, symbol+ i z e d by t h e n o t a t i o n u. u, , r e p r e s e n t i n g m u l t i p l e l o c a t i o n s - e x p l i c i t e l y ,

Ui

1 J

i s t h e i n i c i a l v e r t e x and 4

V j t h e t e r m i n a l one i n t h e l o c a t i o n

Vi

of

w i l l be c a l l e d an o u t g o i n g edge a t V ; , and j: an i n g o i n g one a t V j . Extremecases w i l l be ( i ) those i n which e v e r y b i l o c a >>t i o n UiUj i s symmetric ( t h a t i s , t h e e x i s t e n c e o f ViVj implies t h a t o f >i n t h e g r a p h ) . and ( i i ) those i n which edges a r e t o t a l l y absent, mulVjVi into U

ViVj

t i p l e l o c a t i o n t h e n b e i n g reduced t o s i m p l e l o c a t i o n ( p o i n t s becoming

v e r t i c e s o r sets o f v e r t i c e s without d i r e c t e d l i n k s ) . I n general,

mere

points

h e r e w i l l be graphs, f i n i t e o r i n f i n i t e , and t h e p a t t e r n o f d i r e c t e d

edges

of a g i v e n p o i n t w i l l r e p r e s e n t t h e network o f m u l t i p l e l o c a t i o n s i n t r i n s i c t o t h a t p o i n t . A p o i n t , then, w i l l have s t r u c t u r e , i t s v e r t i c e s and d i r e c t ed edges making up i t s i n t e r n a l c o n s t i t u t i o n . F u r t h e r , i t i s e s s e n t i a l

to

the notion o f multiple location that t h i s internal c o n s t i t u t i o n n o t

be

closed, b u t open t o enlargement and a d d i t i o n a l s t r u c t u r a l a r t i c u l a t i o n . T h i s r e q u i r e s t h a t p o i n t s n o t be s e a l e d elements t o be c o l l e c t e d ;

rather;

they

must be s e t s o f some k i n d t h a t can be i n c l u d e d , embedded i n o t h e r l a r g e r p o i n t s . P o i n t s a r e n o t t o be t a k e n as i r r e d u c i b l e members o f a s e t , immodif i a b l e t o p o l o g i c a l atoms i n a neighborhood, b u t as e n t i t i e s

at

the

same

l o g i c a l l e v e l as t h a t o f any s e t o r neighborhood which c o n t a i n s them.Indeed, h e r e p o i n t s w i l l themselves be s e t s o f a s p e c i a l k i n d , and i n t u r n a

topo-

l o g i c a l space w i l l sometimes be one o r more p o i n t among o t h e r s . From f o r m a l i z i n g m u l t i p l e l o c a t i o n i t f o l l o w s t h a t no f i g u r e h a s a s i n g l e geometric s t r u c t u r e

-

an " a b s o l u t e appearence", t o use a p a r a d o x i c a l

e x p r e s s i o n t h a t d e s c r i b e s o u r o r d i n a r y , n a i v e i d e a o f form. The t o p o l o g i c a l

27

F O R M A L I Z I N G M U L T I P L E LOCATION

a s p e c t o f a f i g u r e i s r e l a t i v e t o v a r i o u s p o i n t s o f view, a m a t t e r o f topol o g i c a l perspective; t h a t i s , considered from d i f f e r e n t v e r t i c e s

within

p o i n t i n a f i g u r e ( t h o u g h t o f as an assemblage o f v e r t i c e s ) , d i f f e r e n t

a

topo-

l o g i c a l c o n f i g u r a t i o n s d e s c r i b e t h a t same f i g u r e . F u r t h e r , t h e r e a r e v e r t i ces f r o m which t h e f i g u r e cannot be d e s c r i b e d by any t o p o l o g i c a l c o n f i g u r a t i o n ; a l s o , t h e r e a r e v e r t i c e s w i t h o u t neighborhoods, as w e l l as v e r t i c e s

A t o r u s i s n o t a t o r u s f r o m a l l v i e w p o i n t s . I f t h i s appears b e w i l d e r i n g a t

f r o m which no t o p o l o g y a t a l l can be b u i l t (see examples i n S e c t i o n 4 ) . f i r s t , l e t us s t o p t o t h i n k : Why s h o u l d a f i g u r e have a u n i q u e

topological

configuration? I n t h e physical world t h e singleness o f a f i g u r e ' s i s a macroscopic p r e c o n c e p t i o n , a m a t t e r o f choosing f r o m

a

topology

w e a l t h o f ap-

pearences whose p r i m a r y o r secondary c h a r a c t e r depends o n v i e w p o i n t . Indeed, i t i s t h e conclusions o f c u r r e n t ultramicroscopic physics t h a t

f o r c e us t o

acknowledge t h i s b a s i c p e r s p e c t i v i s m o f space as a r o u t i n e p r o p e r t y o f matter.

I , A SET THEORY BASED ON

THE

P~OTION

INCLUSION,

OF

1, INCLUSION, The p r i m i t i v e i d e a s a r e t h o s e o f s e t , i n c l u s i o n ,

and b i n a r y r e l a t i o n .

C a p i t a l l e t t e r s s t a n d f o r s e t s and t h e i n c l u s i o n r e l a t i o n i s denoted by S . I n a d d i t i o n we have an u n l i m i t e d number o f symbols f o r b i n a r y

( R , F,

6,

relations

g, ...). A l s o , l e t us assume t h e f o r m a l a r i t h m e t i c o f non-negative

integers, including ordinary induction. DEFINITION 1. AXIOM 1 .

(EQuLLL~Y.) X = Y

stands f o r ( Z ) ( Z S X :ZS Y ) .

( E x t e ~ b i ~ ~ & y . ) (X)(Y)(X=Y->(Z)(XE

Z E YC Z ) ) .

Def. 1 e s t a b l i s h e s t h a t equal s e t s a r e t h o s e h a v i n g t h e same s u b s e t s and no o t h e r s , whereas Ax. 1 determines f u r t h e r t h a t equal s e t s a r e subsets o f t h e same s e t s . Obviously, DEFINITION 2. AXIOM 2 .

X = X.

( P m p e h inclubion.)

(NuR b&.)

(3x)(Y)(Y$

X c Y

stands f o r

X= Y & X

X & (Z)(Z # X E X E Z)).

# Y

.

28

F. G . ASENJO

T h e r e e x i s t s a s e t w i t h o u t subsets which i s i n c l u d e d i n e v e r y o t h e r s e t

.

e x c e p t i t s e l f . By Def. 1 and Ax. 1, t h i s s e t i s u n i q u e sented by @

.

I t w i l l be r e p r e -

Since t h e axiom o f e x t e n s i o n a l i t y guarantees t h a t a s e t i s u n i q u e l y determined by i t s subsets, t h e (where

Y,

X,

Z,

... a r e

n o t a t i o n S = {X,

Y,

...I

Z,

i s then i n order

a l l t h e subsets o f S i n f i n i t e o r i n f i n i t e number).

O f course, f o r e v e r y s e t S # @ , S i t s e l f and @ a r e t o be l i s t e d b e t w e e n b r a c k e t s , and because [@I i s meaningless, so i s t h e e q u a t i o n S = { @ I F u r t h e r , i t i s never t h e case t h a t

X = {XI.

AXIOM 3. (Re.~.l?eXivLty,Antinymrn&y, ( X ) ( X # @ +X

(X)(Y)(Z)(X

GX) & (X)(Y)(X

C Y & Y

=z

---$

and Tnam.iAvLty a6 l n d w i o n . )

cY

AXIOM 4. ( S e p c v r a t i ~ ~ . )( X ) ( 3 Y ) [ Y (V)(Z)((Z where

EX & 0 ( Z )

->z

& Y E X > -

x

= Y) &

X CZ).

SV) > -

& ( Z ) ( Z E X & $ ( Z ) ->Z

E X

Y SV)]

&

EY)

,

@(Z) i s any w f f w i t h one f r e e v a r i a b l e . N o t i c e t h a t Y may a l s o con-

t a i n s e t s U such t h a t

1 0 ( U ) . Obviously, i f @ ( X ) , t h e n X i t s e l f s a t i s f i e s

Ax. 4. I f 1 0 ( X ) , Ax. 4 guarantees t h e e x i s t e n c e o f a l e a s t s e t i n c l u d e d i n X t h a t c o n t a i n s a l l subsets o f X w i t h t h e p r o p e r t y 0 ( p l u s any o t h e r

sets o f

x

w i t h o u t such p r o p e r t y b u t n o t s e p a r a b l e f r o m Y because o f

subtheir

b e i n g i n c l u d e d i n some subset Z o f X w i t h t h e p r o p e r t y 0). The n o t a t i o n Y = ( 2 : Z c X & @ ( Z ) > i s now j u s t i f i e d : Y i s t h e l e a s t subs e t o f X t h a t c o n t a i h s a l l t h e subsets o f X t h a t s a t i s f y

A X I O M 5 . (Expamian.) ( X ) ( ~ Y ) ( ~ Z ) ( cX Y

&

z

EY &

@(Z).

z$x

&

x $ z).

As a consequence o f t h i s axiom t h e r e i s no c l a s s o f a l l s e t s . there e x i s t s a t l e a s t a countable i n f i n i t y o f sets. I n f a c t , i n f i n i t y o f c o u n t a b l y i n f i n i t e sequences o f s e t s .

Also,

there i s

an

Ax. 5 c a n b e a p p l i e d suc-

c e s s i v e l y t o a s s e r t t h e e x i s t e n c e o f n e s t e d sequences o f d ' i s t i n c t s e t s , each properly included i n the following ones(chains),aswell

as t o a s s e t t h e ex-

i s t e n c e o f s e q u e n c e s o f s e t s t h a t a r e p a i r w i s e incomparable w i t h r e s p e c t t o i n c l u s i o n (antichains).These a r e t h e two extreme p o l e s i n t h e spectrum o f a l l t h e p o s s i b l e a r b i t r a r y sequences o f s e t s whose e x i s t e n c e d e r i v e s f r o m t h i s axiom. L e t

Expl(X,U) i n d i c a t e t h a t t h e s e t U i s o b t a i n e d by a p p l y i n g Ax. 5

t o X once, U b e i n g e i t h e r a s e t t h a t p r o p e r l y c o n t a i n s X o r a s e t incompable to X

. Let

Expk(X,U) i n d i c a t e t h a t U i s o b t a i n e d f r o m X a f t e r k a p p l i -

29

FORMALIZING MULTIPLE LOCATION

c a t i o n s o f Ax.5 (where k i s a non-negative i n t e g e r and Exp,,(X,Y)

denotes X

i t s e l f ) , and where t h e k s u c c e s s i v e c h o i c e s a r e made e i t h e r b y f o l l o w i n g some r e c u r s i v e schema o r a t random. (In&kLty).

AXIOM 6 .

(X)(3Y)(Z)(U)(X=

= Y)):

Expk+l(X,w

Y & (€xpk(X,Z) E Y >-

The i n f i n i t e s e t Y (denoted Exp ( X ) ) , whose e x i s t e n c e i s a s s e r t e d

by

t h i s a x i o m , c o l l e c t s a l l t h e s e t s o b t a i n a b l e by a f i n i t e number o f s e q u e n t i a l a p p l i c a t i o n s o f Ax. 5 t o a g i v e n s e t X. N o t i c e t h a t Exp ( X ) does n o t c o l l e c t a l l t h e s u p e r s e t s o f X, b u t a t most o n l y a c o u n t a b l e sequence o f them ( p l u s a l l t h e subsets o f each t e r m o f such sequence). Let

Seq ( X ) denote a p a r t i c u l a r i n f i n i t e sequence X,X1,X2,

o f s e t s o b t a i n e d by successive a p p l i c a t i o n s o f Ax.5 s e t X o f any such sequence,but

...

X,,...

($3 c o u l d b e t h e i n i t i a l

i t c o u l d n o t o c c u p y any o t h e r p l a c e i n t h e s e -

quence). L e t Seqk(X,U) i n d i c a t e

the

k - t h term o f s u c h s e q u e n c e

with

S e q o ( X , U ) = X.

AXIOM 7 .

(Union

06

a sequence) >-

Seqk+l

( X ) ( j Y ) ( Z ) ( U ) ( X 5 Y & (Seq,(X,Z) C Y

'

('9

*

O b v i o u s l y t h e u n i o n o f t h e terms o f a sequence (denoted U s e 4 ( X ) ) a subset o f

is

Exp ( X ) .

AXIOM 8 .

(Union).

AXIOM 9 .

(Iiit('h)eCfiuit)

(X)(Y)(3Z)(U)(U

S X V

c

Y

=

U S Z).

( X ) ( Y ) ( 3 Z ) ( U ) ( U C X & U S Y E U C 2).

Union and i n t e r s e c t i o n , which a r e u n i q u e l y determined, w i l l be denoted by X

u Y and X n Y , r e s p e c t i v e l y .

It i s c l e a r t h a t both operations are

a s s o c i a t i v e and s a t i s f y t h e d i s t r i b u t i v e laws.

2 ELEMENTHOODAND DEFINITION 3.

MEMBERSHIP I

(EYement) E

(X)

stands f o r

(X

# $3) I (Y)(Y # 0 >-

Y$XVY=X). Elements a r e nonempty s e t s w i t h o u t nonempty p r o p e r subsets. s e t i s n o t an element.

The n u l l

30

F. G. ASENJO

AXIOM 1 0 .

(RegLLeatLity)

(X)(X

# 0 ->(3Y)(Y

E X & E(Y)))

Every nonempty s e t c o n t a i n s a t l e a s t one element

(eventually itself

only).

AXIOM 1 1 .

(EYcme.nt expamion)

Y & Z r Y &

(X)(3Y)(3Z)(X=

z

$X &

E (Z)):

Hence, t h e r e i s no s e t o f a l l elements, and t h e r e i s a t l e a s t a c o u n t a b l e i n f i n i t y o f them.

AXIOM 1 2 .

(Paihing)

(X)(Y)(3Z)(E

u

v u

= X

= Y

( X ) & E ( Y ) ->

(U)(U 5

Z :

v u =@I).

There e x i s t s t h e s e t t h a t c o n t a i n s e x c l u s i v e l y a g i v e n p a i r o f e l e m e n t s ( p l u s 0). The b r a c k e t n o t a t i o n { x , y l i s now i n order;small elements, and

.

Ix.xl

letters indicate

{x,gl i s t h e u n i q u e s e t t h a t c o n t a i n s x, y , and 0.

DEFINITION 4.

1x1

is

S ( X ) stands f o r X U Y where Y i s any s e t such

(SucceAboh)

t h a t E ( Y ) and

Y

4 X.

The successor o f a s e t i s n o t u n i q u e l y determined, b u t by Ax. 11 Ax. 8, c o u n t a b l y i n f i n i t e sequences o f s e t s can be assumed t o e x i s t that, begining with a given set, every s e t t h a t follows i s the

and such

successor

o f t h e p r e c e d i n g one.

DEFINITION 5.

(Membetbkip) X E Y

X E Y & E (X)

stands f o r

Only elements a r e members.

THEOREM 1 . PROOF:

( X ) ( E(X) ->

X E X)

Since e v e r y s e t i s a subset o f i t s e l f , e v e r y element i s a member

of itself.

THEOREM 2. PROOF:

$'

X # Y ->X

e

Y & Y

X.

Antisymmetry o f i n c l u s i o n and Def. 3.

AXIOM 1 3 . where

E (X) & E (Y) &

(Compfiehenbion)

(3X)(Y)(Y € X

= @(Y)),

i s any w f f w i t h one f r e e v a r i a b l e . The axiom a s s e r t s t h e e x i s t e n c e

o f a s e t c o n t a i n i n g a l l t h e elements t h a t have t h e p r o p e r t y

@

.

31

FORMALIZING M U L T I P L E LOCATION

THEOREM 3 .

The n e t

06 aU

e l m e n t n X which ate not membenn

06

.thm&vhen

0 empty.

PROOF: T h e o . 1 f o r

X # pI, a n d t h e f a c t t h a t

3, CARTESIANPRODUCT, DEFINITION 6.

lE(@) b y D e f . 3 .

FUNCTIONS, CARDINALITY, ORDER,

(Cahtedian ptoduct)

Given a n y t w o

sets

A and 8,

t h e i r CatLtebian phoduct i s t h e b i n a r y r e l a t i o n d e f i n e d as f o l l o w s : ( A x B)(X,Y) 5 X C A & YE 23). ( C a r t e s i a n p r o d u c t s a r e n o t s e t s . ) 0 x 0 h o l d s f o r no p a i r o f s e t s ,

(0,pI)

included.

DEFINITION 7.

(Comhenpondenchen) Given a C a r t e s i a n p r o d u c t ( A x B), comhenpondence between A and B i s any b i n a r y r e l a t i o n R t h a t s a t i s f i e s R (X,Y) ->(A

a

x B)(X,Y).

DEFINITION 8. (FunCtion6) A 6unCtion o n A i n t o B i s a correspondence F between A and B such t h a t f o r each X E A t h e r e i s one and o n l y one Y 5 B such t h a t F ( X , Y ) . (X)(XCA

I n symbols:

((3Y)f(X,Y)

> -

Iff o r each Y E B

& ( Y ) ( Z ) ( F (X,Y)

& F(X,Z)+Y

= Z))).

f o r which t h e r e i s an X E A such t h a t F(X,Y)

there

i s o n l y one such X, t h e f u n c t i o n i s c a l l e d monomorphic. I f f o r e v e r y

Y EB

t h e r e i s a t l e a s t one X = A such t h a t f ( X , Y ) ,

the functionF i s calledsur-

j e c t i ve.

DEFINITION 9. n a l i t y (denoted

Two s e t s A and B have t h e same c a r d i -

(Catdinality)

IAI

= 181) i f t h e r e e x i s t s a mononiorphic a n d

surjective

f u n c t i o n on A i n t o B . I f t h e r e e x i s t s a monomorphic and s u r j e c t i v e

function

on A i n t o a subset o f 8, b u t n o t one o n 8 i n t o a s u b s e t o f A , t h e n A i s s a i d t o have l e s s e r c a r d i n a l i t y t h a n B ( d e n o t e d I A ( < 181). Obviously, f o r a l l X,

1x1 5 lusty AXIOM 14.

(XI1

5 IExp

(X)I.

(ToaM otdeh)

&

( X ) ( 3 R ) ( ( Y ) ( Y C-X ->R(Y,Y)

( Y ) ( Z ) ( Y C X & Z C X +(R

(Y,Z)

& R(Z,Y)

( Y ) ( Z ) ( U ) ( Y C - X & Z C X & U =X->(R(Y,Z) (Y)(Z)(Y E X & Z

=X

->

Every s e t can be t o t a l l y ordered.

R(Y,Z)

V R(Z,Y))).

->Y

= Z))

&

& R(Z,U) -+R(Y,U))

&

32

F.

G. ASENJO

11, THE TOPOLOGYOF MULTIPLE 4

0

A

LOCATION,

GRAPH TOPOLOGY I

Henceforward, t h e n o t i o n s o f s e t , element, i n c l u s i o n , and sequence a r e those p r e s e n t e d i n P a r t I.A t o p o l o g i c a l space helative .to a uehtex V s h a l l be a d i r e c t e d graph X V

, t h e p r o d u c t graph o f a l l t h e graphs l a b e l e d p o i n t s and a sequence TV o f

r e l a t i v e t o V ( n o t e v e r y subgraph o f Xy i s a V - p o i n t ) , subsets o f XV

, called

neighborhoods, t h a t s a t i s f y

the definition

axioms g i v e n below. Graphs a r e a r r a y s o f v e r t i c e s (elements)

and

and

directed

w?

edges ( i n t r o d u c e d i n t h e usual way, though n o t as a C a n t o r i a n o r d e r e d p a i r o f elements, b u t as elements themselves t h a t a r e symbolized ). It i s understood t h a t e v e r y graph t h a t c o n t a i n s >V i and V j , a l t h o u g h n o t n e c e s s a r i l y V . V . which o f i t s subsets a r e V-points.

J 1

ViVj

as an e l e m e n t a l s o c o n t a i n s

. Given X V ,

it i s

determined

I t i s assumed, f u r t h e r , t h a t i t i s always

p o s s i b l e t o a s c e r t a i n f o r a g i v e n v e r t e x i n a g i v e n subgraph w h e t h e r t h e number o f o u t g o i n g s edges i s g r e a t e r , equal, o r l e s s t h e n t h e number o f i n g o i n g edges ( o r whether t h o s e two numbers a r e incomparable).

Note t h a t , i n

accordance w i t h P a r t I, a s e t o f graphs i s t h e i r own p r o d u c t graph,

which

i n c l u d e s a l l t h e new graphs t h a t can be formed w i t h t h e a s s o r t e d v e r t i c e s and edges o f t h e g i v e n graphs. The t o p o l o g i c a l space XV s h a l l be, t h e n , b o t h a d i r e c t e d graph and a set,. t r u e a l s o o f p o i n t s and neighborhoods. D E F I N I T I O N 1 . G i v e n a v e r t e x W i n XV. a n e i g h b o h h o o d o f W

(denoted

N v ( W ) ) i s any p o i n t o r p r o d u c t graph o f p o i n t s o f X V t h a t

( i ) c o n t a i n s lo,

edges o f W i n

N v ( W ) i s greater

and such t h a t

(ii)t h e number o f o u t g o i n g

t h a n o r equal t o t h e number o f i t s i n g o i n g edges. (Note t h a t j u s t as n e i t h e r t h e subgraph

nor thesupergraph o f a p o i n t

a r e n e c e s s a r i l y p o i n t s , n e i t h e r a r e t h e subgraph

n o r t h e supergraph

of

a

neighborhood n e c e s s a r i l y neighborhoods.) L e t TV be a sequence whose terms a r e a l l neighborhoods and such t h a t e v e r y neighborhoods o f XV i s a subset o f a t e r m o f T V

. Since the

nance o f o u t g o i n g edges i s , p r e s e r v e d by f i n i t e o r i n f i n i t e unions, i s t e n c e o f TV f o l l o w s . TV i s c a l l e d a Ropology r e l a t i v e t o lowing are s a t i s f i e d . AXIOM 1 .

XV i s a .term o f TV

V

predomit h e ex-

i f the

fol-

z

33

FORMAL I I NG MULT I PLE LOCAT I ON

AXIOM 2. Given two neighborhoods Nv(W) and N;(W), t h e i r i n t e r s e c t i o n i s also a

.

V-neighborhood o f W

L e t us l o o k a t two v e r y s i m p l e examples g i v e n h e r e m e r e l y t o add

some

w,

m:

i n t u i t i v e i n t e r p r e t a t i o n t o t h e p r e v i o u s concepts. Consider t h e g r a p h X , > composed o f t h e v e r t i c e s V1, V 2 , V 3 , and t h e edges V3V2: and >L e t t h e p o i n t s of Xul be V1;V2.; V1V2 ( i . e . , t h e subgraph composed o f t h r e e >elements V1, V 2 , V I V , > ) ; and V3V2 L e t t h e p o i n t s o f Xy2 be V3; and -> >>V3V1 F i n a l l y , l e t t h e p o i n t s o f Xy3 be V 1 ; V 2 ; V 1 V 2 ; VjV2;and The

VIVl;

.

w;.

.

n e i g h b o r h o o d s i n X V ( t h e p r o d u c t graph o f i t s p o i n t s ) a r e t h e following. 1 Neighborhoods o f V 2 : V 2 ; V 2 , V1: V1; Vlr V2; and V1V2> >V3V2;,U3, v2, >-v1v2. These e i g h t n e i g h V1. Neighborhoods o f V 3 : v3u;; vl,

.

Neighborhoods o f

borhoods c o n s t i t u t e a t o p o l o g y f o r t h e space Xyl

. The

a t o p o l o g y , a l t h o u g h V 2 has no neighborhoods. Xu3 space. L e t us now c o n s i d e r t h e graph v e r t i c e s and o f a l l t h e edges

is

space Xy2 a l s o has a not

a

toPological

X , composed o f a l l p o s i t i v e i n t e g e r s as

k , k+m>

f o r a l l p o s i t i v e i n t e g e r s k, m 2 1.

F o r each v e r t e x k , l e t t h e p o i n t s o f xk be t h e edges

k,k+m> f o r

e v e r y m > 1.

Only k has neighborhoods i n Xk, b u t t h e s e f o r m a t o p o l o g y f o r xk.

5 , CLOSURE,

D E R I V E D SET, BOUNDARY,

DEFINITION 2 . Given a V - p o i n t p and a neighborhood N V ( W ) , N ~ ( W ) i s c a l l e d a V-neighborhood o f P i f f P E NV(W) . DEFINITION 3. L e t

X Y be a t o p o l o g i c a l space and S a subset o f X y ,

p o i n t P i s s a i d t o be a L i m i t point o f S i f f e v e r y

a

V-

V - n e i g h b o r h o o d o f P con-

t a i n s a t l e a s t one v e r t e x W o f S n o t i n P.

DEFINITION 4. The p r o d u c t graph o f a l l c a l l e d t h e dekiwed beX o f S,

DEFINITION 5. A s e t

S E Xu

c a l l e d d o h e d . We s h a l l c a l l

l i m i t points o f a set

S EXv

is

denoted S ' .

S

u

t h a t contains a l l i t s l i m i t S' =

s t h e d o b w r e o f S.

DEFINITION 6. T h e boundafiy o f a s e t

points i s

S s X ~(denoted B d ( S ) )

i s the

p r o d u c t graph o f t h o s e p o i n t s common t o t h e c l o s u r e o f S and t h e c l o s u r e o f

34

F . G . ASENJO

XV --S,

t h e l a t t e r b e i n g t h e graph spanned by a l l v e r t i c e s and edges n o t i n

S ( n o t e t h a t a l t h o u g h S and X V - S

have no edges i n common, t h e y can have

some v e r t i c e s i n common).

DEFINITION 7. The i n t e t L i o h o f a s e t

is S

S =Xu

-

Bd(S)

Obviously, i f S 1 c S 2 , t h e n S; ES;. Hence, i f Sl a n d S2 a r e b o t h c l o s e d (and t h e r e f o r e S ' c S and S; c S,), t h e n (S1n S,)'=S; and (S1f l S2)l 1- 1 c S; . B u t e v e r y l i m i t p o i n t o f S 1 n S 2 i s a l s o a l i m i t p o i n t o f S1 as w e l l as a l i m i t p o i n t o f S 2 , t h e r e f o r e (S1 n S 2 ) ' s S; Il S;. We t h e n have t h e following. The intemec-tion

THEOREM 1 .

(thehc6ote Rhc boundatry

06

6, HOMEOMORPHISM,

05

a A&

a 6 i n i t e sequence

c l o s e d h e h h closed

CONNECTEDNESS, COMPACTNESS

DEFINITION 8. Given two t o p o l o g i c a l spaces X

06

.LA c t o s e d ) .

X V and Xw f r o m t h e same graph

( o r X V and Yw f r o m d i f f e r e n t graphs X and Y ) , a f u n c t i o n 6 : X V 3 X w ( o r

6:XV > -

Yw r e s p e c t i v e l y ) which maps v e r t i c e s i n t o v e r t i c e s , and edges i n t o

edges o f c o r r e s p o n d i n g v e r t i c e s i n a d i r e c t i o n - p r e s e r v i n g manner ( i . e . , w i t h >each edge V1V2 mapped i n t o t h e edge d(Vl)d(V2<) i s s a i d t o be continu-

ow a t a v e r t e x V O 6-'(NM(6(VO)))

= XV i f f f o r

every

W-neighborhood Nw(6(V0)) of

o f a l l v e r t i c e s and edges whose image under continuous i f

~(VO),

i s a V-neighborhood o f V o , where 6 - l ( N w ( 6 ( V 0 ) ) ) i s t h e graph

6

6

i s Nw(d(Vo)).

6

i s s a i d t o be

i s c o n t i n u o u s a t each v e r t e x o f XV.

DEFINITION 9. Two t o p o l o g i c a l spaces a r e s a i d t o be h o m e o m o t p h i c i f f t h e r e e x i s t i n v e r s e f u n c t i o n s 6:XV ->Xw g:Yw ->Xv

r e s p e c t i v e l y ) such t h a t

6

( o r 6:XV ->

Yw) and g:Xw ->XV(or

and g a r e c o n t i n u o u s ;

6

andg arethen

c a l l ed h a m e o m o ~ p ~ ~ .

DEFINITION 10. Two nonempty s e t s S1 and s a i d t o be nepcmzted i f f S1 ,n S 2 = S1 n S' 2

S2 o f a t o p o l o g i c a l space XV a r e =

S'1 n S2

=

@

.

DEFINITION 11. A s e t which i s n o t t h e u n i o n o f two separated s e t s i s c a l l ed

connected.

35

FORMALIZING MULTIPLE LOCATION

A cloned b e t S 0 connected 4 6 6 .it 0 not t h e u n i o n nonempty d i b j o . i n t cloned nu%.

THEOREM 2 .

PROOF: Assume S = A U 8 where A and 8 a r e nonempty =

0 and A

fl 8' = A

n

S fl 8' 5 A f l 8 =

0

two

and d i s j o i n t c l o s e d

A ' fl S E A and 8' n S C 8. Then A' il 8 = A ' ll S

s e t s . We have

06

. Therefore, A

and 6

8 5 A fl 8

are

sepa-

r a t e d and S i s n o t connected. I f S i s n o t connected, t h e n S = A u 8 where A and 8 a r e separated sets. Then A' fl S = A ' fl ( A U 8 ) = ( A '

n

A ) U (A'

n B)

= A ' fl A

5 A , which means

t h a t A i s c l o s e d . S i m i l a r l y , so i s 8. THEOREM 3 . The cConme

06 a connected b e t

S 0 connected.

PROOF: By c o n t r a d i c t i o n : assume S n o t connected. Since nonempty subsets o f separated s e t s a r e c o r r e s p o n d i n g l y separated, i t i s easy t o show t h a t

then

S c o u l d n o t be connected ( a g a i n s t t h e premise o f t h e theorem).

COROLLARY 4 .

mted

he%%0

A connected b e t t h a t 0 contained in t h e u n i o n contained i n one 06 t h e .two b u % .

06

.two bepa-

C E F I N I T I O N 1 2 . A s e t S i s compact i f f i t i s f i n i t e (composed of a f i n i t e number o f v e r t i c e s and edges) o r i f e v e r y i n f i n i t e s u b s e t o f S has a

non-

empty d e r i v e d s e t . THEOREM 5 . I h e nubnet

a 6.inite numbeh

06

06 a compact net 0 compact. Fuhthet, t h e

compact

PROOF: L e t S = S1 U S

befi

*...u Sn

union

06

0 compact. be t h e u n i o n of n c0mpac.t s e t s . I f T i s an

i n f i n i t e subset o f S, t h e n a t l e a s t one of t h e s e t s T n S j i s i n f i n i t e . S i n c e T

n Si C_Sj , and

Si i s compact, T possesses a t l e a s t one l i m i t p o i n t .

(NOTE: As t h e s e d e f i n i t i o n s and theorems demonstrate, general t o p o l o gy can be c o n s t r u c t e d w i t h o u t e x c e s s i v e d i f f e r e n c e s o n a s e t t h e o r y based on i n c l u s i o n ; t h i s shows t h a t membership i s n o t i n i t s e l f an i n d i s p e n s a b l e r e l a t i o n s h i p f o r t o p o l o g y . Furthermore,to base t o p o l o g y on i n c l u s i o n a l s o i n d i c a t e s a way i n which t o make t o p o l o g i c a l n o t i o n s p l a y a b a s i c graph t h e o r y - g e n e r a l

t o p o l o g y and graph t h e o r y b e i n g d i s c i p l i n e s

f a r have v e r y l i t t l e t o do w i t h one a n o t h e r . )

role in t h a t so

F . G . ASENJO

36

REFERENCES

I

Asenjo, F. G. 1962, El Todo y las Partes, E d i t o r i a l Tecnos, Madrid. Whitehead, A. N. 1967, Science and the Modern World, T h e F r e e P r e s s ,

New Y o r k .

Department o f M a t h e m a t i c s University of Pittsburgh P i t t s b u r y h , Pennsylvania, U.S.A.