Domination in Analysis

P UBLICATIONS DU D ÉPARTEMENT DE MATHÉMATIQUES DE L ^YON

D ^ENNY G ^ULICK

Domination in Analysis

Publications du Département de Mathématiques de Lyon, 1973, tome 10, fascicule 3 , p. 35-64

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

(1973, B o r d e a u x ) p. 189-218

Département de M a t h é m a t i q u e s Lyon 1973 1.10-3

D o m i n a t i o n i n A n a l y s i s

Denny G u l i c k

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

L e t T "be a c o m p l e t e l y r e g u l a r H a u s d o r f f s p a c e⁹ and l e t T

R = t h e s e t o f a l l r e a l - v a l u e d f u n c t i o n s on T .

L S C ( T ) = t h e s e t o f a l l l o w e r s e m i - c o n t i n u o u s f u n c t i o n s i n R T

C ( T ) = t h e s e t o f a l l c o n t i n u o u s f u n c t i o n s i n R •

C o n s i d e r t h e f o l l o w i n g t h r e e c o n d i t i o n s on an a r b i t r a r y such T . C o n d i t i o n A . F o r e a c h h€R , t h e r e i s an f £ C ( T ) such t h a t f ^ h . C o n d i t i o n B . F o r e a c h g € L S C ( T ) , t h e r e i s an f é C ( T ) such t h a t f * g .

C o n d i t i o n C . F o r e a c h h*R , t h e r e i s a g c L S C ( T ) such t h a t g ^ h . When d o e s T s a t i s f y e a c h o f t h e j u s t - m e n t i o n e d c o n d i t i o n s ? I f we r e p l a c e ^ by = i n t h e c o n d i t i o n s , t h e n t h e r e s p o n s e s a r e s i m p l e . I n d e e d , C o n d i t i o n A b e c o m e s C ( T ) = R , C o n d i t i o n B b e - comes C ( T ) = L S C ( T ) , and C o n d i t i o n C b e c o m e s L S C ( T ) = R^T, and e a c h o f t h e s e o c c u r s p r e c i s e l y when T i s d i s c r e t e .

H o w e v e r , f i n d i n g t h o s e T w h i c h s a t i s f y t h e C o n d i t i o n s A , B , and C a s a c t u a l l y p o s e d d o e s n o t i n v o l v e such a t r i v i a l t a s k , and we d e v o t e t h i s a r t i c l e t o an a t t e m p t a t c h a r a c t e r i z i n g i n t e r m s

o f T when C o n d i t i o n s A , B , o r C h o l d .

Of c o u r s e C o n d i t i o n s A , B, and C h o l d w h e n e v e r T i s d i s c r e t e . O n e⁹s f i r s t i m p r e s s i o n m i g h t b e t h a t i f T i s ' n o t d i s c r e t e t h e n T w o u l d n o t s a t i s f y any o f t h e c o n d i t i o n s . The s i t u a t i o n w h i c h o c c u r s

35

when T = [o^flJ r e i n f o r c e s t h i s i m p r e s s i o n * I n d e e d , l e t ( * ⁿ ) ^ i enumerate t h e r a t i o n a l s i n [ 0, l ] , and l e t

i

n, t = t f o r some n<= W 0, a l l o t h e r t€ J p, l ] f l / t , O ^ t ¿ 1

« b C t ) - /

C O , f o r t « 0

Then h.Q £ R and i s unbounded on e v e r y non-empty open s u b s e t o f [0,1], and g Q £ : L S C ( T ) and i s unbounded on [ o, l ] • T o g e t h e r t h e s e f a c t s i m p l y s t r a i g h t a w a y t h a t T = jj0,lj s a t i s f i e s none o f t h e C o n d i t i o n s A^f B , o r C.

I n S e c t i o n 2 we show t h a t i t i s c o n s i s t e n t w i t h t h e u s u a l axioms o f s e t t h e o r y t o assume t h a t no n o n - d i s c r e t e T e x i s t s s a t i s f y i n g C o n d i t i o n A . We t h e n t a c k l e t h e p r o b l e m o f t h o s e T w h i c h s a t i s f y C o n d i t i o n B, and show t h a t i f T d o e s s a t i s f y C o n -

d i t i o n B, t h e n T must be a P - s p a c e . H o w e v e r , a l l t h o s e n o n - d i s c r e t e P - s p a c e s we know do n o t s a t i s f y C o n d i t i o n B, and i t r e m a i n s u n a n s w e r - ed w h e t h e r o r n o t any n o n - d i s c r e t e o n e s e x i s t . We end S e c t i o n 2 by s t u d y i n g t h o s e T w h i c h s a t i s f y C o n d i t i o n C. I n p a r t i c u l a r we show t h a t i f T i s d e n u m e r a b l e , o r i f T i s d i s c r e t e e x c e p t f o r one e l e m e n t , t h e n i t s a t i s f i e s C o n d i t i o n C, w h i l e i f T c o n t a i n s no i s o l a t e d p o i n t s and e i t h e r i s n o n - m e a g e r s a t i s f y i n g t h e s e c o n d

c o u n t a b i l i t y axiom o r i s l o c a l l y c o m p a c t , t h e n T d o e s n o t s a t i s f y C o n d i t i o n C#

I n S e c t i o n 3 * e d i s c u s s C o n d i t i o n C i n a l o c a l l y c o n v e x s p a c e s e t t i n g . I f C⁰( T ) d e n o t e s t h e s p a c e o f r e a l - v a l u e d c o n t i n u o u s f u n c t i o n s on T^t endowed w i t h t h e s i m p l e c o n v e r g e n c e t o p o l o g y , t h e n we show t h a t T s a t i s f i e s C o n d i t i o n C p r e c i s e l y when t h e b i d u a l C " ( T ) o f C(T) i s i d e n t i f i e d w i t h E^T as a l o c a l l y c o n v e x s p a c e ,

s s

We a l s o d e l v e i n t o v a r i o u s p r o p e r t i e s o f C » ( T ) and C " ( T ) , i n c l u d - i n g when t h e y a r e b a r r e l e d o r when t h e y a r e b o m o l o g i c a l *

We c o n c l u d e t h e p a p e r lay c o l l e c t i n g t o g e t h e r some o p e n q u e s t i o n s . 36

The p r e s e n t p a p e r has grown out o f o u r r e c e n t a r t i c l e [ 5 ] , i n w h i c h we c o n c e n t r a t e d on t h e d u a l i t y t h e o r y f o r C^{t t}( T ) . A t t h e

C o n f e r e n c e i n Bordeaux we d i s c u s s e d and o u r a t t e m p t t o r e n d e r t h a t t a l k i n a somewhat more g e n e r a l c o n t e x t f o r t h e s e P r o c e e d i n g s r e s u l t e d i n t h e p r e s e n t p a p e r . The o v e r l a p b e t w e e n t h e two a r t i c l e s r e s i d e s p r i m a r i l y i n t h e e a r l y p a r t o f S e c t i o n 3 o f t h e p r e s e n t

p a p e r , w h e r e we m e r e l y o u t l i n e c e r t a i n o f t h e d u a l i t y p r o p e r t i e s o f C ( T ) . N e v e r t h e l e s s , i t i s our i n t e n t i o n t h a t t h e t w o p a p e r s c a n "be r e a d i n d e p e n d e n t l y , and t h a t t h e y s h o u l d s e r v e as companions o f one a n o t h e r *

B e f o r e we b e g i n t h e p a p e r p r o p e r we d e s c r i b e our n o t a t i o n a l c o n v e n t i o n s . T h r o u g h o u t t h e p a p e r T s t a n d s f o r a c o m p l e t e l y r e g u l a r H a u s d o r f f s p a c e . I f S and S¹ a r e s u b s e t s o f T , t h e n

S \ S ^f » j t € S : t < £ S^fJ , w h i l e d e n o t e s t h e c h a r a c t e r i s t i c f u n c t i o n o f S i n T • We r e c a l l f r o m [V] t h a t T i s e x t r e m a l l y d i s c o n n e c t e d i f e v e r y open s e t i n T has an open c l o s u r e i n T , a n d T i s a P - s p a c e p r o v i d e d t h a t c o u n t a b l e i n t e r s e c t i o n s o f open

s e t s i n T a r e a l w a y s open ( a n d P - p o i n t s a r e d e s c r i b e d a c c o r d i n g l y ) . A s i n [W], £T i s t h e S t o n e - C e c h c o m p a c t i f i c a t i o n o f T , w h i l e

v T i s t h e r e p l e t i o n ( o r t h e H e w i t t r e a l - c o m p a c t i f i c a t i o n ) o f T . We w r i t e h > 0 f o r t h e s t r i c t l y p o s i t i v e h € B , meaning t h a t h ( t ) > 0 f o r a l l t £ T • I f ne N ( w i t h N t h e s e t o f p o s i t i v e

i n t e g e r s ) , t h e n t h e n t h t r u n c a t i n g f u n c t i o n & ⁿ i s d e f i n e d on t h e r e a l s by

f r , i f I r U n

0

( r )

n Lnr/|r( , i f | r t > n

F o r compact T t h e s p a c e M ( T ) i s t h e c o l l e c t i o n o f a l l r e a l - v a l u e d bounded Radon m e a s u r e s on T , and t h e p o i n t mass a t t £ T i s .

I f £ i s a l o c a l l y c o n v e x s p a c e and A Q B , t h e n co b a l A i s s h o r t h a n d f o r t h e c o n v e x b a l a n c e d h u l l o f A , and

co b a l A - j | o ^a f ⁿ : f^a 6 A , ¿ 1 ^ 1 * 1 , and m€ N } .

The c o l l e c t i o n o f a l l r e a l - v a l u e d c o n t i n u o u s l i n e a r forms on E i s t h e d u a l E^f o f E , and we c l o t h e B^f i n t h e d u a l t o p o l o g y , w h i c h i s t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e on t h e bounded s u b s e t s o f E . The b i d u a l o f E i s E" , and by d e f i n i t i o n E" i s t h e d u a l o f E^f w i t h i t s d u a l t o p o l o g y . I f A ^ E , t h e n t h e p o l a r A^ i n B ^F o f A i s g i v e n by t h e f o r m u l a

A^ mfye** : [ < p ( f ) | * 1 , f o r a l l f e A ^ •

I f В С Е¹ , t h e n t h e p o l a r B? i n E o f Б i s g i v e n by t h e f o r m u l a B? = ^ f € E : | < p ( f ) | ^ l , f o r a l l <p6 •

F i n a l l y , i n our p r e s e n t a t i o n we employ " i f f " f o r the more cumbersome " i f and o n l y i f " .

2. A n a l y s i s o f C o n d i t i o n s A , B , and C

I n o r d e r t o d i s c u s s C o n d i t i o n A , we f i r s t r e c a l l t h a t a s e t T i s m e a s u r a b l e ( " n o n - m o d e r e " i n F r e n c h £2]) i f f t h e r e e x i s t s a n o n - t r i v i a l c o u n t a b l y a d d i t i v e measure on t h e power s e t o f T w h i c h assumes o n l y t h e v a l u e s 0 and 1 and such t h a t j * - ({t$ = 0 f o r a l l t € T ( s e e [ * + ] ) • A c a r d i n a l i s c a l l e d m e a s u r a b l e i f f t h e r e

e x i s t s a m e a s u r a b l e s e t T w i t h t h e same c a r d i n a l i t y . The f o l l o w i n g t h e o r e m p r e s e n t s s e v e r a l s t a t e m e n t s e q u i v a l e n t t o t h e s t a t e m e n t t h a t t h e s e t T i s m e a s u r a b l e , and r e s u l t s i n p a r t f r o m a d i s c u s s i o n w i t h M. Haddad, t o whom we w o u l d l i k e t o e x p r e s s o u r a p p r e c i a t i o n . We n o t e b e f o r e s t a t i n g Theorem 1 t h a t c e r t a i n o f t h e s t a t e m e n t s i n i t a r e a l r e a d y known t o be e q u i v a l e n t ( s e e C h a p t e r 12 o f [*+J).

T H E O R E M 1. L e t T be a s e t , l e t T^d be t h e s e t T w i t h t h e d i s ­ c r e t e t o p o l o g y , and l e t T^Q = T U ^ Q J , w i t h " t^Q^ T . Then t h e f o l ­ l o w i n g s t a t e m e n t s a r e e q u i v a l e n t :

a . T h e r e e x i s t s a n o n - d i s c r e t e t o p o l o g y t on TQ such t h a t TQ s a t i s f i e s C o n d i t i o n A .

b . T h e r e e x i s t s a n o n - d i s c r e t e t o p o l o g y t on TQ such t h a t T i s d i s c r e t e f o r t , and such t h a t i f ^{(un) ^ L i i s a Q y} p a r t i t i o n

m

o f T , t h e n U Uⁿ i s a d e l e t e d n e i g h b o r h o o d o f t Q , f o r some m.

c . T h e r e e x i s t s a n o n - d i s c r e t e t o p o l o g y ^ on TQ such t h a t T i s d i s c r e t e f o r X , and such t h a t i f (^{u n}) n = l ^{i s} p a r t i t i o n o f T , t h e n U^m i s a d e l e t e d n e i g h b o r h o o d o f t Q , f o r some m.

d . T h e r e e x i s t s a n o n - d i s c r e t e t o p o l o g y on T^Q w h i c h r e n d e r s T^Q an e x t r e m a l l y d i s c o n n e c t e d P - s p a c e .

e . A f r e e u l t r a f i l t e r c l o s e d u n d e r c o u n t a b l e i n t e r s e c t i o n s e x i s t s on T .

f . T i s m e a s u r a b l e . g . - D T^D * T^d.

3 9

P r o o f * T o p r o v e t h a t ( a ) = > ( b ) , l e t (^{u n}) n _ i ^{b e a} P a r t i t i o n o f T^f and e q u i p T^Q w i t h the t o p o l o g y o f ( a ) . D e f i n e h on T^Q by

f n . a l l t e U , a l l ne N

h ( t ) = { ⁿ

(.0, t = t^Q

By ( a ) t h e r e e x i s t s an f e C ( T^Q) such t h a t f ^ h . S i n c e f i s c o n t i n u o u s a t t^Q , we know t h a t f ^ m on some n e i g h b o r h o o d o f t^Q . By t h e d e f i n i t i o n o f h . Q U i s t h e r e f o r e a d e l e t e d n e i g h b o r h o o d

o f t^Q , so t h a t t h e t o p o l o g y f o f ( a ) i s t h e t o p o l o g y we s o u g h t i n o r d e r t o p r o v e ( b ) . Now assume ( b ) and l e t be t h e n e i g h ­ b o r h o o d f i l t e r o f t^Q i n t . I f & = £F \ ( t ⁰ } : F e S ^ } , t h e n l e t

be an u l t r a f i l t e r i n T c o n t a i n i n g t h e f i l t e r « 2 • L e t % d e n o t e t h e t o p o l o g y on T^Q d e r i v e d from f by s u b s t i t u t i n g t h e

n e i g h b o r h o o d f i l t e r = JF u \ t^Q] : F 6 o f t^Q i n f ^{f o r} ^ • L e t ( ^{u n} ) ^ l i *>^{e a} p a r t i t i o n o f T . By ( b ) t h e r e i s an m such t h a t O ^3* c ^ ^# S i n c e A i s an u l t r a f i l t e r , one o f t h e s e s e t s i n (^{U N}) J[ - l » ^{8 a} y ^um » ^{i s i n} ^ ^# Thus w i t h r e s p e c t t o X^t we have ( c ) . To p r o v e t h a t ( c ) = ^ ( d )^t l e t U be open i n t h e n o n -

d i s c r e t e t o p o l o g y on T^Q o f ( c ) . I f t⁰ £ U , t h e n U = U • I f t ^Q£ U , t h e n [u, T \ u | i s a p a r t i t i o n o f T , so t h a t by ( c )

e i t h e r U o r T \ U i s a d e l e t e d n e i g h b o r h o o d o f t^Q • E i t h e r way U i s open i n TQ , whereupon TQ i s e x t r e m a l l y d i s c o n n e c t e d . To

show t h a t T^Q i s a P - s p a c e , i t s u f f i c e s t o show t h a t t^Q i s a

P - p o i n t . To t h a t e n d , l e t ( ^{u n}) ^ - i ^{b e a} d e c r e a s i n g s e q u e n c e o f <FCUW n e i g h b o r h o o d s o f t^Q , each minus the p o i n t t^Q , and l e t = T • Then t h e c o l l e c t i o n j A U ^f ju V I T - . 1 T i l f o r m s a p a r t i t i o n o f

I * « J n * i n U+L> n = J J ^

T, so must by ( c ) c o n t a i n a d e l e t e d n e i g h b o r h o o d o f t Q . E v i d e n t l y t h e o n l y p o s s i b i l i t y i s t h a t C\ Uⁿ be t h i s d e l e t e d n e i g h b o r h o o d , p r o v i n g t h a t t^Q i s a P - p o i n t . N e x t we p r o v e t h a t

( d ) = M e ) .

F o r t h a t we l e t T^A be t h a t o f ( d ) , and l e t 9, be t h e n e i g h b o r h o o d f i l t e r o f t^Q i n T^Q and l e t ^ » J Ps \T^Q\: P e . Then 2r i s a f i l t e r on T , and s i n c e T^Q i s a P- s p a c e , ^ i s c l o s e d u n d e r

c o u n t a b l e i n t e r s e c t i o n s . N e x t we show t h a t & i ^{S a}n u l t r a f i l t e r . L e t S S T . Then T^Q = S u T \ S , so assume t h a t t ^g 6 S • S i n c e T^Q i s e x t r e m a l l y d i s c o n n e c t e d , and s i n c e S i s open i n TQ , we know t h a t S* = S U - [ t⁰j i s open i n T^Q , w h i c h means t h a t S t By c o n s t r u c t i o n $f i s f r e e ( s e e \k]) • Thus ( e ) h o l d s . We assume n e x t t h e e x i s t e n c e on T o f t h e u l t r a f i l t e r < ^ o f ( e ) and we

d e f i n e on t h e power s e t o f T by ( l , i f ( 0^f i f

By t h e p r o p e r t i e s o f ^ i t i s i m m e d i a t e t h a t ^A- i s t h e d e s i r e d measure g i v i n g us ( f ) . The e q u i v a l e n c e o f ( f ) and ( g ) i s p r e ­

c i s e l y the c o n t e n t o f Theorem 1 2 . 2 i n IV]. A somewhat d i f f e r e n t p r o o f t h a t ( f ) = ^ ( g ) g o e s as f o l l o w s . L e t yu. be t h e measure on

T

T a s s o c i a t e d w i t h the m e a s u r a b i l i t y o f T . L e t hfeR , l e t p<£N, and d e f i n e U ^{n p} = | t € T : ( n - l ) / 2^p£ h ( t ) <: n / 2^p j, f o r a l l n&Z

( w h e r e Z d e n o t e s t h e c o l l e c t i o n o f a l l i n t e g e r s ) . Then ( ^{u n p} ) ^{n 6} z p a r t i t i o n s T , and by t h e d e f i n i t i o n o f / * - we s e e t h a t f o r each pe N t h e r e e x i s t s an n £ Z such t h a t / « - ( ^{u n p} ) = 1# Because ju- i s

P P

T

c o u n t a b l y a d d i t i v e , l i m h ( U ) e x i s t s , so we d e f i n e tf on R by

p - > o o npp

^ > ( h ) = l i m h ( U ) , a l l h € R^{T t}

S u r e l y y> i s w e l l - d e f i n e d , and a f t e r a l i t t l e c o m p u t a t i o n we s e e T

t h a t <^ i s a p o s i t i v e l i n e a r f o r m on R • By H e w i t t ' s Theorem

[ 7 . Theorem 2 2 j , y c o r r e s p o n d s t o a measure s u p p o r t e d on a compact s u b s e t K o f v T ^ • Because t h e o n l y compact s u b s e t s o f T^d a r e f i n i t e , and b e c a u s e our o r i g i n a l JJU a n n i h i l a t e s a l l f i n i t e s u b s e t s o f T , we know t h a t K ^ T • Thus * » T^d £ T^d , w h i c h p r o v e s ( g ) . F i n a l l y we show t h a t ( g ) = M a ) . To do i t , l e t ^t o ^{€ v T} d ^{X T} d 9 8 1 1 1 ( 1

l e t T^Q « ^T u{ t ^Q ] h a v e t h e t o p o l o g y i n d u c e d f r o m v T ^d . I f h i s d e f i n e d on T^Q , t h e n g =* h |^T i s i n C ( T^d) , so g has a c o n t i n -

uous e x t e n s i o n g d e f i n e d on T^Q . L e t t i n g f = | g | + ^ i i C *⁰) | , we o b s e r v e t h a t f £ C ( T^Q) and f^[h)^t c o m p l e t i n g t h e p r o o f t h a t

< g ) - » ( a ) . |

We remark a t t h i s p o i n t t h a t t h e r e a r e e q u i v a l e n t ways o f s a y ­ i n g t h a t T i s measurable o t h e r than t h o s e m e n t i o n e d i n Theorem 1 ; s e e f o r i n s t a n c e [ 2 ] . However t h e ones s t a t e d h e r e a r e t h e o n e s w h i c h p l a y a r o l e i n t h e p r e s e n t a r t i c l e * The f o l l o w i n g two c o r o l ­

l a r i e s t e l l us d e f i n i t i v e l y when t h e r e e x i s t T s a t i s f y i n g Con­

d i t i o n A .

COROLLARY 2 . T h e r e e x i s t s a n o n - d i s c r e t e T s a t i s f y i n g C o n d i t i o n A i f f t h e r e e x i s t s a m e a s u r a b l e c a r d i n a l .

P r o o f . I f T i s n o n - d i s c r e t e and s a t i s f i e s C o n d i t i o n A , t h e n w i t h any s t r o n g e r n o n - d i s c r e t e t o p o l o g y i t s a t i s f i e s c o n d i t i o n A . So t a k e any s t r o n g e r n o n - d i s c r e t e t o p o l o g y on T f o r which a l l but

one p o i n t o f T i s i s o l a t e d . That space i s t h e T^Q o f s t a t e m e n t ( a i n Theorem 1 . S i n c e ( a ) » ^ ( f ) ^f t h e r e e x i s t s a measurable c a r d i n a l . On t h e o t h e r hand, ( f ) = » ( a ) i n Theorem 1 y i e l d s the c o n v e r s e . | COROLLARY 3 . I t i s c o n s i s t e n t w i t h t h e Z e r m e l o - F r a e n k e l axioms t o assume t h a t no T e x i s t s s a t i s f y i n g C o n d i t i o n A .

P r o o f . C o r o l l a r y 2, t o g e t h e r w i t h Theorem 1 2 . 5 and t h e remarks i n 1 2 . 6 o f [ > ] t y i e l d s t h e r e s u l t . I

I n c o n t r a s t t o C o r o l l a r i e s 2 and 3, i t i s not known ( a t l e a s t t o u s ) i f i t i s c o n s i s t e n t t o assume t h a t a T s a t i s f y i n g C o n d i ­ t i o n A — o r e q u i v a l e n t l y a T o f m e a s u r a b l e c a r d i n a l — d o e s e x i s t . I f we assume t h a t such a c a r d i n a l e x i s t s , i t must be h u g e . Such a c a r d i n a l ^ must be e x t r e m e l y u l t r a - s u p e r - g i a n t i n a c c e s s i b l e^f and i n d e e d t h e r e must e x i s t c< i n a c c e s s i b l e c a r d i n a l s l e s s t h a n <* , by a theorem o f T a r s k i [ l 2 ] . I n a d d i t i o n , i f we assume t h a t such a c a r d i n a l e x i s t s , t h e n c u r i o u s phenomena o c c u r i n s e t t h e o r y QlcT] »

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On t h e o t h e r hand, more r e c e n t l y J. S i l v e r p r o v e d t h a t i f t h e Z e r m e l o - F r a e n k e l a x i o m s , t o g e t h e r w i t h t h e axiom o f c h o i c e and

t h e axiom o f t h e e x i s t e n c e o f a m e a s u r a b l e c a r d i n a l , a r e c o n s i s t e n t , t h e n i t i s c o n s i s t e n t t o assume i n a d d i t i o n t h a t the g e n e r a l i z e d continuum h y p o t h e s i s h o l d s [ l l ] •

We t u r n our a t t e n t i o n t o C o n d i t i o n B , and b e g i n w i t h a p r e l i m ­ i n a r y r e s u l t .

LEMMA L e t - A . d e n o t e t h e o r d i n a l s l e s s than some i n f i n i t e l i m i t o r d i n a l , and l e t ±Q€ T • Assume t h a t t h e r e e x i s t s a c o l l e c t i o n

( U ^ ) J ^ J ^ o f open n e i g h b o r h o o d s o f such t h a t ( i ) U ^ U j f o r a l l / y A * J L v i t h / ~ c j l .

( i i ) Whenever V i s a n e i g h b o r h o o d o f t^Q t h e r e i s a A ^v e JL such t h a t f o r each ^ >^y an m e N e x i s t s f o r w h i c h

T h e n T does n o t s a t i s f y C o n d i t i o n B .

P r o o f , F o r any n o n - l i m i t o r d i n a l \ e-J[_^t l e t n^ e- N such t h a t n^ - n i s a l i m i t o r d i n a l ( o r 0 ) , and l e t , A ^ = ^ \eJJ = l £ » D e f i n e g by

A « A » » i A ^{+ n} /n*i X+m O

I f A i A ^ A ^ w i t h t h e n ^u ^ i + ⁿ ^ ^ ^u ^ ^{+ m} » ^{8 0 w c h a v e} e a s i l y t h a t g ( t ) a 0 » f o r a l l * € U ^ and a l l n 6 N • Thus g i s r e a l -

v a l u e d . Because ^ ^{u A + m} i s a l w a y s c l o s e d i n T and U ^ +ⁿ a l w a y s i s o p e n , we know t h a t g £ L S C ( T ) . L e t V be a n e i g h b o r h o o d o f t^Q , l e t \*JL^awith <\>>y*y 1 and l e t n e N • By h y p o t h e s i s t h e r e e x i s t s a

t d ( V n U ^ ^{+ n} ) \ ( V n U ^a ^ ^{n + D i} ) f o r some m £ N ^# Then g ( t ) ^ n , so t h a t g i s unbounded on e v e r y such n e i g h b o r h o o d V o f t^Q , w h i c h i s t r u e o f no f € C ( T ) . Thus T does not s a t i s f y C o n d i t i o n B * | THEOREM 5' I f T s a t i s f i e s C o n d i t i o n B , t h e n T i s a P - s p a c e . P r o o f . Assume t h a t " t⁰£ T i s not a P - p o i n t . Then t h e r e e x i s t s a

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sequence (^un) ^ L i o f open n e i g h b o r h o o d s o f t^Q such t h a t fj Uⁿ i s n o t a n e i g h b o r h o o d o f t g , and we can as w e l l assume t h a t ^{u n} + i — ^ ⁿ f o r a l l n . S i n c e f*) U i s n o t a n e i g h b o r h o o d o f t^A , i t i s c l e a r t h a t f o r any n e i g h b o r h o o d V o f t g and any n , t h e r e e x i s t s an mⁿ> n such t h a t ^{V A} ^ ⁿ * ^{V r ,} ^ ^m • Thus t h e h y p o t h e s e s o f Lemma h

n

a r e s a t i s f i e d , and we can d e r i v e t h e d e s i r e d c o n c l u s i o n . I

U n f o r t u n a t e l y t h e c o n v e r s e to Theorem 5 i s f a l s e . I n d e e d , l e t T be t h e n o n - l i m i t o r d i n a l s l e s s t h a n t h e f i r s t u n c o u n t a b l e c^>, ,

w i t h a;, a d j o i n e d , and c l o t h e T i n t h e o r d e r t o p o l o g y . Then T i s a P - s p a c e . I t i s e a s y t o s e e t h a t o>, f u l f i l l s the h y p o t h e s e s on t^ i n Lemma h⁹ where - A . = [l⁹co^t)* C o n s e q u e n t l y T d o e s n o t s a t i s ­ f y C o n d i t i o n B .

That C o n d i t i o n A i m p l i e s C o n d i t i o n B means t h a t i f we assume t h e e x i s t e n c e o f m e a s u r a b l e c a r d i n a l s , t h e n we know t h a t t h e r e e x i s t T^fs w h i c h s a t i s f y C o n d i t i o n B. However, w i t h o u t the e x i s t e n c e o f

m e a s u r a b l e c a r d i n a l s we do n o t know any T w h i c h s a t i s f y C o n d i t i o n B, and e v i d e n t l y none e x i s t s w h i c h has but one n o n - i s o l a t e d p o i n t ( l i k e t h e T Q i n Theorem 1 ) . C o n c e i v a b l y no such T e x i s t s w i t h o u t t h e e x i s t e n c e o f m e a s u r a b l e c a r d i n a l s . T h i s p o s s i b i l i t y becomes p l a u s i b l e when we n o t e t h a t e v e r y P - s p a c e i s b a s i c a l l y d i s c o n n e c t e d ( P r o b l e m hK oi M ) , w h e r e a s a n o n - d i s c r e t e e x t r e m a l l y d i s c o n n e c t e d s p a c e e x i s t s o n l y i f m e a s u r a b l e c a r d i n a l s e x i s t ( P r o b l e m 12H o f jV]) — a s l i g h / t l y s t r o n g e r r e s u l t than ( d) = M f ) o f our Theorem 1 . I n o t h e r w o r d s , t o f i n d a n o n - d i s c r e t e space w h i c h s a t i s f i e s C o n d i t i o n B but w h i c h has n o n - m e a s u r a b l e c a r d i n a l i t y , one must l o c a t e a T o f r e a s o n a b l e c a r d i n a l i t y which i s more d i s c o n n e c t e d t h a n b a s i c a l l y d i s c o n n e c t e d but w h i c h i s n o t q u i t e e x t r e m a l l y d i s c o n n e c t e d . T h a t j o b may n o t toe so e a s y . A n o t h e r way o f l o o k i n g a t t h e s i t u a t i o n i s as f o l l o w s . I f we assume t h a t a l l c a r d i n a l s a r e n o n - m e a s u r a b l e , then a n o n - d i s ­ c r e t e s p a c e T s a t i s f i e s C o n d i t i o n B o n l y i f e v e r y p o i n t w i s e bounded f a m i l y i n C ( T ) i s dominated by a c o n t i n u o u s f u n c t i o n but t h e r e e x -

44

i s t s a f a m i l y (^fx ^ U J L — ^C ^ ^{S U} ° ^{h t h &} t 1 5 1 1 5 ^ ^C^^T* ^ ^{s e e} P r o b l e m 3N i n M ) .

A n o n - d i s c r e t e compact T cannot s a t i s f y C o n d i t i o n B b e c a u s e a compact P - s p a c e i s n e c e s s a r i l y f i n i t e , by P r o b l e m *fK o f [h].

But a n o n - d i s c r e t e r e p l e t e T s a t i s f y i n g C o n d i t i o n B i s n o t r u l e d o u t a p r i o r i * What we know i s the f o l l o w i n g r e s u l t *

PROPOSITION 6. I f T s a t i s f i e s C o n d i t i o n B, t h e n v T a l s o s a t i s - f i e s C o n d i t i o n B*

P r o o f * L e t g £ L S C ( v T ) . I f g = g |^T , t h e n g ^ L S C ( T ) , so t h a t by h y p o t h e s i s t h e r e e x i s t s an f € C ( T ) w i t h f ^ g • Now l e t t g € - t f T N T . P o r each £ > 0 , t h e r e e x i s t s a n e i g h b o r h o o d U^£ o f t^Q on which g ^ g ( t^Q ) - L • The c o n t i n u o u s e x t e n s i o n f o f f on t>T g i v e s f ( t ^Q ) > g ( t^Q ) • Thus f > g • I

Prom now on we c o n c e n t r a t e on C o n d i t i o n C. U n l i k e the p r o b l e m s c o n f r o n t i n g us i n f i n d i n g n o n - d i s c r e t e T s a t i s f y i n g e i t h e r Con-

d i t i o n A o r C o n d i t i o n B , i t i s t r i v i a l t o f i n d n o n - d i s c r e t e T s a t i s f y i n g C o n d i t i o n C , and we can d e s c r i b e w i t h no e f f o r t two t y p e s o f such T •

I n t h e f i r s t p l a c e , i f T i s denumerable, w i t h any c o m p l e t e l y r e g u l a r H a u s d o r f f t o p o l o g y w h a t e v e r , t h e n T s a t i s f i e s C o n d i t i o n C.

m

To s e e t h i s , j u s t l e t T = ("k^^Li 8 1 1 ( 1 l e t h< £ R • M e r e l y d e f i n e g on T by g ( t ⁿ) = m a x £ l h ( t^m) | : m = l , * . * , n } - , f o r e a c h n e N . Then g é L S C ( T ) and g ^ h .

Por t h e s e c o n d t y p e , we n o t e t h a t t h e r e a r e n o n - d i s c r e t e T o f a r b i t r a r i l y l a r g e c a r d i n a l i t y which s a t i s f y C o n d i t i o n C. Por i f T i s any space which c o n t a i n s but one n o n - i s o l a t e d e l e m e n t , t h e n

T

any h£ R i s l o w e r s e m i - c o n t i n u o u s e x c e p t p o s s i b l y a t tQ , so such an h can be m a j o r i z e d by a s u i t a b l e g £ L S C ( T ) .

On t h e o t h e r hand, i f T = £ o, l ] , t h e n T d o e s n o t s a t i s f y C o n d i t i o n C , as we saw a t t h e o u t s e t o f t h e paper* Because i n some

45

sense i n any space s a t i s f y i n g C o n d i t i o n C t h e r e seem t o be l o w e r s e m i - c o n t i n u o u s f u n c t i o n s which o s c i l l a t e a r b i t r a r i l y w i l d l y , l e t us make the f o l l o w i n g s l i g h t l y p i c t o r i a l d e f i n i t i o n ( a s i n e f f e c t we di d i n •

DEFINITION 7- The space T i s w i l d l y o s c i l l a t o r y i f f T s a t i s f i e s C o n d i t i o n C ( i . e . , i f f o r each h € E , t h e r e e x i s t s a g e L S C ( T )

such t h a t g ^ h ) • I f T i s not w i l d l y o s c i l l a t o r y , we say t h a t T i s m i l d l y o s c i l l a t o r y .

I n a n a l y z i n g which T a r e w i l d l y ( o r m i l d l y ) o s c i l l a t o r y , we w i l l r e f e r t o what we c a l l e v e r y w h e r e unbounded f u n c t i o n s . We say

T

t h a t h6 E i s e v e r y w h e r e "unbounded i f f h i s unbounded on each non-empty open subset o f T . A f t e r a moment's r e f l e c t i o n y o u w i l l

a g r e e t h a t T i s m i l d l y o s c i l l a t o r y i f on t h e one hand L S C ( T ) c o n t a i n s no p o s i t i v e e v e r y w h e r e unbounded f u n c t i o n w h i l e on t h e

T

o t h e r hand R does c o n t a i n an e v e r y w h e r e unbounded f u n c t i o n . ( P e r h a p s the c o n v e r s e i s v a l i d t o o , f o r t h o s e T which c o n t a i n no i s o l a t e d p o i n t s . )

What t y p e o f T admit no p o s i t i v e e v e r y w h e r e unbounded f u n c ­ t i o n i n L S C ( T ) ? R e c a l l t h a t T i s non-meager i f f T i s not the c o u n t a b l e u n i o n o f c l o s e d nowhere dense s u b s e t s [8, p . 2 1 3 ) • Then we can s t a t e and p r o v e t h e f o l l o w i n g p r o p o s i t i o n .

PROPOSITION 8. T i s non-meager i f f L S C ( T ) admits no p o s i t i v e e v e r y w h e r e unbounded f u n c t i o n .

P r o o f . I f T i s non-meager and i f g 6 L S C ( T ) i s p o s i t i v e , l e t Sⁿ = | t 6 T : g ( t ) £ n } , f o r each n £ N .

Then each S i s c l o s e d i n T and Q s ^ = T . Because T i s

u ^{« « 1} u

non-meager, t h e r e i s an i n t e g e r n such t h a t c o n t a i n s a non-empty open s u b s e t . F o r t h e c o n v e r s e , assume t h a t L S C ( T ) a d m i t s no

p o s i t i v e e v e r y w h e r e unbounded f u n c t i o n . L e t ( S ) ° * - » be an i n c r e a s - n n = l

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i n g sequence o f c l o s e d s u b s e t s w i t h u n i o n T, and f o r e a c h n € , N l e t g ( t ) = n f o r a i l t ^ S n ^ S ^ i ( w h e r e S^Q = 0 ) . Then g £ L S C ( T ) and i s p o s i t i v e , and c o n s e q u e n t l y by h y p o t h e s i s t h e r e e x i s t s an mGN and an open subset U C T such t h a t g ( t ) £ m f o r a l l t e U . Thus U C \JS . Now t h e r e g u l a r i t y o f T y i e l d s a

*»( n

non-empty open subset V o f T such t h a t V C U and a l s o a p < m such t h a t V C S^p . |

S i n c e a l l B a i r e s p a c e s , and i n p a r t i c u l a r a l l l o c a l l y compact s p a c e s , a r e non-meager, P r o p o s i t i o n 8 a t t e s t s t h a t f o r any such T^f t h e space L S C ( T ) c o n t a i n s no p o s i t i v e e v e r y w h e r e unbounded f u n c t i o n .

I f T c o n t a i n s no n o n - i s o l a t e d p o i n t s , t h e n the n o n - e x i s t e n c e o f an e v e r y w h e r e unbounded f u n c t i o n on T i s c l o s e l y r e l a t e d t o C o n d i t i o n ( b ) o f Theorem 1 . I n f a c t , T admits no e v e r y w h e r e unbounded f u n c t i o n p r e c i s e l y when f o r e a c h p a r t i t i o n (^xJnLl °*

T t h e r e e x i s t s a n o n - i s o l a t e d t € T and an m. e N such t h a t (b I L i s a d e l e t e d n e i g h b o r h o o d o f t • As i s e v i d e n t , t h i s l a t t e r c o n d i ­ t i o n i s a m o d e s t l y weakened v e r s i o n o f c o n d i t i o n ( b ) , and i s e x a c t l y c o n d i t i o n ( b ) i n t h e e v e n t t h a t T has e x a c t l y one n o n - i s o l a t e d e l e m e n t .

Remember t h a t Theorem 1 and C o r o l l a r y 3 t e l l us i n e f f e c t t h a t i t i s c o n s i s t e n t w i t h t h e u s u a l axioms o f s e t t h e o r y t o assume t h a t any T w i t h one n o n - i s o l a t e d p o i n t (and i n d e e d any n o n - d i s c r e t e T ) does n o t s a t i s f y c o n d i t i o n ( b ) . T h e r e f o r e , w i t h a l i t t l e o f t h e p o l l y a n n a i n us we c o n j e c t u r e t h a t i t i s c o n s i s t e n t w i t h t h e u s u a l axioms o f s e t t h e o r y t o assume t h a t e v e i y T which c o n t a i n s no i s o l a t e d p o i n t s admits an e v e r y w h e r e unbounded f u n c t i o n . I f our c o n j e c t u r e happens t o be t r u e , then at l e a s t f o r t h o s e T w i t h o u t i s o l a t e d p o i n t s a c r i t e r i o n f o r T t o be m i l d l y o s c i l l a t o r y ( o r t o n o t s a t i s f y C o n d i t i o n C) would e v o l v e : such a space would be m i l d l y o s c i l l a t o r y i f T w e r e n o n - m e a g e r .

47

W i t h o u t a p r o o f o f t h e c o n j e c t u r e , we can i n any c a s e p r o v e t h a t many T admit e v e r y w h e r e unbounded f u n c t i o n s . We b e g i n w i t h t h e f o l l o w i n g s i m p l e o b s e r v a t i o n .

PROPOSITION 9- I f T i s s e p a r a b l e , t h e n T admits an e v e r y w h e r e unbounded f u n c t i o n .

P r o o f . I f ( *ⁿ) ^ j [ ^{i s} dense i n T ^f t h e n l e t b ( tⁿ ) « n, f o r a l l n £ N , and l e t h ( t ) = 0 f o r a l l o t h e r t € T . |

As a r e s u l t o f P r o p o s i t i o n 9, i f T i s s e p a r a b l e and non-

m e a g e r — l i k e t h e r e a l s — t h e n T i s m i l d l y o s c i l l a t o r y . M o r e o v e r , by u s i n g Hamel b a s e s we can e a s i l y show t h a t any t o p o l o g i c a l v e c t o r space T o v e r t h e r e a l s a d m i t s an e v e r y w h e r e unbounded f u n c t i o n , and m o r e o v e r such a T i s a l s o m i l d l y o s c i l l a t o r y .

Next we w i l l show t h a t each l o c a l l y compact space w i t h o u t i s o l a t e d p o i n t s admits an e v e r y w h e r e unbounded f u n c t i o n . We w i l l u t i l i z e t h e n o t i o n o f what might be c a l l e d " h o m o c a r d i n a l i t y " . We

say t h a t a space T i s h o m o c a r d i n a l i f f c a r d U = c a r d T f o r any non-empty open U i n T ( w h e r e " c a r d " s t a n d s f o r " c a r d i n a l i t y " ) . We are t h e n r e a d y t o s t a t e and p r o v e P r o p o s i t i o n 1 0 , t h e b a s i c i d e a f o r whose p r o o f i s due t o J. Saint-Raymond.

PROPOSITION 1 0 . L e t T have no i s o l a t e d p o i n t s , and l e t S ^ T w i t h S = T . L e t ^^{e a} p a i r w i s e d i s j o i n t c o l l e c t i o n o f homo-

c a r d i n a l open s u b s e t s o f T .such t h a t

( a ) IJT

^A

- T

( b ) F o r each X±JL^% t h e r e e x i s t s a base IX ^X « ( U ^ L _ o f o p e n

^ a o ire ^I

s u b s e t s f o r T^A such t h a t c a r d UL^ ^ c a r d ( U ^AT N S ) , f o r a l l v*r^A $ and such t h a t i f c a r d ( U j^yn S ) =TQ^0$ t h e n we have c a r d ( U^{A Y}N S ) s c a r d .

Then S (and hence T ) s u p p o r t s an e v e r y w h e r e unbounded f u n c t i o n .

48

P r o o f . F i x \>*Jt and l e t "U^x - (Uj ^{s a}" f c i s f y p r o p e r t y ( b ) . D e s i g n a t e by <t t h e o r d i n a l such t h a t we may (and w i l l ) i d e n t i f y

P^v w i t h t h e o r d i n a l s l e s s than c< • I f c a r d (XL ^v H S ) * TflL f o r some Y^r. I t h e n by assumption ( b ) we know t h a t c a r d U^{N v} = W . But i s h o m o c a r d i n a l , so t h i s means t h a t c a r d T^A = 7 ^ . Con­

s e q u e n t l y we can d e f i n e h. on T. as i n P r o p o s i t i o n 9 . H e n c e - f o r t h we can t h e r e f o r e assume t h a t c a r d (U^ ^yR \ S) ^ ?fy f o r each V £ p , and f o r t h e moment we f i x V £ p ^# Then

c a r d y^o < oc = c a r d l c a r d (U^ ^y n S ) .

U t i l i z i n g t h e s e i n e q u a l i t i e s , we can by t r a n s f i n i t e i n d u c t i o n f i n d a d i s t i n c t sequence ( t . ^ U. ^ S such t h a t

^nA^T * ^m\>* ^{1 f o r a l l 1 r 6} ^ suck t h a t ff*^, and a l l m , n £ N . Define h ^ on T ^ by

f n , i f t s t » ^v , f o r some n e N and V ^ P , .

h> ( t ) - J ^{n >}° * ° ^A°

^0, a l l o t h e r t 6 T^ .

S i n c e (U* ^v V , « c o m p r i s e s a b a s e f o r open s e t s i n T* , t h i s means t h a t h. i s e v i d e n t l y unbounded on a l l non-empty open sub-

s e t s o f T. r\ S ( a n d hence on a l l t h o s e non-empty open s u b s e t s o f T ^{> o}) . F i n a l l y , u t i l i z i n g t h e f a c t t h a t t h e T ^ ' s a r e p a i r w i s e d i s j o i n t , we can l e t h "be d e f i n e d on T by

V ( t ) , a l l t e T , , a l l AeJt.

h ( t ) = <

0, a l l o t h e r t € T .

Then h i n h e r i t s t h e r e q u i r e d p r o p e r t y from t h e k ^^fs , s i n c e Xj (i^rss) = s" = T . |

The S w h i c h a p p e a r s i n P r o p o s i t i o n 10 i s n o t needed i n o r d e r t o d e r i v e t h e f o l l o w i n g p r o p o s i t i o n , but i t w i l l a p p e a r i n Example 1 2 and w i l l p l a y an e s s e n t i a l r o l e i n P r o p o s i t i o n 1 7 i n S e c t i o n 3 .

The n e x t r e s u l t i s e s s e n t i a l l y due t o J . S a i n t - R a y m o n d .

PEOPOSITION 1 1 . I f T i s l o c a l l y compact and c o n t a i n s no i s o l a t e d p o i n t s , then T s u p p o r t s an e v e r y w h e r e unbounded f u n c t i o n — a n d thus T i s m i l d l y o s c i l l a t o r y .

P r o o f . F i r s t , by Z o r n ' s Lemma l e t (T^ be a maximal c o l l e c t i o n o f p a i r w i s e d i s j o i n t open homocardinal s u b s e t s o f T . I f by chance

U TV * T , t h e n t h e r e would e x i s t an open V £ ( T \ UY^A) w i t h minimal c a r d i n a l i t y * T h i s V would be h o m o c a r d i n a l , c o n t r a d i c t i n g t h e m a x i m a l i t y o f (T* ) . » . Thus LTTT = T . F i x A * ^ , and f o r a p a i r s , t € T ^ , l e t U g ^ C T ^ be r e l a t i v e l y compact and open, and such t h a t s £ U ^{s t} but t ^ . To show t h a t ^VL^ ^^Us t ^ s t € T i s a b a s i s f o r open s e t s o f T^ , l e t U be any non-empty open subset o f T^ , l e t s ^ U , and l e t t^Q € T^\[s}. Then U ^ . \ U i s compact. E i t h e r I T . C U or e l s e t h e r e e x i s t s an n such t h a t

s x^Q

s xn *^{m | s t n s t} 0

which means t h a t

U' s t " ^ ^U s t " ft ^TTAT ^ ^U >

s x0 »^m* ^{s x}n »*o ^{s t} n

c o n f i r m i n g t h a t i s a base f o r open s e t s o f T^ . M o r e o v e r , card 1A^> a card T^ , so t h a t by l e t t i n g S = T i n P r o p o s i t i o n 1 0 , we demonstrate the e x i s t e n c e of an e v e r y w h e r e unbounded f u n c t i o n on T . That T i s t h e r e f o r e m i l d l y o s c i l l a t o r y now f o l l o w s from P r o p o s i t i o n 8. |

I n p r e p a r a t i o n f o r P r o p o s i t i o n 17$which a p p e a r s l a t e r o n , we g i v e now an example o f a T which a d m i t s an e v e r y w h e r e unbounded f u n c t i o n on a s p e c i a l , n o n - t r i v i a l dense subset S .

EXAMPLE 1 2 . L e t T = If and l e t S denote t h e c o l l e c t i o n o f a l l P - p o i n t s o f T . I f we assume t h e continuum I j y p o t h e s i s , t h e n S i s dense i n T by Problem 6V o f D + ]. We w i l l show t h a t S admits an e v e r y w h e r e unbounded f u n c t i o n . To t h a t end, l e t U be

50

non-empty and open i n T , so t h a t U / ^ S * 0 . I f ( U n S ) c. ( 8ⁿ) ^ L ^ >

then l e t s ^ U n S , and f o r a l l n ^ 2 l e t VⁿC U be a n e i g h b o r h o o d o f sⁿ such t h a t s ^ V . Then V = ^ V i s a non-empty G.

l n ^ n m - i n o

c o n t a i n i n g s^g S , so V i s a n e i g h b o r h o o d o f s^ • But Vr\ S ^ { s - ^ and s^ i s n o t i s o l a t e d , which means t h a t S cannot be dense i n T , c o n t r a d i c t i n g our p r e v i o u s a s s e r t i o n . Thus f o r e v e r y non-empty U i n T , we know t h a t c a r d ( U r i S ) ^ ^ . S i n c e N i s l o c a l l y compact, t h e space T i s compact by 6.9d o f {h] , so t h a t by t h e p r o o f o f

P r o p o s i t i o n 1 1 t h e r e e x i s t s a c o l l e c t i o n (T^ o f p a i r w i s e d i s ­ j o i n t , h o m o c a r d i n a l , open s u b s e t s o f T such t h a t = T .

P r o b l e m 6S o f [*f] t e l l s us t h a t t h e r e i s a b a s i s o f open s e t s i n T (and hence i n each T^ ) o f c a r d i n a l i t y • Then P r o p o s i t i o n 10 f i n i s h e s the p r o o f t h a t t h e r e e x i s t s an e v e r y w h e r e unbounded f u n c t i o n on S.

I n showing t h a t a g i v e n T i s w i l d l y o r m i l d l y o s c i l l a t o r y , we have had t o l e a n h e a v i l y on t h e h y p o t h e s i s t h a t T c o n t a i n no

i s o l a t e d p o i n t s . As y e t we have found no method f o r c i r c u m v e n t i n g t h i s h y p o t h e s i s , e v e n when T i s l o c a l l y compact and v e r y down t o e a r t h . F o r e x a m p l e , i f T i s t h e o r d i n a l s l e s s than t h e f i r s t

u n c o u n t a b l e o>, ( o r l e s s than any to* >c*>,), t h e n T i s l o c a l l y compact and i s m o r e o v e r e m i n e n t l y s t r u c t u r e d . I s T w i l d l y o s c i l l a t o r y ? I t seems t o us t h a t w h a t e v e r be t h e r e p l y , i t s p r o o f cannot be c o n s t r u c t i v e . Could t h e answer c o n c e i v a b l y e n t a i l a new axiom?

51

3* W i l d l y o s c i l l a t o r y spaces and l o c a l l y convex s p a c e s

I n t h i s s e c t i o n we d e t e r m i n e which T a r e w i l d l y ( o r m i l d l y ) o s c i l l a t o r y i n the c o n t e x t o f l o c a l l y c o n v e x s p a c e s . To b e g i n w i t h , we l e t C U T ) d e n o t e t h e space C ( T ) w i t h t h e t o p o l o g y o f s i m p l e s c o n v e r g e n c e on the e l e m e n t s o f T. I t i s the a n a l y s i s i n [ l ] o f v a r i o u s s p e c i a l l o c a l l y convex space p r o p e r t i e s o f C^S( T ) w h i c h o r i g i n a l l y drew our a t t e n t i o n t o C U T ) .

s

C e r t a i n c h a r a c t e r i s t i c s o f C U T ) a r e e v i d e n t . I n t h e f i r s t s

p l a c e , C U T ) i s a l w a y s dense i n R when R c a r r i e s i t s p r o d u c t T

t o p o l o g y . M o r e o v e r , C U T ) = R i f f T i s d i s c r e t e , as we men- s

t i o n e d at t h e o u t s e t o f the a r t i c l e . Because t h e supremum o f a

s e t o f c o n t i n u o u s f u n c t i o n s i s a l w a y s l o w e r s e m i - c o n t i n u o u s ( t h o u g h p o s s i b l y i n f i n i t e - v a l u e d ) , t h e c o l l e c t i o n (^Bg ) g ^ L S C ( T ) g > 0

forms a base f o r the bounded s e t s o f C U T ) , where by d e f i n i t i o n s

B^g - {t € C^S( T ) : | f k g ] .

The d u a l i t y t h e o r y f o r C _ ( T ) i s by and l a r g e known ( s e e

E5j)»

s

we mention a few p e r t i n e n t p r o p e r t i e s o f C U T ) at t h i s p o i n t . As s

a v e c t o r space t h e dual C U T ) i s i d e n t i f i e d w i t h t h e s e t o f a l l s

f i n i t e l i n e a r c o m b i n a t i o n s o f p o i n t masses ^ , t € . T , and e v i d e n t ­ l y (^Bg ) g 6 I S C ( T ) g > 0 ^{f o r m s a} * >^{a s}i^{s f o r} n e i g h b o r h o o d s o f 0 i n C ^ ( T ) i n t h e dual t o p o l o g y , where i n t h e i d e n t i f i c a t i o n we o b t a i n

B^6g « co b a l | S^t/ g ( t ) : t ^ T j

[5, Lemma 2 ] . As a r e s u l t , t h e o n l y s u b s e t s o f C ^ ( T ) which a r e bounded i n t h e dual t o p o l o g y o f C U T ) a r e t h o s e c o n t a i n e d i n

s

f i n i t e - d i m e n s i o n a l subspaces C o r o l l a r y . An immediate c o n s e ­ quence o f t h i s i s t h e f a c t t h a t C U T ) i s a l w a y s a s e m i - M o n t e l

s s p a c e .

T P r i m a r i l y because t h e ( w e a k ) c o m p l e t i o n o f C^g( T ) i s R , the b i d u a l C U T ) o f C U T ) admits t h e f o l l o w i n g i d e n t i f i c a t i o n :

s s

5 2

C £ ( T ) = [ h e R^T: g?\h\ f o r some g 6 L S C ( T ) ] .

The b i d u a l t o p o l o g y on C " ( T ) t u r n s out t o be t h e t o p o l o g y o f s i m p l e c o n v e r g e n c e , s i n c e the hounded s u b s e t s o f C ^ ( T ) a r e c o n ­ t a i n e d i n f i n i t e d i m e n s i o n a l s u b s p a c e s . ICow the f o r m t h a t C " ( T ) t a k e s as a v e c t o r space may remind y o u o f our o r i g i n a l C o n d i t i o n C.

The r e l a t i o n s h i p w i l l be s p e l l e d out d i r e c t l y i n Theorem 1 6 .

From t h e c h a r a c t e r i z a t i o n o f C " ( T ) , we o b s e r v e t h a t C " ( T )

s s

m fp

l i e s between C ( T ) and R , so t h a t C " ( T ) i s always dense i n R . s s The f a c t t h a t C " ( T ) c o n t a i n s a l l f u n c t i o n s bounded on T p l a y s a

s

d e c i s i v e r o l e i n t h e s t r u c t u r e o f C " ( T ) , as we b e g i n t o see i n the s

f o l l o w i n g lemma.

LEMMA

1 3 .

The c o l l e c t i o n ( ^{D n} ) ^{h € R T} ^ >

0

^{f o r m s a b a s e f o} r t t i e

hounded s u b s e t s o f C £ ( T ) , where I >^h = { k e C £ ( T ) : | k | * h } . M o r e ­ o v e r , i f ( ^ n ) ^ ! ^ ^T » "tben t h e r e e x i s t s a k e D ^ such t h a t k ( tⁿ) m h ( tⁿ) , f o r n = l , . . . , m .

P r o o f . That t h e c o l l e c t i o n forms a base f o r t h e bounded s e t s i s t r i v i a l , s i n c e t h e t o p o l o g y on C^'(T) i s t h e t o p o l o g y o f s i m p l e c o n v e r g e n c e . I f he R^T w i t h h > 0 , and i f ( " ^ J ^ i ^ T » ^{t h e n}

T

d e f i n e k € R by

Ch(t ) , n - 1 , . . . , m k ( t ) - « ⁿ

(.0, a l l o t h e r t £ T

S i n c e C " ( T ) c o n t a i n s a l l the bounded f u n c t i o n s d e f i n e d on T , s

we know t h a t k 6 C ^ ( T ) and hence t h a t k 6. . 1 PROPOSITION Ih a. C'''(T) = ( R^T) » f o r each T .

s

b . C g ( T ) i s a M o n t e l s p a c e , f o r each T . T

P r o o f . P a r t ( b ) f o l l o w s from ( a ) s i n c e ( R ) • i s a M o n t e l space f o r a l l T . Consequently we o n l y need p r o v e ( a ) . H o w e v e r , as a v e c t o r s p a c e , C ? ( T ) = ( R )T ^f , and i n a d d i t i o n , C ' ( T ) i s a s e m i -

s s m

Montel s p a c e , which means t h a t C^CT) = ( R ) ^f as a v e c t o r s p a c e .

5 3

Thus we o n l y have t o make c e r t a i n t h a t C*"(T) has the r i g h t t o p o - s

l o g y . I t i s easy t o check t h a t s e t s o f t h e form co b a l / h ( t ) r t e T } , T

where h € R and h > 0 , form a base o f n e i g h b o r h o o d s o f 0 i n ( R^T )^f. But v i a Lemma

1 3

t h e c o l l e c t i o n ( ^ h ^ R ¹ , h >

0

^{f o r m s}

a base o f n e i g h b o r h o o d s o f 0 i n C ' " ( T ) . I t t h e r e f o r e s u f f i c e s 3

T

t o p r o v e t h a t f o r each h e R w i t h h > 0 , we have ])£ = co b a l £ $^t/ h ( t ) s t f i l } .

On t h e one hand, c l e a r l y D £ 2 co k a l { $^t/ h ( t ) : t € l } . On t h e o t h e r hand, i f C g ( T ) but co b a l | £^t/ h ( t ) : t £ T | , t h e n

r* I n

where the ( ^{c n}) ^ _ i a*¹** "^{t i i e} ^ n ^ n = l ^{a r e a}P P^r° Pr i a : t e> where

J L i

^{c n}

t > l f

and where

l

^{c n}

l > °

>^{f o r} n = 1 , . • . , m . Now l e t k £ C £ ( T ) be d e f i n e d by

k( t ) = / ^{( | C n l / C n } h ( t n )} ' ^{f o r} * - ^ » ⁿ - ^{1 m} 1 0 , a l l o t h e r t £ T

Then k £ D^h w h i l e / « - ( l c ) =

^ L i

^c

n l

^{> : L}

•

^{T h u 3} J^^^Bh ' ^{A s a r e s u l t}

B n ^ co b&l jS^/hit) : t T | , and the p r o o f i s c o m p l e t e .1 As an immediate consequence o f P r o p o s i t i o n Ik we have COROLLARY 1 5 . C " " ( T ) = R^T f o r each T .

s

I t i s apparent by v i r t u e o f C o r o l l a r y 1 5 t h a t a l t h o u g h C^a( T ) s i s r e f l e x i v e o n l y when T i s d i s c r e t e , on t h e o t h e r hand, C g^M( T ) i s a l w a y s r e f l e x i v e .

For each T , t h e i n c l u s i o n s

c^s( T ) £ c ; ( T ) c ^CJ^M( T ) ~ R ^T

h o l d . I n a d d i t i o n , C ( T ) = C " ( T ) i f f T i s d i s c r e t e . I t i s S O

t h e r e f o r e v e r y n a t u r a l t o ask when C " ( T ) = fi^A ( w h i c h i t t u r n s o u t i s the same as a s k i n g when C " ( T ) i s r e f l e x i v e ! ) . The answer i s

s

5 4

c o n t a i n e d i n Theorem 1 6 , w h i c h i s c l o s e l y r e l a t e d t o Theorem 9 i n [ 5 ] . B e f o r e we s t a t e Theorem 1 6 we r e c a l l t h a t a l o c a l l y c o n v e x space f o r which a l l weak* compact s u b s e t s i n i t s d u a l a r e e q u i c o n t i n u o u s i s c a l l e d a s t r i c t l y Mackey space ( s e e £3])»

THEOREM 1 6 . F o r an a r b i t r a r y T^f t h e f o l l o w i n g s t a t e m e n t s a r e e q u i v a l e n t :

a. C J ( T ) - R^T.

b . T i s w i l d l y o s c i l l a t o r y ( i . e . , T s a t i s f i e s C o n d i t i o n C ) • c . C ' ( T ) i s b o r n o l o g i c a l .

s

d. C U T ) i s u l t r a b o r n o l o g i c a l .

©• C ! ( T ) i s i n f r a b a r r e l e d . s

f . C*^Q(T) i s a M o n t e l s p a c e , g . C » ( T ) i s b a r r e l e d .

s

h . C ^ ( T ) i s a s t r o n g Mackey space*

i . T h e r e e x i s t s a c o m p l e t e l y r e g u l a r H a u s d o r f f space U and a homeomorphism tp z C " ( T ) — > C^a( U ) which i s onto and w h i c h p r e -

• s s

s e r v e s e i t h e r t h e n a t u r a l o r d e r o r t h e n a t u r a l p o i n t w i s e m u l t i p l i c a t i o n o f b o t h s p a c e s .

P r o o f . The f a c t t h a t ( a ) ^ = ^ ( b ) i s o b v i o u s . Assume t h a t ( b ) h o l d s , and l e t B C C ' ( T ) be c o n v e x and b a l a n c e d and a b s o r b a l l

s

bounded s u b s e t s o f C ^ ( T ) . Then B 2 co b a l { 5^ / h ( t ) : t £ T j f o r T

some h e R w i t h h > 0 • By ( b ) we know t h a t t h e r e i s a s t r i c t l y p o s i t i v e g £ L S C ( T ) such t h a t h ^ g , so t h a t

B 2 co b a l j $^t/ g ( t ) : t f i l j a •

C o n s e q u e n t l y B i s a n e i g h b o r h o o d o f 0 i n C U T ) , v e r i f y i n g t h a t s

C ^ ( T ) i s b o r n o l o g i c a l . Thus ( c ) h o l d s . Because t h e bounded s u b s e t s o f C^S( T ) a r e r e l a t i v e l y compact i n C J C T ) , we have

( c ) ^ = 5 > ( d ) and a l s o <^e) * > - * < f ) « - * ( g ) ( s e e [ 8 , p . 2 3 l ] ) . I n any l o c a l l y c o n v e x s p a c e , ( d ) = - ^ ( e ) and ( g ) — M h ) • Now t o show

55

t h a t ( h ) = ^ ( a ) , assume t h a t ( a ) i s f a l s e . L e t h€ R ^T \ C " ( T ) , and l e t D = f o } U [ h ( t ) % ^ : t e TJ • N o t e t h a t D i s compact ( t h u s a u t o m a t i c a l l y weak* c o m p a c t ) i n C ^ ( T ) s i n c e any n e i g h b o r h o o d o f 0 i n C " ( T ) c o n t a i n s a l l but f i n i t e l y many e l e m e n t s o f D.

However, i f D w e r e e qui c o n t i n u o u s , then t h e r e would n e c e s s a r i l y be a g € l S C ( T ) such t h a t g > 0 and such t h a t

D⁹ 2 B* = co b a l \ S^t/ g ( t ) : te ?} .

But t h e n | h l < g , which i s f a l s e s i n c e h ^ C J J ( T ) • Thus C ^ ( T ) i s not a s t r o n g Mackey s p a c e , which c o m p l e t e s t h e p r o o f t h a t

( h ) « = » ( a ) • That ( a ) = £ > ( i ) comes from l e t t i n g U b e T w i t h IT T

t h e d i s c r e t e t o p o l o g y , so t h a t C^S( U ) = R = R • Then <{ i s m e r e l y the i d e n t i t y f u n c t i o n . F i n a l l y , t o p r o v e t h a t ( i ) — 5 * ( a ) , we

o b s e r v e t h a t i f f ( g ) = 1 , t h e n <p(D ) = B¹ . However, D i s compact i n C £ ( T ) , w h i l e B-j^ i s compact i n C^S( U ) i f f U i s d i s c r e t e . Thus C^B( U ) i s r e f l e x i v e , and c o n s e q u e n t l y C*I(T) i s

s s

rp

a l s o r e f l e x i v e , so by C o r o l l a r y 1 5 , we o b t a i n C " ( T ) « R , which s

p r o v e s ( a ) • |

S e v e r a l comments a r e i n o r d e r w i t h r e s p e c t t o Theorem 1 6 . F i r s t o f a l l , t h e e q u i v a l e n c e o f ( a ) and ( b ) f u l f i l l s o u r promise t o c h a r a c t e r i z e i n l o c a l l y c o n v e x space terms t h o s e T which a r e w i l d l y o s c i l l a t o r y . S e c o n d l y , t h e ^ o f ( i ) need

not be l i n e a r . However, i f ^ happens t o be a l i n e a r homeomorphism onto ^CS ( U ) , t h e n we c o n j e c t u r e t h a t need n o t p r e s e r v e e i t h e r the o r d e r o r t h e m u l t i p l i c a t i o n o f t h e s p a c e s e n t a i l e d i n o r d e r f o r ( i ) t o be e q u i v a l e n t t o ( a ) .

F i n a l l y , the c o n d i t i o n " C U T ) i s a Mackey s p a c e " i s c o n - s

s p i c u o u s l y a b s e n t from t h e a r r a y o f s t a t e m e n t s i n Theorem 1 6 , and f o r a v e i y g o o d r e a s o n . We s i m p l y do n o t know f o r what T t h e space C* ( T ) i s a Mackey s p a c e . C o n c e i v a b l y C * ( T ) i s a Mackey

S 8

space i f f T i s w i l d l y o s c i l l a t o r y , but t o p r o v e i t one would i n 56

a l l l i k e l i h o o d need a much more p e n e t r a t i n g u n d e r s t a n d i n g o f t h e convex compact s u b s e t s o f C £ ( T ) than we now h a v e . I n f a c t , f o r us t o f i n d e v e n one T f o r w h i c h C^f( T ) i s n o t a Mackey s p a c e has

s

been no t r i v i a l t a s k . We p r e s e n t t h e example as P r o p o s i t i o n 1 7 . PROPOSITION 1 7 . I f T = ^ 3 N \ N and i f we assume t h e continuum h y p o t h e s i s , t h e n C * ( T ) i s not a Mackey s p a c e .

s

P r o o f . L e t S be t h e s e t o f P - p o i n t s i n T . By Example 1 2 we know t h a t S i s dense i n T , and t h a t t h e r e e x i s t s a ( p o s i t i v e )

T

€ R which i s e v e r y w h e r e unbounded on S • S i n c e T i s compact, P r o p o s i t i o n 8 t e l l s us t h a t no p o s i t i v e g d L S C ( T ) i s e v e r y w h e r e unbounded. T h e r e f o r e , i f h = ^ o^ g ^{o n T |} ^ ^{h e n} k ^{i s n o t} m a j o r ­

i z e d by any g £ L S C ( T ) . Now l e t

A = { h ( * ) f y t ) ^{5 % 6 T} ) »

so t h a t A C C " ( T ) , and l e t D = co b a l A ^R . E v i d e n t l y D i s T s

T

compact i n R • U s i n g t h e f a c t t h a t each p o i n t o f S i s a P - p o i n t i n T , we w i l l show t h a t D S C £ ( T ) • W i t h an argument s i m i l a r t o one found i n Example 1 1 o f [ 6 ] , we w i l l f i r s t show t h a t i f k e D , t h e n k ( t ) * 0 f o r a t most c o u n t a b l y many t £ T ( w h i c h such t

T

o f c o u r s e must be i n S ) . I n t h a t d i r e c t i o n , l e t k e R and assume t h a t k ( s ) * 0 f o r uncountably many 8 6 S • F o r each r > 0 , l e t

S^p = j s 6 S : h ( s ) £ r } ,

so t h a t U S ^W a S . Then t h e r e n e c e s s a r i l y e x i s t s an e > 0 and r>o r

r ^f > 0 such t h a t ( k ( t ) f > £ f o r u n c o u n t a b l y many (and hence c o u n t a b l y many) e l e m e n t s o f S^{p f} . N e x t , l e t p € N such t h a t p € . / 2 > r^f

and l e t C ^ ) ] ^ C S ^{f t} such t h a t ) k ( s^m) | > €. , f o r a l l m€,N • F i n a l l y , l e t

V = { d € R^T :

| K s

^m

) | ^ e / 2

, m ~ l , . . . , p } ,

so t h a t V i s a n e i g h b o r h o o d o f 0 i n R^T. Now i f J 6 . V , t h e n

5 7