HEWITT-NACHBIN SPACES
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NORTH-HOLLAND MATHEMATICS STUDIES
17
Notas de Matematica (57) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester
Hewitt-Nachbin Spaces
MAURICE D. W E I R Naval Postgraduate School Monterey, California USA
1975
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM OXFORD AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK
0 NORTH-HOLLAND
PUBLISHING COMPANY
- 1975
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
Library of Congress Catalog Card Number: 14 2899 1 North-Holland ISBN .for this Series: 0 7204 2700 2 North-Holland ISBN for this Volume: 0 1204 21 18 5 American Elsevier ISBN: 0 444 10860 2
Publishers :
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NORTH-HOLLAND PUBLISHING COMPANY, LTD. - OXFORD Sole distributors for the U.S.A. and Canada: AMERICAN ELSEVIER PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017
PRINTED I N THE NETHERLANDS
PREFACE
T h i s book i s a d d r e s s e d t o t h e g r a d u a t e s t u d e n t who, having completed t h e s t a n d a r d f i r s t c o u r s e i n g e n e r a l topology, w i s h e s t o l e a r n a b o u t more recent developments i n t h i s f i e l d . T h i s book i s a l s o i n t e n d e d a s a r e f e r e n c e f o r t h o s e who a r e c a r r y i n g on m a t h e m a t i c a l r e s e a r c h . My o b j e c t i v e i s t o expose t h e t h e o r y of Hewitt-Nachbin
s p a c e s (also known a s r e a l c o m p a c t o r
Q-spaces) i n a cohesive
f a s h i o n which t a k e s i n t o a c c o u n t t h e many s y n e r g i s t i c p o i n t s of view from which t h e s e s p a c e s may b e i n v e s t i g a t e d .
The
major emphasis i s p l a c e d on t h e s t u d y of Hewitt-Nachbin s p a c e s from a t o p o l o g i c a l p e r s p e c t i v e u t i l i z i n g f i l t e r s on t h e s p a c e under i n v e s t i g a t i o n v i c e t h e a l g e b r a i c p e r s p e c t i v e u t i l i z i n g i d e a s of t h e r i n g C ( X ) of a l l r e a l - v a l u e d c o n t i n u o u s
X
f u n c t i o n s on
X
c a l ve ct or space.
o r the consideration of
C ( X ) a s a topologi-
Although I a p p e a l t o much of t h e t h e o r y of
R i n q s o f Continuous F u n c t i o n s a s developed by L . Gillman and M.
Jerison,
t h e n e c e s s a r y t o o l s f o r t h i s book a r e f u l l y d e v e l -
oped h e r e . The c o n t e n t s o f t h i s book f a l l n a t u r a l l y i n t o f o u r p a r t s . Chapter 1 m o t i v a t e s t h e n o t i o n o f a Hewitt-Nachbin s p a c e i n t h e more g e n e r a l s e t t i n g o f
E-compact s p a c e s .
That p o i n t o f
view i s a l s o c o n c e p t u a l l y u s e f u l b e c a u s e i t p r o v i d e s t h e prop-
e r s e t t i n g i n which t o view Hewitt-Nachbin s p a c e s from a c a t e g o r i c a l p e r s p e c t i v e . I n Chapter 2 t h e p r o p e r t y o f H e w i t t Nachbin c o m p l e t e n e s s i s f o r m u l a t e d i n t e r m s o f z e r o - s e t u l t r a f i l t e r s on t h e s p a c e
X.
A s y s t e m a t i c s t u d y of t h e p r o p e r t i e s
and known c h a r a c t e r i z a t i o n s of Hewitt-Nachbin s p a c e s then ens u e s from t h a t s t a n d p o i n t .
H e r e a l s o i s developed t h e H e w i t t -
Nachbin c o m p l e t i o n , b u t i n t h e g e n e r a l s e t t i n g of WallmanF r i n k t y p e c o m p a c t i f i c a t i o n s and c o m p l e t i o n s .
*
R e c e n t develop-
men t s i n v o l v i n g C-embedding, C -embedding, z- embedding, and u-embedding a r e b r o u g h t i n t o p l a y c o u p l e d w i t h t h e i r a p p l i c a t i o n t o t h e problem of t h e Hewitt-Nachbin c o m p l e t i o n of a product . C h a p t e r 3 r e l a t e s Hewitt-Nachbin c o m p l e t e n e s s t o t h e uniform s p a c e c o n c e p t . Here t h e i m p o r t a n t Nachbin- S h i r o t a Theorem i s evolved and u t i l i z e d t o e s t a b l i s h K a t g t o v ' s r e s u l t
vi
PREFACE
t h a t every paracompact Hausdorff s p a c e of nonmeasurable c a r d i n a l i s Hewitt-Nachbin complate.
The r e c e n t work of Buchwalter
and Schmets, viewing Hewitt-Nachbin s p a c e s i n t h e c o n t e x t o f functional analysis, i s a l s o discussed.
And s e v e r a l c l a s s e s
of s p a c e s , such a s t h e a l m o s t r e a l c o m p a c t and t h e
cb-spaces,
a r e i n v e s t i g a t e d i n t h e i r r e l a t i o n s h i p t o t h e Hewitt-Nachbin spaces. Chapter 4 s t u d i e s t h e i n v a r i a n c e and i n v e r s e i n v a r i a n c e of Hewitt-Nachbin completeness under c o n t i n u o u s mappings. Unl i k e t h e p r o p e r t y of compactness, Hewitt-Nachbin c o m p l e t e n e s s i s n o t i n v a r i a n t under an a r b i t r a r y c o n t i n u o u s mapping,
In
f a c t an example i s g i v e n which d e m o n s t r a t e s t h a t t h e p e r f e c t image of a Hewitt-Nachbin s p a c e need n o t be Hewitt-Nachbin complete.
T h i s m o t i v a t e s t h e i n v e s t i g a t i o n of s e v e r a l c l a s s e s
of mappings germane t o t h e i n v a r i a n c e of Hewitt-Nachbin comp l e t e n e s s such a s t h e p e r f e c t mappings, t h e and t h e the
WZ-mappings.
E-perfect,
z - c l o s e d mappings,
These mappings a r e t h e n g e n e r a l i z e d t o
E-closed,
and weakly
g e t h e r with t h e i r a s s o c i a t i o n t o t h e
E-closed mappings toE-compact s p a c e s s t u d i e d
i n Chapter 1. And t h e c i r c l e i s c o m p l e t e . I t i s d i f f i c u l t t o r e c o g n i z e a l l t h o s e who have c o n t r i b u t e d , i n one way o r a n o t h e r , t o the development of t h i s book. F i r s t I am i n d e b t e d t o my two t e a c h e r s , Richard A . Alo and Harvey L. S h a p i r o , who i n s p i r e d m e t o w r i t e t h i s book, r e a d t h e p r e l i m i n a r y v e r s i o n s of t h e m a n u s c r i p t , and offered sugg e s t i o n s and c o r r e c t i o n s t o t h e o r g a n i z a t i o n and t o t h e p r o o f s
too numerous t o s p e c i f i c a l l y mention.
And I a l s o wish t o thank
P r o f e s s o r s W . W i s t a r Comfort, R . E n g e l k i n g , S . F r a n k l i n , H . H e r r l i c h , J . Mack, and S . Mrbwka f o r t h e i r a d d i t i o n s t o my b i b l i o g r a p h y and t h e i r encouragement.
Nancy Colmer d i d a
b e a u t i f u l job i n typing t h e manuscript. F i n a l l y I w i s h t o thank P r o f e s s o r Leopoldo Nachbin €or h i s k i n d h e l p w i t h t h e e d i t i n g , and my d e p a r t m e n t of mathematics f o r p r o v i d i n g res e a r c h s u p p o r t f o r t h e completion o f t h i s p r o j e c t . January 1975
Maurice D . Weir Naval P o s t g r a d u a t e School Monterey, C a l i f o r n i a U . S . A .
vii
TABLE O F CONTENTS PREFACE
.......................................
V
CHAPTER 1
1
EMBEDDING I N TOPOLOGICAL PRODUCTS
1. 2.
3.
4.
5.
......................... T h e E m b e d d i n g L e m m a . . ............................ completely R e g u l a r Spaces . . . . . . . . . . . . . . . . . . . . . . E - C o m p a c t Spaces ................................. A C a t e g o r i c a l Perspective ........................ Notation and Terminology
5
9 15 23 32
CHAPTER 2
41
HEWITT-NACHBIN S P A C E S AND CONVERGENCE
........................
6.
3-Filters a n d C o n v e r g e n c e
7.
H e w i t t - N a c h b i n C o m p l e t e n e s s v i a Ideals, F i l t e r s , and N e t s
8.
C h a r a c t e r i z a t i o n s a n d P r o p e r t i e s of H e w i t t - N a c h b i n
9.
Hewitt-Nachbin Completions
..................................
................................... ....................... a n d v - E m b e d d i n g .....................
Spaces. 10.
z-Embedding
11.
H e w i t t - N a c h b i n C o m p l e t i o n s of P r o d u c t s
41
58 74
96 108
...........
120
CHAPTER 3 HEWITT-NACHBIN S P A C E S , U N I F O R M I T I E S , AND RELATED TOPOLOGICAL S P A C E S 12.
A R e v i e w of U n i f o r m Spaces
.......................
137
...
143
..............
157
13.
H e w i t t - N a c h b i n C o m p l e t e n e s s a n d U n i f o r m Spaces
14.
Almost Realcompact and
cb-Spaces..
136
CHAPTER 4 HEW1 TT- NACHBIN COMPLETENESS AND CONTINUOUS MAPPINGS
1 71
15.
Some C l a s s e s of Mappings .........................
173
16.
P e r f e c t Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
186
17.
C l o s e d Mappings a n d H e w i t t - N a c h b i n
198
18.
WZ-Mappings
213
19.
E-Perfect
...................................... Mappings ...............................
225
.................................
237
BIBLIOGRAPHY. INDEX.
Spaces
........
........................................
261
T h i s book is d e d i c a t e d t o Deo W e i r and F l o r a Beaudin Gale Hempstead Maia Deborah and Rene)e E l i z a b e t h Gary and J e a n e Lonnie, Lynn, and Eva Sam and J u d y Mardie and C r a i g and t o my many t e a c h e r s
Chapter 1 EMBEDDING
2 TOPOLOGICAL PRODUCTS
Some of t h e most i m p o r t a n t r e s u l t s o f c l a s s i c a l a n a l y s i s depend on p r o p e r t i e s p o s s e s s e d by r e a l - v a l u e d c o n t i n u o u s funct i o n s d e f i n e d o v e r compact domains: f o r i n s t a n c e , t h e boundedn e s s o f t h e s e f u n c t i o n s and t h e f a c t t h a t t h e y assume t h e i r maximum and minimum v a l u e s .
I t i s not c u r i o u s , then,
t h a t the
s t u d y of compact s p a c e s h a s been o f c o n s i d e r a b l e i n t e r e s t i n t h e i n v e s t i g a t i o n o f p r o p e r t i e s of g e n e r a l t o p o l o g i c a l s p a c e s . The t h e o r y o f compact s p a c e s was s t u d i e d e x t e n s i v e l y by P . A l e x a n d r o f f and P. Urysohn i n t h e i r 1 9 2 9 p a p e r "MLmoire s u r
l e s Espaces Topologiques Compact."
I n 1 9 3 0 A . Tychonoff
proved t h e i m p o r t a n t a d d i t i o n a l r e s u l t t h a t complete r e g u l a r i t y i s t h e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r a t o p o l o g i c a l s p a c e t o b e a subspace o f some compact Hausdorff s p a c e . The compact s p a c e c o n s t r u c t e d by Tychonoff was s u b s e q u e n t l y U
s t u d i e d by E . Cech i n h i s 1 9 3 7 p a p e r "On bicompact spaces.Il S t i l l l a t e r , i n 1948, P.
Samuel i n v e s t i g a t e d t h e n o t i o n o f compactness i n t h e c o n t e x t o f uniform s p a c e s and t h e t h e o r y o f ultrafilters.
These and f u r t h e r i n v e s t i g a t i o n s have r e v e a l e d
how t h e compact s p a c e s p l a y a c e n t r a l r o l e i n g e n e r a l t o p o l o g y and t h e y have i d e n t i f i e d an i m p o r t a n t r e l a t i o n s h i p between t h e topology of a s p a c e and i t s a s s o c i a t e d r i n g o f r e a l - v a l u e d continuous functions:
i n f a c t t h e t o p o l o g y o f a compact Haus-
d o r f f s p a c e i s e n t i r e l y determined by i t s r i n g o f r e a l - v a l u e d continuous f u n c t i o n s .
T h i s n o t i o n w i l l be f o r m u l a t e d i n a
p r e c i s e way f u r t h e r on i n t h e s e q u e l . The complete m e t r i c s p a c e s , and more g e n e r a l l y t h e comp l e t e uniform s p a c e s , a l s o occupy key p o s i t i o n s i n t h e s t u d y of t o p o l o g i c a l spaces and i t s a p p l i c a t i o n s t o a n a l y s i s .
For
i n such s p a c e s t h e convergence o f s e q u e n c e s o r n e t s i s c h a r a c t e r i z e d by t h e i m p o r t a n t Cauchy p r o p e r t y . Complete m e t r i c s p a c e s w e r e i n t r o d u c e d by M. FrLchet i n h i s 1906 p a p e r "Sur Quelques P o i n t s d u C a l c u l F o n c t i o n n e l " and i t w a s F . H a u s d o r f f who proved i n h i s 1914 book Grundziiqe der Menqenlehre t h a t e v e r y m e t r i c s p a c e h a s a c o m p l e t i o n : h i s proof i s based on
EMBEDDING I N TOPOLOGICAL PRODUCTS
2
t h e f a m i l i a r method of d e f i n i n g t h e i r r a t i o n a l numbers by means o f Cauchy s e q u e n c e s of r a t i o n a l n u m b e r s . W e i l i n h i s p a p e r , "Sur l e s Espaces
A.
e t s u r l a Topologie G&&ale," o f a uniform s p a c e .
'a
Then i n 1937
S t r u c t u r e Uniforme
introduced the g e n e r a l notion
Another approach t o uniform s p a c e s was
developed by J . Tukey i n 1940.
A n e x c e l l e n t s u r v e y o f uniform
s p a c e s a p p e a r s i n t h e 1964 book u n i f o r m Spaces by J . R . I
Isbell.
Now t h e compact s p a c e s and t h e complete s p a c e s a r e w e l l
behaved w i t h i n t h e framework s u p p o r t i n g t h e s t u d y of g e n e r a l topological spaces:
c l o s e d s u b s e t s o f compact ( c o m p l e t e )
s p a c e s a r e themselves compact ( r e s p e c t i v e l y , complete) and t o p o l o g i c a l p r o d u c t s of compact ( c o m p l e t e ) s p a c e s a r e compact (complete).
I n f a c t any compact Hausdorff s p a c e can be c h a r -
a c t e r i z e d a s a s p a c e t h a t i s homeomorphic t o some c l o s e d subs p a c e of a t o p o l o g i c a l p r o d u c t of t h e c l o s e d u n i t i n t e r v a l
[x
x
11 i n t h e r e a l l i n e . I t would seem n a t u r a l t o g e n e r a l i z e t h a t i d e a and c o n s i d e r t h e c l a s s o f t o p o l o g i c a l : 0
s p a c e s t h e members of which a r e homeomorphic t o any c l o s e d subs p a c e o f t o p o l o g i c a l powers of some g i v e n s p a c e
E.
This idea
o r i g i n a t e d i n t h e 1958 p a p e r by R. Engelking and S . Mrdwka, and f u r t h e r i n v e s t i g a t i o n s have a p p e a r e d i n t h e p a p e r s of R. Blefko (1965 and 1 9 7 2 ) , H . H e r r l i c h ( 1 9 6 7 ) , and S . Mrdwka (1966, 1968, and 1 9 7 2 ) .
O n e s p e c i a l i n s t a n c e of t h a t g e n e r a l -
i z a t i o n i s t h e case i n which t h e s p a c e
E
is t h e real l i n e .
T h i s c l a s s of s p a c e s would n e c e s s a r i l y i n c l u d e t h e compact s p a c e s , b u t o t h e r s p a c e s would b e i n c l u d e d a s w e l l , the r e a l l i n e i t s e l f .
such a s
These s p a c e s a r e t h e Hewitt-Nachbin
spaces t h a t a r e t o be i n v e s t i g a t e d i n t h i s book. O r i g i n a l l y known a s
Q-spaces by E . H e w i t t and a s s a t u -
r a t e d s p a c e s by L. Nachbin, many a d j e c t i v e s have been employed naming t h e Hewitt-Nachbin s p a c e s . With p u b l i c a t i o n o f t h e 1960 t e x t , Rinqs of Continuous F u n c t i o n s by L . Giflman and M . J e r i son, t h e s e s p a c e s have most r e c e n t l y b e e n c a l l e d r e a l c o m p a c t spaces.
However i t t u r n s o u t t h a t t h e t e r m " r e a l f ' h a s been
j u s t i f i a b l y o b j e c t i o n a b l e t o numerous m a t h e m a t i c i a n s . Moreover, t h e s e s p a c e s a r e more c l o s e l y r e l a t e d t o t h e i d e a of completen e s s r a t h e r than t h e i d e a of compactness. I n f a c t , a l l of t h e
terms
e-complete,
realcomplete,
f u n c t i o n a l l y c l o s e d , and
3
IX'IRIDLJCl'ION
H e w i t t have been used by v a r i o u s m a t h e m a t i c i a n s i n r e f e r r i n g
t o Hewitt-Nachbin s p a c e s .
Our t e r m i n o l o g y i s j u s t i f i e d by t h e
p r e c e d i n g d i s c u s s i o n and t h e f a c t t h a t t h e s t u d y o f t h e s e s p a c e s was i n i t i a t e d by Edwin H e w i t t and Leopoldo Nachbin i n d e p e n d e n t l y d u r i n g t h e y e a r s 1947-1948.
The work r e c e i v e d
a t t e n t i o n when H e w i t t p u b l i s h e d i n 1948 h i s fundamental and s t i m u l a t i n g paper,
I."
"Rings 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 ,
H e w i t t s t u d i e d h i s s p a c e s w i t h i n t h e framework of t h e
a l g e b r a i c r i n g of 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 d e f i n e d on a t o p o l o g i c a l s p a c e , and h e d e m o n s t r a t e d t h a t t h e s p a c e s s h a r e d many t o p o l o g i c a l p r o p e r t i e s i n common w i t h t h o s e e n joyed by t h e compact s p a c e s .
For i n s t a n c e , t h e Hewitt-Nachbin
p r o p e r t y i s s h a r e d by t h e c l o s e d s u b s e t s a s w e l l a s t h e topoHowever, w e w i l l
l o g i c a l p r o d u c t s o f Hewitt-Nachbin s p a c e s .
see l a t e r t h a t Hewitt-Nachbin s u b s p a c e s need n o t be c l o s e d . Nachbin became i n t e r e s t e d i n what h e then c a l l e d s a t u r a t e d s p a c e s p r i o r t o 1947 from t h e p o i n t of view o f Nachbin uniform s t r u c t u r e s .
The f i r s t r e s u l t s which N a c h b i n p u b l i s h e d
from t h i s p o i n t o f view a p p e a r i n h i s 1950 p a p e r .
(Actually
H e w i t t l e a r n e d of N a c h b i n ' s work i n 1948 and u t i l i z e d t h e
Nachbin approach i n one of t h e p r o o f s a p p e a r i n g i n h i s subs e q u e n t 1950 p a p e r .
W e w i l l i n v e s t i g a t e N a c h b i n ' s p o i n t of
view i n Chapter 3 . )
Nachbin c h a r a c t e r i z e d h i s s a t u r a t e d
spaces i n t e r m s of t h e space ous f u n c t i o n s on t h e s p a c e
C ( X ) of a l l r e a l - v a l u e d c o n t i n u X,
where
C(X)
i s considered a s a
t o p o l o g i c a l v e c t o r s p a c e w i t h t h e topology o f uniform convergence o n compact s e t s .
He showed t h a t each semi-norm
bounded on t h e bounded s e t s o f C(X)
C ( X ) i s continuous
i s b o r n o l o q i c a l ) i f and o n l y i f
t o t h e uniform s t r u c t u r e g e n e r a t e d by
X
that is
(i.e., that
i s complete r e l a t i v e @(X)
.
These l a t t e r
c o n c e p t s w i l l be f u l l y exposed i n t h e f i r s t s e c t i o n of C h a p t e r 3.
C o r o l l a r y 1 3 . 6 ( 1 ) e s t a b l i s h e s Nachbin's c h a r a c t e r i z a t i o n
of Hewitt-Nachbin c o m p l e t e n e s s . I n 1951-1952 T . S h i r o t a , and i n 1957-1958 S . MrGwka, a l s o made numerous and i m p o r t a n t c o n t r i b u t i o n s t o t h e f o u n d a t i o n a l t h e o r y of Hewitt-Nachbin s p a c e s .
The p u b l i c a t i o n o f t h e 1960
Gillman and J e r i s o n t e x t then provided t h e f i r s t s y s t e m a t i c survey o f Hewitt-Nachbin spaces i n c o r p o r a t i n g b o t h t h e H e w i t t
4
EMBEDDING
I N TOPOLOGICAL PRODUCTS
and t h e Nachbin a p p r o a c h e s .
That t e x t was s t i m u l a t e d b y M.
Henriksen, who t o g e t h e r w i t h J . I s b e l l i n 1958, also made v a l uable c ont ri butions i n t h i s a r e a .
R e c e n t l y s e v e r a l books i n
g e n e r a l topology have i n c l u d e d a t l e a s t some mention o f H e w i t t Nachbin s p a c e s ( a l t h o u g h r e f e r r e d t o a s r e a l c o m p a c t s p a c e s ) : n o t a b l y t h e 1968 t e x t by J. Nagata and t h e 1 9 7 0 t e x t by S . Willard. Given t h a t t h e c l a s s o f Hewitt-Nachbin s p a c e s a r i s e s n a t u r a l l y i n t h e i n v e s t i g a t i o n s o f complete and compact spaces, and more g e n e r a l l y from c o n s i d e r a t i o n s o f embedding s p a c e s i n t o t o p o l o g i c a l powers o f some g i v e n s p a c e , one might wonder what r o l e t h e s e s p a c e s p l a y w i t h i n t h e framework o f g e n e r a l topology.
I t t u r n s o u t t h a t t h e Hewitt-Nachbin s p a c e s p l a y a
r o l e w i t h i n t h a t framework t h a t r u n s p a r a l l e l t o t h a t p l a y e d by t h e compact s p a c e s .
Namely, t h e topology of a H e w i t t -
Nachbin s p a c e i s e n t i r e l y determined by i t s r i n g of 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 a l t h o u g h t h a t r i n g may c o n t a i n unbounded f u n c t i o n s .
Moreover, w e w i l l see t h a t t h e H e w i t t -
Nachbin s p a c e s correspond v e r y n e a r l y t o t h e c l a s s o f complete uniform s p a c e s . E v i d e n t l y t h e r e a r e a v a r i e t y o f a p p r o a c h e s t h a t might be s e l e c t e d i n i n i t i a t i n g a n y s t u d y o f Hewitt-Nachbin s p a c e s . T h i s book w i l l b e g i n t h a t s t u d y by c o n s i d e r i n g such a s p a c e a s one which i s homeomorphic t o a c l o s e d subspace of a t o p o l o g i c a l product of real l i n e s .
T h i s approach h a s t h e a d v a n t a g e of
s i m p l i c i t y and immediately exposes t h e c l a s s of Hewitt-Nachbin s p a c e s i n c l o s e a s s o c i a t i o n w i t h t h e p r o p e r t i e s of completen e s s and compactness. I t h a s t h e added a t t r a c t i o n o f prov i d i n g t h e m o t i v a t i o n f o r examining t h e s a l i e n t f e a t u r e s i n t h e g e n e r a l s e t t i n g o f c o n s i d e r i n g t o p o l o g i c a l powers o f some a r b i t r a r y given space
E:
problem i n t o s h a r p f o c u s .
t h i s w i l l bring the nature of t h a t A t t h e n e x t s t a g e Hewitt-Nachbin
completeness w i l l b e t r a n s l a t e d i n t o convergence c r i t e r i a a s s o c i a t e d w i t h c e r t a i n c l a s s e s of f i l t e r s d e f i n e d on t h e space i n q u e s t i o n .
T h i s w i l l s u p p o r t H e w i t t ' s approach t o
Hewitt-Nachbin s p a c e s and s e t t h e s t a g e which b r i n g s t h e a l g e b r a i c r i n g of real-valued continuous f u n c t i o n s i n t o p l a y . Moreover i t w i l l f a c i l i t a t e a r e v e a l i n g c o n s t r u c t i o n t h a t
NOTATION A N D TERMINOLOGY
5
embeds a g i v e n t o p o l o g i c a l s p a c e d e n s e l y w i t h i n an a p p r o p r i a t e Hewitt-Nachbin s p a c e . That c o n s t r u c t i o n a p p e a r s i n t h e p a p e r s
of R . Alo and H . L. S h a p i r o (196819 and 1968B) g e n e r a l i z i n g t h e z e r o - s e t f i l t e r c o n s t r u c t i o n s a s p r e s e n t e d i n C h a p t e r s 6 and 8 of t h e Gillman and J e r i s o n t e x t .
W e w i l l need t o d e v e l o p a
t h e o r y o f g e n e r a l i z e d f i l t e r s i n o r d e r t o implement t h a t development and w e s h a l l do s o i n t h e n e x t c h a p t e r .
Finally
w e w i l l c o n s i d e r Hewitt-Nachbin s p a c e s i n t h e c o n t e x t o f u n i -
form s t r u c t u r e s . Before w e embark on o u r f o r m a l s t u d y o f Hewitt-Nachbin s p a c e s , a few remarks of a g e n e r a l n a t u r e a r e i n o r d e r .
The
n o t a t i o n and terminology employed i n t h i s book w i l l c l o s e l y f o l l o w t h a t o f t h e 1960 L . G i l l m a n a n d M. J e r i s o n t e x t and t h e 1974 R . Alo and H . L . S h a p i r o book.
Other r e f e r e n c e s t h a t a r e
u s e f u l a r e t h e 1955 t e x t , G e n e r a l Topoloqy by J . L . K e l l e y and t h e 1966 t e x t , Topoloqy by J . Dugundji.
A l l of t h e s e books
a r e l i s t e d i n the bibliography.
More precise r e f e r e n c e t o
t h e s e works i s sometimes u s e f u l :
(Gillman and J e r i s o n , 8 . 4 ) ,
f o r example, d e n o t e s a r e f e r e n c e t o S e c t i o n 4 of C h a p t e r 8 o f t h e Gillman and J e r i s o n t e x t . by t h e a u t h o r ' s name and d a t e :
Research p a p e r s a r e r e f e r r e d t o f o r example, " t h e 1957A p a p e r
of S . Mr6wka." T h i s book i s e n t i r e l y s e l f - c o n t a i n e d a l t h o u g h w e w i l l s t a t e ( o f t e n w i t h o u t p r o o f ) a l l of t h e r e s u l t s t h a t a r e needed from t h e f i r s t t h r e e c h a p t e r s of Gillman and J e r i s o n . The r e a d e r who i s u n f a m i l i a r w i t h t h e s e r e s u l t s may f i n d them more l u c i d , a s w e l l a s h i s u n d e r s t a n d i n g of t h e m a t e r i a l i n t h i s book g r e a t l y enhanced., by r e f e r r i n g d i r e c t l y t o t h e G i l l man and J e r i s o n t e x t . S e c t i o n 1:
N o t a t i o n and Terminoloqy
W e assume t h a t t h e r e a d e r h a s a knowledge o f t h e e l e m e n -
t a r y f a c t s c o n c e r n i n g t o p o l o g i c a l s p a c e s and t h e t h e o r y o f a l g e b r a i c r i n g s . However, t h e r e a r e several basic n o t i o n s t h a t c a n be a source of confusion; f o r i n s t a n c e , t h e s e p a r a t i o n axioms and t h e n o t i o n o f a paracompact s p a c e .
We w i l l state
t h e d e f i n i t i o n s o f such t e r m s i n t h i s s e c t i o n i n o r d e r t o a v o i d any c o n f u s i o n . formed.
Only a q u i c k p e r u s a l i s n e c e s s a r y f o r t h e in-
6
EMBEDDING I N TOPOLOGICAL PRODUCTS
If
s e t of B
i s an a r b i t r a r y s e t , t h e n
X
1x1
and
X
denotes the c a r d i n a l i t y of
a r e a r b i t r a r y sets, then
r e l a t i v e complement of
in
A
+
s y s t e m of p o s i t i v e i n t e g e r s by
The n o t a t i o n
f
: X
+
Y
Y.
and codomain
X
and
A
The system of r e a l numbers
B.
IR , t h e subsystem of r a t i o n a l n u m b e r s by
domain
~f
X.
B \ F = ( X F B : x#A) d e n o t e s t h e
R , t h e subsystem o f n o n - n e g a t i v e r e a l numbers
i s denoted by
by
P(X) d e n o t e s t h e power
cp, and t h e sub-
.
N
stands f o r a function The f u n c t i o n
f
with
is surjective
f
i f and o n l y i f t h e image
Y;
f ( X ) = ( f ( p ) : P E X ] i s t h e codomain i t i s i n j e c t i v e provided f ( x ) = f ( y ) i m p l i e s x = y . The
symbols
f ( A ) and
f - l ( A ) d e n o t e , r e s p e c t i v e l y , t h e image and
i n v e r s e image of a s e t functions f ( g ( x )) g
.
f
and
g
A
under
f.
i s denoted by
W e assume t h a t t h e image
i s a s u b s e t of t h e domain of
The composition of t h e f o g , where ( f 0 9 ) ( x ) = g ( X ) of t h e domain
of
X
f.
A t o p o l o q i c a l space i s a p a i r
( X , T ) where
d e n o t e s t h e f a m i l y of a l l open s u b s e t s o f i s u n l i k e l y w e w i l l d e n o t e ( X , T ) by simply
X.
X
#
and
r
When c o n f u s i o n
When i t i s
X.
d e s i r e d t o c a l l p a r t i c u l a r a t t e n t i o n t o t h e t o p o l o g y T o f X, o r when t h e u n d e r l y i n g p o i n t - s e t i s t o be p r o v i d e d w i t h more than one topology, w e s h a l l r e f e r t o X a s " t h e t o p o l o g i c a l
( x , ~. I)t
space
noted by by
The c l o s u r e of a s u b s e t
A
of
w i l l be de-
X
c l A , o r , when t h e r e i s a p o s s i b i l i t y of c o n f u s i o n ,
c 1 3 ; the i n t e r i o r of
A
int A
w i l l b e d e n o t e d by
or
int?. A collection
the closed sets --of members o f
63
of c l o s e d s u b s e t s o f
i f every closed set i n
63.
E q u i v a l e n t l y , 63
s e t s i f t h e r e i s a member
BE^
X
X
is a base for
i s an i n t e r s e c t i o n
i s a base f o r the closed
satisfying
F
C
B
and
x,dB
F
i s a c l o s e d s e t t h a t d o e s n o t c o n t a i n the p o i n t x . A subbase f o r t h e c l o s e d s e t s i s a c o l l e c t i o n of c l o s e d s e t s , t h e f i n i t e u n i o n s o f which form a b a s e for t h e c l o s e d
whenever
sets. 1.1 DEFINITION.
space
Let
11
b e an e l e m e n t i n t h e t o p o l o g i c a l
1i = (U : acG) b e a f a m i l y o f s u b s e t s of a i s l o c a l l y f i n i t e a t p i f there e x i s t s a
X , and l e t
The f a m i l y
p
X.
7
NOTATION AND TERMINOLOGY
neighborhood
Ua
@
I7 G =
of
G
p
and a f i n i t e s u b s e t
a{J.
f o r every
The family
i f t h e r e e x i s t s a neighborhood such t h a t IK/ family
1
n
Ua
and
of
H
H = @
L p
J c G
such t h a t
at
is discrete
K c G
and a s u b s e t
f o r every
The
a/K.
is locally f i n i t e (respectively, discrete) i f it is
L
x.
l o c a l l y f i n i t e ( r e s p e c t i v e l y , d i s c r e t e ) a t every p o i n t of A set
if
i n a t o p o l o g i c a l space
G
G -set
6-
A set i s
F - s e t if i t can be w r i t t e n a s a c o u n t a b l e union of
c a l l e d an
u-
closed s e t s . if
is called a
X
i s a c o u n t a b l e i n t e r s e c t i o n of open s e t s .
G
p
A subset
F
i s s a i d t o be r e q u l a r c l o s e d
X
C
These c o n c e p t s w i l l prove t o be very u s e -
F = cl(int F).
f u l i n t h e study of Hewitt-Nachbin
spaces.
acG) of s u b s e t s of a s e t x i s s a i d t o cover X i f a : ~ E G ] . The f a m i l y L i s s a i d t o be open ( r e s p e c t i v e l y , c l o s e d ) i f Ua i s open (reA non-empty family
L = (U
a X = U(U
s p e c t i v e l y , c l o s e d ) f o r each
:
li = ( V
If
acG.
a n o t h e r non-empty family of s u b s e t s of refine
1(
( o r be a refinement
of
: DEB)
is
Ir i s s a i d t o PEB) = i s a s u b s e t of some
X,
L) i f
P
then
U{Vp
:
li i s s a i d t o have t h e f i n i t e i n t e r s e c t i o n property ( r e s p e c t i v e l y , countable i n t e r s e c t i o n U(U,
: a c G ) and i f each element of
element of
The family
i .
Li
p r o p e r t y ) i f t h e i n t e r s e c t i o n of every f i n i t e ( r e s p e c t i v e l y ,
i s non-empty.
c o u n t a b l e ) subfamily of
Next we d e f i n e , f o r purposes of completeness and r e f e r ence, t h e t o p o l o g i c a l s e p a r a t i o n axioms.
Note t h a t t h e
T1-
s e p a r a t i o n axiom i s n o t p a r t of t h e d e f i n i t i o n of a completely r e g u l a r space, normal space, and s o f o r t h a s i s taken by s o m e
writers 1.2
( f o r example, J . Dugundji i n h i s 1966 t e x t ) .
DEFINITION.
s a i d t o be a
If
i s a t o p o l o g i c a l space, then
X
T1-space
provided t h a t f o r each
singleton ( x ) i s closed.
x,ycX XEX
sets
XCU
and
ycv.
and each c l o s e d s e t
U
and
v
such t h a t
The space
F
with
XEU
completely r e q u l a r i f f o r each with
xjfF
X
xjfF and
XEX
is
X
the
space i f f o r each
x # y , t h e r e a r e d i s j o i n t open s e t s
with
such t h a t
I t i s a Hausdorff
xcX U
and
V
i s r e q u l a r i f f o r each t h e r e a r e d i s j o i n t open
F c V.
x
The space
and each c l o s e d s e t
t h e r e i s a continuous r e a l - v a l u e d f u n c t i o n
f
is F
on
8
X
EMBEDDING I N TOPOLOGICAL PRODUCTS
such t h a t
f(x) = 0
and
f(y) = 1
f o r every
ycF.
A
T1-space . i s s a i d t o b e a Tychonoff s p a c e .
completely r e g u l a r
i s s a i d t o be normal i f f o r e a c h p a i r F1,F2 of d i s j o i n t c l o s e d s e t s t h e r e e x i s t d i s j o i n t open s e t s U and V w i t h F1 C U and F2 C V . I t i s p e r f e c t l y normal i f X is The s p a c e
X
X
normal and i f e v e r y c l o s e d s u b s e t o f
X
is a
G6.
The s p a c e
i s s a i d t o b e c o l l e c t i o n w i s e normal i f f o r e v e r y d i s c r e t e
3 = (Fa
acG] o f c l o s e d s u b s e t s o f X t h e r e i s a f a m i l y S = f G a : a c G ] of p a i r w i s e d i s j o i n t open s u b s e t s of X such t h a t Fa c Ga f o r every a c G . Next w e d e f i n e t h e v a r i o u s n o t i o n s o f compactness. If X i s a t o p o l o g i c a l s p a c e , then X i s a compact s p a c e i f e v e r y open cover o f X h a s a f i n i t e s u b c o v e r . By a c o m p a c t i f i c a t i o n of X i s meant a compact s p a c e i n which X i s d e n s e ( u p t o homeomorphism). The s p a c e X i s c o u n t a b l y compact i f e v e r y c o u n t a b l e open c o v e r of X h a s a f i n i t e s u b c o v e r . I t i s l o c a l l y compact i f e v e r y p o i n t of X h a s a compact neighborhood. I t i s 0-compact i f X can b e w r i t t e n a s t h e u n i o n of c o u n t a b l y many compact s u b s e t s . The s p a c e X i s pseudocompact i f e v e r y c o n t i n u o u s r e a l - v a l u e d f u n c t i o n on X i s family
:
-
bounded.
I t i s zero-dimensional
i f t h e r e i s a base f o r t h e
topology c o n s i s t i n g of open and c l o s e d s u b s e t s of Lindelb'f s p a c e i f e v e r y open c o v e r o f cover.
The s p a c e
X
X
X.
It is a
h a s a c o u n t a b l e sub-
i s paracompact i f e v e r y open c o v e r of
h a s a l o c a l l y f i n i t e open r e f i n e m e n t .
I t i s c o u n t a b l y para-
compact i f e v e r y c o u n t a b l e open c o v e r o f f i n i t e open r e f i n e m e n t .
The s p a c e
X
X
X
has a locally
i s s e q u e n t i a l l y compact
i f e v e r y sequence o f
X h a s a c o n v e r g e n t subsequence. Many well-known r e l a t i o n s h i p s e x i s t between t h e v a r i o u s
compactness n o t i o n s .
A good summary of
t h o s e t h a t a r e impor-
t a n t t o o u r development o c c u r s i n t h e 1 9 7 0 t e x t by S . W i l l a r d . W e do assume t h a t t h e r e a d e r i s f a m i l i a r w i t h such n o t i o n s a s
a s e p a r a b l e s p a c e , f i r s t c o u n t a b l e s p a c e , second c o u n t a b l e s p a c e , t h e i d e a of a p s e u d o m e t r i c , topoloqies.
and t h e p r o d u c t and g u o t i e n t
W e remark t h a t t h e d e f i n i t i o n o f paracompactness
g i v e n above i s t h e one f o r m u l a t e d by Kuratowski. I t d i f f e r s from t h e o r i g i n a l d e f i n i t i o n g i v e n by J. DieudonnL i n t h a t Dieudonnd r e q u i r e s a paracompact s p a c e t o be H a u s d o r f f .
The
9
THE EMBEDDING LEMMA
d e f i n i t i o n o f Kuratowski p r o v i d e s f o r e v e r y p s e u d o m e t r i c s p a c e (A proof o f t h i s o c c u r s
t o b e paracompact. K e l l e y ' s book.
i n Chapter 5 of J.
I t i s a l s o shown t h a t a paracompact Hausdorff
space i s r e g u l a r and t h a t a paracompact r e g u l a r s p a c e i s normal.) Given two s p a c e s
and
X
of a l l continuous f u n c t i o n s
n,
the r e a l l i n e
then
C ( X , E ) denote the s e t
let
E,
from
f
c(X,R )
into
X
If
E.
is
E
i s an a l g e b r a i c r i n g r e l a t i v e
t o t h e o p e r a t i o n s of a d d i t i o n and m u l t i p l i c a t i o n of f u n c t i o n s
c(:ij : t h e s u b r i n g o f *
and w i l l be denoted more simply by
C ( X ) w i l l be denoted by
bounded f u n c t i o n s of constant function f o r any
re=.
functions
f
: X
If V
g
f and
R
-3
and f
i s d e f i n e d by
g
g
Pi
belong t o
C
(X)
.
The
~ ( x =) r
(xEX)
then the
C(X),
a r e d e f i n e d by
( f V 9 ) ( x ) = max( f ( x ) , q ( x ) 1
and
( f A 9 ) ( x ) =: m i n ( f ( x ) , g ( x ) ) . I t i s s t r a i g h t f o r w a r d t o show t h a t i f
f
and
q
t h e n t h e same h o l d s t r u e f o r t h e f u n c t i o n s
C(X),
belong t o f V g
and
f A q: f v q
= T1 ( f +
g
+
If
-
91)
+fg
-
If
-
91)
and
1 f A g = ~ (
Thus, a c c o r d i n g t o t h e above t e r m i n o l o g y , a s p a c e pseudocompact i f and o n l y i f
C(X) = C
*
(X).
X
is
I t is not d i f f i -
c u l t t o e s t a b l i s h t h a t e v e r y c o u n t a b l y compact s p a c e i s pseudocompact. T h i s s e c t i o n w a s i n t e n d e d o n l y a s a b r i e f summary o f t h e b e t t e r known n o t i o n s c o n c e r n i n g t o p o l o g i c a l s p a c e s i n o r d e r t o f a c i l i t a t e t h e development i n s u b s e q u e n t s e c t i o n s .
Lesser
known i d e a s and r e s u l t s w i l l be d e f i n e d and e s t a b l i s h e d i n t h e s e q u e l a s needed. Section 2:
The Embeddinq Lemma
I n t h i s s e c t i o n w e w i l l i n v e s t i g a t e t h e two problems t h a t a r e n a t u r a l l y a s s o c i a ted w i t h t o p o l o g i c a l p r o d u c t s :
( a ) given
EMBEDDING I N TOPOLOGICAL PRODUCTS
10
a space
f i n d a l l s p a c e s t h a t a r e homeomorphic t o s u b s p a c e s
E
of t o p o l o g i c a l powers of
E , and ( b ) g i v e n an
E
find a l l
s p a c e s t h a t a r e homeomorphic t o c l o s e d s u b s p a c e s of t o p o l o g i c a l powers of
( a ) i s a g e n e r a l i z a t i o n of t h e n o t i o n
Property
E.
of complete r e g u l a r i t y and p r o p e r t y ( b ) g e n e r a l i z e s compact-
ness.
A t h i r d problem i s t h a t o f homeomorphically embedding a
given space
Y
s i o n space P,
a s a d e n s e subspace of some t o p o l o g i c a l e x t e n -
X
t h a t p o s s e s s e s some d e s i r e d t o p o l o g i c a l property
such a s compactness, m e t r i z a b i l i t y , c o m p l e t e n e s s , o r H e w i t t -
Nachbin c o m p l e t e n e s s .
T h i s problem was s t u d i e d i n t h e 1968
paper by J . Van d e r S l o t coupled w i t h t h e c o n s i d e r a t i o n of e x t e n d i n g c o n t i n u o u s f u n c t i o n s on with property
X
i n t o a codomain s p a c e
Y.
t o t h e extension space
P
J
I n h i s 1966 p a p e r S . Mrowka p r o v i d e s a g e n e r a l i z e d form of t h e Embedding Lemma t h a t a p p e a r s i n t h e 1955 t e x t by J . L . K e l l e y (Lemma 5 , c h a p t e r 4 ) .
T h i s lemma i s f o u n d a t i o n a l w i t h
r e s p e c t t o t h e problems under d i s c u s s i o n .
Moreover, a s w e
have a l r e a d y i n d i c a t e d , t h e Embedding Lemma p r o v i d e s a n a t u r a l s e t t i n g f o r i n t r o d u c i n g t h e c o n c e p t of a Hewitt-Nachbin s p a c e . We begin w i t h t h e s t a t e m e n t of t h e Embedding Lemma. Let
b e an a r b i t r a r y t o p o l o g i c a l s p a c e , and l e t
X
IXa : a 4 ) b e a non-empty f a m i l y o f t o p o l o g i c a l s p a c e s . each
a&,
and l e t
let
fa
b e an a r b i t r a r y mapping from
d e n o t e t h e f a m i l y (fa : asG).
F
X
For
into
Xa,
There i s then a s s o -
F a n a t u r a l mapping u from X i n t o n(Xa : acG) d e f i n e d b y u ( p ) = ( f , ( ~ ) ) ~ ~ ~ .
c i a t e d with t h e family the product space The mapping associated 2.1
u
i s c a l l e d t h e p a r a m e t r i c o r e v a l u a t i o n mappinq
with
F.
THE EMBEDDING LEMMA (Kelley-MrAwka)
and -
.
If
X,
a r e qiven a s i n t h e preceding paraqraph,
Xa (acG), F then t h e
followinq statements a r e t r u e :
(1)
mappinq fa
(2)
u
c o n t i n u o u s i f and o n l y i f each
i s continuous.
The mappinq u is p a i r of p o i n t s
exists 2 (3)
is
fa
The mappinq
p
i n j e c t i v e i f and o n l y i f f o r e a c h g in X with p # q t h e r e
p J
in F such t h a t f,(p) # f a ( q ) . u & 2 homeomorphism i f and o n l y i f i t
THE EMBEDDING LEMMA
i s continuous,
-f i e s the
i n 7 e c t i v 2 , and t h e c l a s s
pcX\A
al
satis-
F
followii3q c o n d i t i o n :
For e v e r y c l o s e d s u b s e t f
11
A c X
and f o r e v e r y
therrz e x i s t s a f i n i t e s u b c o l l e c t i o n
of F >...’fan -
such t h a t t h e p o i n t
( p ) , . . . , f a ( p ) ) does not l i e i n th2 c l o s u r e al n o f t h e set [ (fa ( a ) , . . , f a ( a ) ) : aEA), where 1 n t h e c l o s u r e i s taken i n t h e p r o d u c t s p a c e x x . . . x xa . al n Assume t h a t t h e s p a c e s Xu a r e a l l Hausdorff and t h a t u & 2 homeomorphism. Then u ( X ) i s c l o s e d i n t h e p r o d u c t s p a c e n(Xa : a d ) i f and o n l y i f t h e -(f
(i)
---
(4)
.
f o l l o w i n q c o n d i t i o n i s s a t i s f i e d by t h e c l a s s I f there ---
ins
in
(ii)
Y
X F
exists
Hausdorff s p a c e
Y
F:
contain-
d e n s e l y such t h a t e v e r y f u n c t i o n admits a continuous e x t e n s i o n
into xa,
then
fa f & from
Y = X.
S t a t e m e n t s (1) and ( 2 ) of t h e above lemma a r e due t o K e l l e y (1955, Lemma 4 . 5 ) , and s t a t e m e n t s ( 3 ) and ( 4 ) a r e due t o Mrdwka (1966, Theorem 2 . 1 ) .
The importance of t h e Embedding
Lemma i s t h a t i t r e d u c e s t h e problem o f embedding a t o p o l o g i c a l space “Xu
:
a&)
homeomorphically i n t o a p r o d u c t s p a c e
X
t o t h a t of f i n d i n g a “ r i c h enough“ f a m i l y o f
c o n t i n u o u s f u n c t i o n s from
X
i n t o each
Xa.
Before p r o v i n g t h e Embedding Lemma w e s h o u l d l i k e t o d i s c u s s t h r e e well-known a p p l i c a t i o n s o f i t :
Urysohn’ s m e t r i -
V
z a t i o n theorem, t h e Stone-Cech c o m p a c t i f i c a t i o n , and t h e comp l e t i o n of a Hausdorff uni-form s p a c e . I n t h e c a s e of m e t r i z a b i l i t y w e b e g i n w i t h a r e g u l a r T1-space t h a t i s second countable.
Because o f t h e second c o u n t a b i l i t y , i t i s e a s y t o
d e t e r m i n e a c o u n t a b l e c o l l e c t i o n of c o n t i n u o u s f u n c t i o n s from
x
i n t o t h e u n i t i n t e r v a l [0,1] t h a t s a t i s f i e s the c o n d i t i o n s
o f t h e lemma.
Using t h e f a c t t h a t a c o u n t a b l e p r o d u c t o f
m e t r i c s p a c e s i s m e t r i z a b l e , t h e embedding t e c h n i q u e y i e l d s a m e t r i z a t i o n of t h e g i v e n space (see K e l l e y , Theorem 1 6 , Chap-
ter 4 f o r the d e t a i l s ) .
12
EMBEDDING I N TOPOLOGICAL PRODUCTS
v
For t h e Stone-Cech c o m p a c t i f i c a t i o n of a Tychonoff s p a c e
X, t h e complete r e g u l a r i t y of X i n s u r e s t h a t t h e f a m i l y * C (X) of bounded 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 X i s s u f f i c i e n t l y r i c h i n t h e s e n s e of t h e lemma. embedding t e c h n i q u e , w e embed
X
Hence, using the
homeomorphically i n t o a p r o 6
u.
u c t of c l o s e d i n t e r v a l s v i a t h e p a r a m e t r i c mapping
Since
t h e p r o p e r t y of compactness i s c l o s e d - h e r e d i t a r y and product i v e , t h e c l o s u r e of u ( X ) i n t h e p r o d u c t s p a c e i s a compact Hausdorff s p a c e c o n t a i n i n g a d e n s e homeomorphic copy of
X.
V
T h i s compact Hausdorff s p a c e i s known a s t h e Stone-Cech com-
pX.
pX i s c h a r a c t e r i z e d a s t h e u n i q u e compact Hausdorff s p a c e c o n t a i n i n g X d e n s e l y f o r which e v e r y bounded c o n t i n u o u s r e a l - v a l u e d pactification of
X
f u n c t i o n on
X,
and i s d e n o t e d by
The s p a c e
admits a continuous extension t o
pX
i n the
following sense:
If
&2
Y
compact Hausdorff s p a c e c o n t a i n i n q
X
densely
---
and such t h a t e v e r y bounded c o n t i n u o u s r e a l - v a l u e d f u n c t i o n
on
X a d m i t s a c o n t i n u o u s e x t e n s i o n to Y, then Y is homeomorp h i c t o pX under a homeomorphism t h a t i s t h e i d e n t i t y on X (see K e l l e y , Theorem 2 . 4 , Chapter 5 ) . shown t h a t
the
function rinqs
C*(X)
Moreover, i t can be
and
C(@X) are alqebrai-
c a l l y isomorphic (see Gillman and J e r i s o n , Theorem 6 . 6 ( b ) f o r t h e d e t a i l s concerning t h i s r e s u l t )
.
F i n a l l y , i n t h e c a s e of t h e completion o f a Hausdorff uniform s p a c e , t h e f a m i l y o f r e a l - v a l u e d u n i f o r m l y c o n t i n u o u s f u n c t i o n s on
s a t i s f i e s t h e c o n d i t i o n s o f t h e lemma (see J .
X
I s b e l l ' s 1964 book, Theorem 1 3 , page 7). As was t h e c a s e i n o u r p r e c e d i n g d i s c u s s i o n , s i n c e t h e p r o p e r t y of c o m p l e t e n e s s i s c l o s e d - h e r e d i t a r y and p r o d u c t i v e , t h e c l o s u r e o f i~ ( X ) i n t h e product space of real l i n e s i s the d e s i r e d completion, d e n o t e d by
If X
Y
Moreover, YX
yX.
i s unique i n t h e f o l l o w i n g sense:
i s a complete Hausdorff uniform s p a c e c o n t a i n i n q
densely, then t h e r e e x i s t s a uniformly continuous b i j e c t i o n
from -
yX
onto
Y
t h a t leaves
X
p o i n t w i s e f i x e d and whose
i n v e r s e is a l s o uniformly continuous.
Moreover, e v e r y u n i -
formly c o n t i n u o u s r e a l - v a l u e d f u n c t i o n
on
X
admits a uni-
13
THE EMBEDDING LEMMA
formly c o n t i n u o u s e x t e n s i o n
to
yX
W e p o i n t o u t t h a t a u n i f o r m l y c o n t i n u o u s b i j e c t i o n whose
i n v e r s e i s a l s o uniformly c o n t i n u o u s i s c a l l e d a uniform
+-
morphism. I t i s a l s o p o s s i b l e t o o b t a i n a c o m p l e t i o n o f a nonHausdorff uniform s p a c e .
'The c o n s t r u c t i o n f o r such comple-
t i o n s i s g i v e n i n Theorem 2 7 and Theorem 2 8 of C h a p t e r 6 of Kelley
.
Proof of ---
t h e Embedding Lemma:
P a r t s (1) and ( 2 ) o f t h e lemma
a r e w e l l known and w e o m i t t h e p r o o f s h e r e ( s e e , f o r example, 4 . 5 on page 116 o f K e l l e y € o r d e t a i l s ) .
The f o l l o w i n g p r o o f s
of p a r t s ( 3 ) and ( 4 ) a r e due t o Mrdwka. ( 3 ) : Assume t h a t a i s c o n t i n u o u s and i n j e c t i v e and t h a t t h e c l a s s F s a t i s f i e s c o n d i t i o n ( i ) . L e t A be a c l o s e d s u b s e t of X . For each f i n i t e s e t a l , a 2 , ..., an o f i n d i c e s i n G , l e t T ( a 1 , a 2 , . . , a n ) d e n o t e t h o s e p o i n t s e of t h e p r o d u c t Z = n ( X a : a c G ) such t h a t T~ ( e ) = f (p) for i ai some pcA and f o r i = 1 , 2 , . . , n . Then c o n d i t i o n ( i ) i s equivalent t o t h e f a c t t h a t u ( A ) i s the i n t e r s e c t i o n of a l l
Part
.
.
s e t s o f t h e form
.
u ( X ) fl clZT(al,a2,,. , .,a ) where a l , a 2 , . . , a
n
r a n g e s o v e r a l l f i n i t e s e t s o f e l e m e n t s of closed i n
u ( X ) and
u
G.
n
Thus, u ( A ) i s
i s t h e r e f o r e a homeomorphism.
u
C o n v e r s e l y , assume t h a t
i s a homeomorphism.
be a c l o s e d s u b s e t o f X and l e t P E X M . I t f o l l o w s t h a t t h e r e i s a b a s i c open s e t
Let
A
Then o ( p ) f! c l z u ( A ) . - 1 (G1) n T
U =
n...n
a,
n - l ( G n ) i n t h e p r o d u c t Z , where Gi i s open i n an s u c h t h a t u ( p ) E U and U fl u ( A ) = @. For each i =
T - ~ ( G ~ ) a2
xa,,
1,2,
. . . ,n
t h e mapping g i v e n by
,. . .
f a ,f
and t h e f i n i t e system
1.
= ~~~o
fa
i ,f
a2
(T
belongs t o
F,
s a t i s f i e s the requirean
m e n t s of c o n d i t i o n ( i ) .
Part n(Xa
(4):
u ( X ) i s closed i n t h e product
Assume t h a t
: acG).
Let
b e a Hausdorff s p a c e c o n t a i n i n g
Y
d e n s e l y such t h a t each sion
f:
: Y
-$
Xu.
Let
fcx i n
cry : Y
X
admits a continuous exten-
F -$
2 =
Z
denote t h e parametric
14
EMBEDDING I N TOPOLOGICAL PRODUCTS
*
u.
e x t e n s i o n of
*
u (Y) =
mapping g i v e n by
x.
u (Y) = u I n o t h e r words, u
( f a ( Y ) 1 acG.
i s dense i n
Since
X
(ClYX)
c c l z o (X) =
*
maps
quently, i f we set
g(p) = p
i s dense i n
t i o n and
u
Then s i n c e
Z.
Y
superspace
2
X
*
fa = into
T
0
a Xa
u ( X ) and
Moreover, Y
u
satisfied.
*
p ~ x . Since
i s t h e i d e n t i t y func-
i s a homeomorphism t h e r e e x i s t s a
u
*
i s homeomorphic t o
Y
t h a t extends
topological relations between densely.
i s a con-
o ( X ) f a i l s t o be c l o s e d i n t h e
such t h a t
under a homeomorphism t h o s e between
Conse-
X
Thus c o n d i t i o n ( i i ) i s e s t a b l i s h e d .
Y = X.
C o n v e r s e l y , assume t h a t product
u(X).
f o r every
g
i t follows t h a t
Y
i s an
= U(X).
g :Y
then
*
i t follows t h a t
ClZU(X)
i n t o t h e image
Y
g = u - l o u‘,
tinuous function satisfying X
Y,
*
c
u
clearly
and
X
clzu(X).
Y
Thus
is H a u s d o r f f .
u.
Clearly the
are i d e n t i c a l t o Y
contains
a&.
X
F i n a l l y , t h e formula
d e f i n e s a continuous extension of
f o r each
clZu(X)
fa
from
Y
Thus c o n d i t i o n ( i i ) f a i l s t o b e
T h i s c o m p l e t e s t h e p r o o f of t h e Embedding Lemma.
For a f u r t h e r d i s c u s s i o n of t h e p a r a m e t r i c mapping and r e s u l t s r e l a t i n g t o t h e Embedding Lemma w e r e f e r t h e i n t e r /
e s t e d r e a d e r t o S e c t i o n I1 o f Mrowka’s 1968 p a p e r . The Embedding Lemma i s a l s o f o u n d a t i o n a l t o t h e s t u d y of Tychonoff s p a c e s b e c a u s e t h e s e a r e p r e c i s e l y t h e s p a c e s t h a t a r e homeomorphic t o s u b s p a c e s o f a p r o d u c t of u n i t i n t e r v a l s . An examination o f t h e proof o f t h a t r e s u l t i n K e l l e y (Theorem 7 , page 118) o r i n Dugundji (Theorem 7 . 3 , page 1 5 5 ) q u i c k l y
r e v e a l s t h a t t h e d e s i r e d homeomorphism i s t h e p a r a m e t r i c mapping a s s o c i a t e d w i t h t h e c o l l e c t i o n of c o n t i n u o u s mappings from t h e s p a c e i n t o [0,1]. /
I n 1958 R. Engelking and S . Mrowka i n i t i a t e d t h e s t u d y o f a g e n e r a l i z e d n o t i o n o f complete r e g u l a r i t y a s w e l l a s compact-
ness.
These i n v e s t i g a t i o n s w e r e c o n t i n u e d by Mr6wka i n 1966,
1968, and 1 9 7 2 .
work.
R.
B l e f k o a l s o make c o n t r i b u t i o n s t o t h a t
I n h i s 1967B p a p e r H . H e r r l i c h s t u d i e d s i m i l a r g e n e r a l -
i z a t i o n s of complete r e g u l a r i t y and compactness d i s c u s s e d w i t h -
15
E- COMPLETELY REGULAR SPACES
i n t h e framework o f c a t e g o r i c a l t o p o l o g y . We w i l l f o c u s o u r a t t e n t i o n on some of t h e s e i d e a s i n t h e n e x t s e v e r a l s e c t i o n s a s they emerge a s a n a t u r a l outgrowth o f o u r c o n s i d e r a t i o n s c o n c e r n i n g embeddings i n t o p o l o g i c a l p r o d u c t s . T h i s w i l l l e a d q u i c k l y t o t h e n o t i o n of a Hewitt-Nachbin s p a c e . Section 3:
E-Completely Reqular Spaces
The n o t i o n o f an
E-completely r e q u l a r s p a c e o r i g i n a t e d
i n t h e 1958 paper by Engelking and Mrdwka.
The d e f i n i t i o n
g e n e r a l i z e s t h e c h a r a c t e r i z a t i o n of a Tychonoff s p a c e a s one t h a t i s homeomorphic t o a subspace o f a p r o d u c t o f u n i t intervals. 3.1
DEFINITION.
spaces.
Then
X
vided t h a t c a l power
X
Let
and
X
E
i s s a i d t o be
b e two g i v e n t o p o l o g i c a l E-completely r e q u l a r pro-
i s homeomorphic t o a subspace of t h e t o p o l o g i -
for some c a r d i n a l number
Em
m.
E-completely r e g u l a r s p a c e s i s d e n o t e d by
The c l a s s of a l l The c l a s s
@(E).
B
of t o p o l o g i c a l s p a c e s i s c a l l e d a c l a s s o f complete r e q u l a r i t y i f t h e r e e x i s t s a space
E
with
6 = B(E) .
6([0,1]) = @(R)
I t i s c l e a r from t h e d e f i n i t i o n t h a t
corresponds t o t h e c l a s s of a l l completely r e g u l a r s p a c e s .
We
s h a l l p r o v i d e add t i o n a l examples of c l a s s e s of complete regul a r i t y f u r t h e r on i n t h e development o f t h i s s e c t i o n .
The
f o l l o w i n g r e s u l t s a r e immediate consequences o f t h e d e f i n i t i o n and w e s t a t e them w i t h o u t p r o o f . 3.2
THEOREM.
Then t h e --
J &
E
b e two g i v e n t o p o l o g i c a l spaces.
following a r e t r u e :
(1) The s p a c e (2)
and
X
If
X
morphic
&a
E
is
E-completely r e q u l a r .
E-completely r e g u l a r subspace
of
X,
and
then
Xo Xo
i s homeoE-=-
pletely reqular. (3)
The t o p o l o q i c a l p r o d u c t o f a n a r b i t r a r y c o l l e c t i o n of E-completely r e q u l a r s p a c e s is E-completely reqular.
(4)
If
El
is
t o p o l o q i c a l space, then
6 ( E ) c @(El)
16
EMBEDDING I N TOPOLOGICAL PRODUCTS
i s e q u i v a l e n t to (5)
E
@(El).
E
m,
For e v e r y c a r d i n a l
@(E) = @(Em)
The f o l l o w i n g c h a r a c t e r i z a t i o n of
E-complete r e g u l a r i t y
was g i v e n by Engelking and Mrowka i n t h e i r 1958 p a p e r .
.
THEOREM (Engelking and Mro/wka)
3.3
A space
p l e t e l y r e q u l a r i f and o n l y i f t h e f o l l o w i n q
-
E-=-
X
two
conditions
are satisfied: (a)
For e v e r y
p,q
belonqinq Q
with
X
e x i s t s g continuous f u n c t i o n
# q
p
there
f E C(X,E) satisfyinq
.
f(P) # ffq) For every closed s u b s e t
(b)
A c X
t h e r e e x i s t s 2 -f i n i t e number n -function is
X
morphism
h
a
o h
j!
c l f(A)
.
(p)
#
X
I - ~ O h(g)
f o r some
Thus,
Next, suppose t h a t
pcX\F.
is
A
i s open and
h
Since
Now
T T ~ t h, e
a t h coordinate space.
and t h a t
m.
f o r some c a r d i n a l
s a t i s f i e s condition ( a ) .
i n j e c t i v e the p o i n t
h ( p ) b e l o n g s t o t h e open s e t
h(X)\h(A)
n
Therefore, t h e r e e x i s t s a f i n i t e p o s i t i v e integer
Em.
h ( p ) b e l o n g s t o t h e b a s i c open s e t
such t h a t
U TI h ( A ) =
with
I - ~ h O
into the
Em
a closed subset of
in
h ( x ) c Em
such t h a t
p r o j e c t i o n of I-
and a c o n t i n u o u s
f(p)
E-completely r e g u l a r , then t h e r e e x i s t s a homeo-
h ( p ) # h ( q ) so t h a t f =
with
C(X,En)
E
PEX\F
F i r s t w e e s t a b l i s h t h e n e c e s s i t y of t h e c o n d i t i o n s .
Proof. If
f
and p o i n t -
a.
p r o d u c t of t h e maps
Define ~~0
h
f :
x
3
En
i = 1,2,
€or
c o n t i n u o u s (see f o r example, Theorem 2 . 5 ,
by t a k i n g
.. . , n .
f
as the
Then
€
is
page 1 0 2 o f Dugundji)
and t h e p o i n t
belongs t o h(q)
E
G1
X
G2 X...x
T T ~ hO( q )
# G ~ .Therefore
f ( p ) does n o t b e l o n g t o En.
Gn.
h(A) t h e r e e x i s t s a
k
f (A)
Moreover, g i v e n any p o i n t such t h a t
n
1
[ G x~ G~ x . .
k
.x
n Gn]
and =
and
c l f ( A ) where t h e c l o s u r e i s t a k e n i n
Thus c o n d i t i o n (b) is s a t i s f i e d .
17
E- COMPLETELY REGULAR SPACES
C o n v e r s e l y , suppose t h e two c o n d i t i o n s a r e s a t i s f i e d and
let
.
F = C(X,E)
Then s t a t e m e n t ( 2 ) o f t h e Embedding Lemma i s
clearly satisfied.
To o b t a i n statement
observe t h a t i f
i s a c l o s e d s u b s e t of
A
then t h e r e e x i s t s a p o s i t i v e i n t e g e r f : X fk =
En
such t h a t
Of
where
-+
7rk
f(p) Then
E.
the f i n i t e s u b c o l l e c t i o n dition
n
with
X
fk
and a f u n c t i o n
fl, f 2 , .
En
. ., f n
( i i ) of t h e Embedding Lemma.
of
into its
kth
. ., n
and
s a t i s f i e s con-
F
m =
Thus, l e t t i n g
I
i t i s c l e a r t h a t t h e p a r a m e t r i c map a s s o c i a t e d w i t h F i s a homeomorphism o f X IC(X,E)
Set
k = 1,.
f o r each
F
E
pcX\F,
( A ) by h y p o t h e s i s .
i s t h e p r o j e c t i o n of
7rk
c o o r d i n a t e space
p cl f
( 3 ) of t h e Lemma,
u
: X
into
+
Em
This
Em.
completes the p r o o f .
I n h i s 1968 p a p e r Mrdwka remarks t h a t i f space,
is a
X
T 0
then c o n d i t i o n ( a ) o f t h e p r e v i o u s r e s u l t may be This i s because i n t h a t c a s e c o n d i t i o n (b) i m p l i e s
omitted.
c o n d i t i o n ( a ) (see MrJwka (1968) Theorem 2 . 3 f o r t h e d e t a i l s ) . Moreover Engelking and Mr4wka (1958) have shown t h a t i t i s i n s u f f i c i e n t t o consider only f u n c t i o n s
(b 1
f
: X
+
i n condition
E
. Blefko (1965) h a s a l s o p r o v i d e d a c h a r a c t e r i z a t i o n o f
R.
E-completely r e g u l a r s p a c e s i n t h e p r e s e n c e of the ward s o w e omit i t h e r e . space
X
&
The s t a t e m e n t i s a s f o l l o w s :
--l e n t t o the
converqence
function
E
C(X,E).
c a n n o t be o m i t t e d . E
is a
fi
To-
E-completely r e q u l a r i f and o n l y i f t h e conver-
qence o f any n e t [ x n : n c D ) f
To-sepa-
The proof t o h i s r e s u l t i s q u i t e s t r a i g h t f o r -
r a t i o n axiom.
of
in
t o a point
X
( f ( x n ) : nED)
f ( p ) for every
The c o n d i t i o n t h a t
I n fact, i f
X
i s ecfuiva-
p
X
be a
To-space
i s a n i n d i s c r e t e s p a c e and
To-space t h e n e v e r y c o n t i n u o u s
f
: X
3
E
is a
c o n s t a n t and t h e n e t c o n d i t i o n i s always s a t i s f i e d . 3.4
EXAMPLE.
(O,l).'
Let
A space
X
D
is
denote t h e two-point d i s c r e t e space D-completely
r e q u l a r i f and o n l y i f i t
i s a z e r o - d i m e n s i o n a l T -space. To see t h i s , suppose f i r s t 0 t h a t X i s D-completely r e g u l a r . L e t p and g d e n o t e
_ I
d i s t i n c t p o i n t s of
x.
By c o n d i t i o n (a) of 3 . 3 t h e r e e x i s t s
18
f
EMBEDDING I N TOPOLOGICAL PRODUCTS
C ( X , D ) such t h a t
E
set
f(p) = 0
f-l(O) contains
Next, suppose t h a t
space. pcG.
Let
Since
Dn
n
and
c l o s e d ) and hence
X
Thus t h e open
so t h a t
X
is a
i s a n open s u b s e t of
G
T
-
0
and
X
f
C(X,Dn)
E
f(p) f cl f(A).
such t h a t
i s d i s c r e t e , f ( A ) i s c l o p e n ( i . e . , b o t h open and b e l o n g s t o t h e clopen subset
p
which i s c o n t a i n e d i n for
f ( q ) = 1. q
By c o n d i t i o n ( b ) o f 3 . 3 t h e r e e x i s t s a
A = X\G.
f i n i t e number
and
and m i s s e s
p
i s now c l e a r , and c o n s e q u e n t l y
The c o n v e r s e i s e q u a l l y s i m p l e . t h e r e i s a clopen s e t t i o n d e f i n e d by
G
satisfying
f ( G ) c [ O ) and
d i t i o n ( a ) of 3 . 3 .
X\f-’(f(A))
The r e q u i r e d b a s e of c l o p e n s e t s
G.
i s zero-dimensional.
X
If
p
d
c l ( q ) , then
peG c X \ c l ( y ) .
The func-
f(X\G) c (1) s a t i s f i e s con-
C o n d i t i o n ( b ) i s s a t i s i f e d i n an e n t i r e l y
a n a l a g o u s manner y i e l d i n g t h e
D-complete r e g u l a r i t y .
A proof v e r y s i m i l a r t o t h a t p r o v i d e d above can be used
t o show t h a t i f
Dc
denotes
the
connected dyad ( i . e . , t h e
two-point s p a c e [ O , l ) whose o n l y p r o p e r non-empty open s e t i s ( O ] ) , then t h e c l a s s
@(Dc)
precisely t h e c l a s s of
T 0
spaces.
I n h i s 1968 p a p e r , Mrdwka comments t h a t n e i t h e r t h e c l a s s of Hausdorff s p a c e s nor t h e c l a s s o f r e g u l a r
T1-spaces
is a
I n a n u n p u b l i s h e d r e s u l t by
c l a s s of complete r e g u l a r i t y .
B i a l y n i c k i - B i r u l a i n 1958 i t w a s shown t h a t t h e r e i s no space
E
such t h a t
@(E) contains
T1-
Hausdorff s p a c e s .
H.
H e r r l i c h (1965) o b t a i n e d a s t r o n g e r r e s u l t showing t h a t t h e r e
i s no
T - s pa c e
E
such t h a t
@(E) c o n t a i n s
reqular
Hausdorff s p a c e s . O n e of t h e f a s c i n a t i n g a s p e c t s of a c o m p l e t e l y r e g u l a r
s p a c e ( i n t h e u s u a l sense where
E = 7 R ) i s t h a t i t can b e
c h a r a c t e r i z e d i n c o n n e c t i o n w i t h t h e zero- s e t s a s s o c i a t e d w i t h
i t s r i n g of 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 (Gillman and Jerison, 3.2-3.8).
These r e s u l t s have b e e n g e n e r a l i z e d by
/
In o r d e r t o view t h a t development i n i t s p r o p e r p e r s p e c t i v e w e
Mrowka (1968) and w e w i l l p r e s e n t t h a t development h e r e .
c o n s i d e r t h e known r e s u l t s f o r c o m p l e t e l y r e g u l a r s p a c e s . These r e s u l t s occur i n t h e f i r s t t h r e e c h a p t e r s o f t h e Gillman and J e r i s o n t e x t .
E- COMPLETELY REGULAR SPACES
3.5 f
E
DEFINITION.
If
i s a t o p o l o g i c a l space and i f
X
c ( x ) , then t h e s e t
--zero-set
of
-s e t of f . -cozero-set) sets { Z ( f ) collection
= {xtX : f ( x ) = 0 ) i s c a l l e d t h e
Z ( f ) i s c a l l e d t h e cozero-
I f S c X , then S i s a z e r o - s e t ( r e s p e c t i v e l y , i n case S = Z ( f ) (respectively, s = x \ z ( f ) ) f o r
f E C(X)
some
Z(f)
The complement of
f.
19
:
.
For
C'
C
t h e c o l l e c t i o n of a l l zero-
C(X)
fEC' ) i s denoted by
For s i m p l i c i t y t h e
Z(C').
Z ( C ( X ) ) of a l l z e r o - s e t s i n
I t is clear that
i s denoted by Z ( X ) .
X
ncm.
Z ( f ) = Z ( / f l ) = Z ( f n ) f o r every
Also,
demonstrate t h a t
Z(X)
i s c l o s e d under t h e formation of f i n i t e
unions and f i n i t e i n t e r s e c t i o n s .
I n fact
Z ( X ) i s closed
under countable i n t e r s e c t i o n s : OD
z ( g ) = n[z(fn) : nEN )
where
Z
g(x) =
If,/
A
2-".
n=l
shows t h a t every z e r o - s e t i s a G - s e t . ( I n a normal space, b every closed G 6 i s a z e r o - s e t . See Gillman and J e r i s o n , 3D.3.)
The following important r e s u l t r e l a t e s t h e s e p a r a t i o n
property of complete r e g u l a r i t y t o t h e c o l l e c t i o n 3.6
THEOREM (Gillman and J e r i s o n ) .
If
X
is 2
Z(X)
.
topoloqical
space, then the followinq s t a t e m e n t s a r e t r u e : space
X
is
collection closed
is
X F
completely r e q u l a r i f and only i f
Z ( X ) of a l l z e r o - s e t s is a base f o r
sets. completely r e q u l a r , then every c l o s e d sub-
i s an i n t e r s e c t i o n of z e r o - s e t neiqhborhoods
EMBEDDING I N TOPOLOGICAL PRODUCTS
20
(3)
of -
F.
If
X
is
c o m p l e t e l y r e q u l a r , then e v e r y neiqhbor-
-hood of a p o i n t the p o i n t . -
-Proof
(1): I f
of
X
i s a c l o s e d s e t and
f(x) = 1
and
x ,k Z ( f ) .
i s completely r e g u l a r , for a l l
Consequently
Then, f o r
suppose
F = cl F
and
such t h a t
Z(f)
3
and
F
Z(X)
i s a b a s e f o r the closed there i s a zero-set
g(x) =
g(y) = 0
Hence
F
xf'F,
x ,d Z ( f ) .
Then t h e f u n c t i o n
0.
g ( x ) = 1 and
then whenever
f E C (X)
yfF.
of
i s a base f o r t h e closed s e t s .
Z(X)
F c Z ( f ) and
Z ( f ) with
r #
there exists
x/F
f(y) = 0
On t h e o t h e r hand,
sets.
c o n t a i n s a z e r o - s e t neiqhborhood
for a l l
Let
1
r
r = f ( x ) so that
f ( x ) belongs t o
~ E F . Therefore
C(X)
X
,
is
completely r e g u l a r . The p r o o f s t o ( 2 ) and ( 3 ) a r e e n t i r e l y s i m i l a r and w e l e a v e them t o t h e r e a d e r , Next we w i l l p r e s e n t a r e s u l t p a r a l l e l t o 3.6(1) i n t h e c a s e of
E-completely r e g u l a r s p a c e s .
The f o l l o w i n g d e f i n i /
t i o n s and r e l a t e d r e s u l t s a p p e a r i n t h e 1968 p a p e r o f Mrowka. 3.7
A set
DEFINITION.
provided t h a t f o r some f i n i t e
-1
n
and a c o n t i n u o u s f u n c t i o n
T C En
A = f
is s a i d t o be
A C X
(T).
The s e t
A
is
E-closed i n
X
there e x i s t s a closed subset f
E
C(X,E")
such t h a t
E-open i f and o n l y i f
X/A
is
E- c l o s e d .
The importance of t h e above d e f i n i t i o n l i e s i n t h e f a c t tha;
R - c l o s e d s e t s are p r e c i s e l y t h e z e r o - s e t s of
the
continuous r e a l - v a l u e d f u n c t i o n s .
i n t h e c a s e of
The same s t a t e m e n t i s t r u e
1 - c l o s e d sets, where
11 = [0,1]. I t i s n o t
d i f f i c u l t t o show t h a t any f i n i t e union and f i n i t e i n t e r s e c E-closed s e t s i n
t i o n of fact, Em
if
&
m
X
is again
E-closed i n
X.
In
i s a c a r d i n a l number and e v e r y c l o s e d s u b s e t o f then t h e i n t e r s e c t i o n of m E - c l o s e d sub& E - c l o s e d i n X. T h i s r e s u l t g e n e r a l i z e s t h e
E-closed,
-s e t s of
X
f a c t t h a t t h e i n t e r s e c t i o n o f c o u n t a b l y many z e r o - s e t s i s a zero-set stated e a r l i e r . interest.
The f o l l o w i n g r e s u l t i s of p r i m a r y
E-COMPLETELY REGULAR SPACES
3.8
THEOREM (Mrdwka)
. A
T 0- s p a c e
21
E-completely requ-
X
l a r if and o n l v i f t h e c l a s s of a l l
E-closed s u b s e t s
--I_
i s a b-----a s e f o r t h e c l o s e d s e t s of X. Proof. Necessity. Suppose X i s E-completely Then whenever
is a closed s e t i n
F
e x i s t s a f i n i t e number f
C(X,En)
E
Then
with
n
X
and
and
X
regular.
~ E X \ F there
and a c o n t i n u o u s f u n c t i o n
p cl f ( F ) by 3 . 3 ( b ) . S e t p p f - l ( T ) . Consequently,
f(p)
F C fP1(T)
of
T = cl f(F).
the c l a s s of
E-closed s e t s i s a b a s e . Sufficiencv.
Suppose t h a t t h e c l a s s of
forms a b a s e f o r X
and
and
PEX\F,
pPA.
Let
Then whenever
X.
t h e r e i s an
F
E-closed s e t
A = f - l ( T ) , where
f
a s p r o v i d e d by t h e d e f i n i t i o n 3.7. f(p) that
Since
T.
j!
is
X
is a
X
E
E-closed s e t s
i s a closed s u b s e t of A such t h a t F c A cl f(A) = T
n
are
and
T - s p a c e i t f o l l o w s from 3 . 3 ( b ) 0
E-completely r e g u l a r c o n c l u d i n g t h e p r o o f .
A theorem o f fundamental importance g i v e n
Jer i s o n
T , and
C(X,En),
Then
in Gillman and
1960, 3 . 9 ) e l i m i n a t e s any r e a s o n f o r c o n s i d e r i n g
r i n g s o f c o n t i n u o u s f u n c t i o n s on o t h e r t h a n c o m p l e t e l y r e g u l a r That theorem a s s e r t s t h a t f o r e v e r y t o p o l o g i c a l s p a c e
spaces. X
t h e r e e x i s t s a completely r e g u l a r space
ous mapp ng f H f
0
7
r
of
X
onto
Y
i s a n isomorphism o f
and a c o n t i n u -
such t h a t t h e mapping C(Y)
onto
p a p e r Mrdwka g e n e r a l i z e s t h i s r e s u l t f o r spaces.
Y
I n h i s 1968
C(X).
E-completely r e g u l a r
W e s t a t e t h a t r e s u l t h e r e f o r t h e s a k e o f complete-
n e s s a l t h o u g h w e s h a l l n o t have o c c a s s i o n t o r e f e r t o i t l a t e r on i n t h e s e q u e l and hence o m i t t h e p r o o f .
(The i n t e r e s t e d
r e a d e r can see Mrdwka ( 1 9 6 8 ) , 3 . 1 9 f o r t h e d e t a i l s . ) 3.9
THE I D E N T I F I C A T I O N THEOREM (Mrdwka).
For e v e r y s p a c e
--- map T of X o n t o Y such t h a t t h e mappinq -i s a n isomorphism pf C ( Y , E ) onto C ( X , E ) . there e x i s t s an
E-completely r e q u l a r s p a c e
~ U S
Y
X
and a c o n t i n u f M f o r
W e remark t h a t t h e p a r t i c u l a r r e s u l t o f t h e p r e v i o u s
theorem a s s o c i a t e d .with t h e c a s e when d i s c u s s e d by E.
Zech (19371, p. 8 2 6 ) .
E = R V
was originally
Cech a l s o d i s c u s s e d t h e
EMBEDDING I N TOPOLOGICAL PRODUCTS
22
E = D
c a s e i n which
and he s t a t e s t h a t spaces
...
goroff
( i . e . , To-)
C'
'I..
.
t h e connected dyad d e f i n e d p r e v i o u s l y , the theory of general topological
c a n be c o m p l e t e l y reduced t o t h e t h e o r y of Kolmospaces."
Another u s e f u l c o n c e p t r e l a t e d t o t h e i d e a s of t h i s sect i o n i s t h e n o t i o n of c o m p l e t e l y s e p a r a t e d s e t s .
This concept
w i l l b e v e r y i m p o r t a n t t o t h e development of Hewitt-Nachbin spaces. 3.10
TWO s u b s e t s
DEFINITION.
space
Of a topological
B
a r e s a i d t o b e c o m p l e t e l v s e p a r a t e d (from one an-
X
&
other)
and
A
i n case there e x i s t s a function
X
1,
that
0
3.11
THEOREM (Gillman and J e r i s o n ) .
f(A)
and
{O),
C
f(B)
C
If
f E C*(X)
such
{ 13. X
&a
topoloqical
then t h e f o l l o w i n s s t a t e m e n t s a r e t r u e :
space
1)
Two s e t s a r e c o m p l e t e l y s e p a r a t e d
i f thev 2)
a r e contained
in
in
X
i f and o n l y
d i s j o i n t zero-sets.
Completely s e p a r a t e d s e t s have d i s j o i n t z e r o - s e t neiqhborhoods.
If x
(3)
a c h o n o f f space, any two d i s i o i n t c l o s e d
are c o m p l e t e l y m-
s e t s , one o f which i s compact, rated, Proof.
A
(1) Suppose
zero-sets
Z(f) and
and
B
a r e contained i n the d i s j o i n t
Z ( g ), r e s p e c t i v e l y .
no z e r o s and t h e f u n c t i o n
h
Then
1 fl + 1 g /
has
d e f i n e d by
- 1
h(x) = belongs t o and t o
Moreover
C(X).
1 on
Z(g)
Conversely, i f there exists
f
E
:
f(x)
3
A
h
i s equal t o
B
a r e completely s e p a r a t e d then
on
0
Z(f)
2
B.
A
and
equal t o
C(X)
0
on
A
1 on
and
B.
The
disjoint sets
F =
(X
L$)
and
F' = ( x
a r e d i s j o i n t z e r o - s e t neighborhoods o f
A
: f(x)
and
L$)
B,
respectively.
T h i s a l s o p r o v e s (2). (3)
Suppose
A
and
B
a r e d i s j o i n t c l o s e d s e t s , and t h a t
A
E- COMPACT SPACES
i s compact. Zk
For e a c h
Zx
such t h a t
X E A , choose d i s j o i n t z e r o - s e t s
k
j o i n t zero-sets
Z
x1 respectively.
x
i s a neighborhood of
{zXjxEA o f t h e compact s e t , . . . , Z x :. Then A and
cover
23
A
B
U Zx U . . . U Z 2 Xk
and
and
Zx
The
B C Z'. X
h a s a f i n i t e subcover, a r e contained i n the d i s and
Zi
9 2; 1
3..
.n ZJZ , k
2
This proves ( 3 ) .
For any Hausdorff s p a c e
t h e property of normality i s
X,
e q u i v a l e n t t o t h e c o n d i t i o n t h a t any two d i s j o i n t c l o s e d s e t s be c o m p l e t e l y s e p a r a t e d ( s e e Gillman and J e r i s o n , 3 D , page 4 8 ) . Mr6wka g e n e r a l i z e s t h i s n o t i o n i n (1968) by d e f i n i n g t h e notion of
E-normality.
A space
is
X
E-normal p r o v i d e d t h a t any
two d i s j o i n t c l o s e d s e t s a r e c o n t a i n e d i n d i s j o i n t subsets.
completely r e q u l a r .
E-closed
is EW e w i l l not d e v e l o p t h e s e i d e a s f u r t h e r
I t can be shown t h a t e v e r y
E-normal
T1-space
and r e f e r t h e i n t e r e s t e d r e a d e r d i r e c t l y t o Mro'wka' s 1968 paper. Section 4:
E-Compact Spaces
W e now f o c u s o u r a t t e n t i o n on t h e second q u e s t i o n on topo-
l o g i c a l embeddings posed i n S e c t i o n 2 ; namely,
t h a t of d e t e r -
mining s p a c e s t h a t a r e homeomorphic t o c l o s e d s u b s p a c e s of t o p o l o g i c a l powers of some g i v e n s p a c e .
The i n v e s t i g a t i o n o f
t h a t problem was i n i t i a t e d i n t h e 1958 p a p e r by E n g e l k i n g and Mrowka a l t h o u g h o u r t r e a t m e n t f o l l o w s t h a t o f Mrdwkal s 1968 paper. 4.1
DEFINITION.
x
s p a c e s . Then
Let
X
and
E
i s s a i d t o be
b e two g i v e n t o p o l o g i c a l X
is
homeomorphic t o a c l o s e d subspace of t h e t o p o l o g i c a l power
Em
f o r some c a r d i n a l number
m.
s p a c e s i s denoted by
.
called classes
of
R(E)
E-compact provided t h a t The c l a s s of a l l
C l a s s e s o f t h e form
R(E) are
compactness.
I t i s c l e a r from t h e d e f i n i t i o n t h a t e v e r y
space i s
E-compact
E-completely r e g u l a r .
Evidently,
E-compact
t h e c l a s s R ( [0,1])
c o r r e s p o n d s t o t h e c l a s s of a l l compact Hausdorff s p a c e s . class
tX(sR)
The
i s t h e c l a s s of Hewitt-Nachbin s p a c e s t o be
s t u d i e d i n t h i s book, b u t w e w i l l examine additional examples of
24
EMBEDDING I N TOPOLOGICAL PRODUCTS
c l a s s e s of compactness i n t h i s s e c t i o n p r i o r t o t h a t s t u d y . 4.2
Let
THEOREM (Mrdwka-Engelking).
Hausdorff spaces.
The
(1) The s p a c e (2)
If
is
X
and
X
E
be two q i v e n
f o l l o w i n s s t a t e m e n t s are t r u e : E-compact.
E
and Xo is
E-compact
of
c l o s e d subspace
X,
is
Xo
then
homeomorphic
9
E-compact.
(3)
The
t o p o l o q i c a l p r o d u c t of a n a r b i t r a r y c o l l e c t i o n E-compact s p a c e s
(4)
of If
El
(5)
If
[Xu
is 2 Hausdorff s p a c e , e q u i v a l e n t to E E R ( E 1 ) . spaces,
(6)
X
be 2 Proof.
E-compact.
: afG)
then
R(E)
i s a non-empty f a m i l y
then n be a n
(Xa :
aEG}
is
of
If
is
@(El)
E-compact
E-compact.
E-compact s p a c e and l e t
c o n t i n u o u s mappinq.
C
Yo
f : X
i s an
-f
Y
E-compact
subspace of Y , then f - l ( Y o ) is E-compact. The r e s u l t s (1)- ( 4 ) a r e s t r a i g h t f o r w a r d consequences
of t h e d e f i n i t i o n o f
E-compactness s o we o m i t t h e i r p r o o f s
here.
n
The i n t e r s e c t i o n
(51
diagonal of t h e product closed i n t h e product.
{Xu : a E G ) i s homeomorphic t o t h e
II(Xa : a E G ) and t h e d i a g o n a l i s These r e s u l t s a r e c l a s s i c a l and t h e
theorem now f o l l o w s from ( 2 ) and ( 3 ) . The i n v e r s e image
(6)
of
f
restricted t o
f-l(Yo)
i s homeomorphic t o t h e g r a p h
fw1(Yo); i . e . ,
t o the set [(x,y) : y =
f (x), XEX, f ( x ) E Yo) which i s c l o s e d i n i s now immediate from ( 2 ) and ( 3 ) .
X
X
Yo.
The r e s u l t
We remark t h a t s t a t e m e n t ( 6 ) of t h e above theorem genera l i z e s t h e r e s u l t on Hewitt-Nachbin s p a c e s g i v e n i n Gillman and J e r i s o n ,
(1960, 8 . 1 3 ) .
A s t r o n g e r s t a t e m e n t t h a n (2) can
b e provided i n t h e c a s e t h a t
E = IR
and t h i s w i l l b e g i v e n
f u r t h e r on i n t h e s e q u e l i n c o n n e c t i o n w i t h t h e development of t h e Wallman-Frink c o m p l e t i o n .
The r e s u l t s on t o p o l o g i c a l prod-
u c t s ( 3 ) and c l o s e d s u b s p a c e s ( 2 ) i n t h e case where
E = R
w e r e o r i g i n a l l y due t o H e w i t t (1948, Theorem 6 2 ) and K a t g t o v (195lA), r e s p e c t i v e l y .
They w i l l be r e s t a t e d i n t h e s e c t i o n
on p r o p e r t i e s of Hewitt-Nachbin s p a c e s f o r p u r p o s e s of e a s y
25
E- COMPACT SPACES
reference. I t is w e l l known t h a t t h e e x i s t e n c e o f a Hausdorff comp a c t i f i c a t i o n of an a r b i t r a r y c o m p l e t e l y r e g u l a r Hausdorff s p a c e was f i r s t e s t a b l i s h e d by A . given space
X
Tychonoff.
H e embedded t h e
i n a p r o d u c t of a s u i t a b l e number o f c o p i e s
of t h e u n i t i n t e r v a l [0,1] and h i s p r o c e s s y i e l d s t h e s p a c e V
pX, t h e Stone-Cech c o m p a c t i f i c a t i o n d i s c u s s e d i n S e c t i o n 2.
I n f a c t , e v e r y c o m p a c t i f i c a t i o n of
can b e o b t a i n e d b y t h i s Whereas Tychonoff seemed i n t e r e s t e d i n k e e p i n g t h e
procedure.
X
V
number of f a c t o r s a s s m a l l a s p o s s i b l e , E . Cech i n 1 9 3 7 recogn i z e d t h e a d v a n t a g e s of t a k i n g l a r g e p r o d u c t s . -4
c o n s t r u c t i o n o f Cech i n o b t a i n i n g y i e l d a more g e n e r a l r e s u l t f o r
PX
The famous
can be d u p l i c a t e d t o
E-compactness.
This r e s u l t
o c c u r s i n t h e 1958 p a p e r by Engelking and Mrdwka and w e w i l l have more t o s a y a b o u t i t s s i g n i f i c a n c e r e l a t i v e t o H e w i t t Nachbin s p a c e s f u r t h e r on i n t h e s e c t i o n . 4.3 X
THE
be a n
E- COMPACTIFICATION THEOREM ( Engelking-Mdwka)
_Then
E-completely r e q u l a r Hausdorff s p a c e .
-a Hausdorff
pEX
E-compactification
. has
X
w i t h t h e f o l l o w i n q prop-
erties : (1) The s p a c e
f
extension (2)
If Y
is dense i n
X
function
: X -+ E
f*
:
i s an
pEx
pEX
and e v e s y c o n t i n u o u s
a d m i t s a unique c o n t i n u o u s +
E.
E-compact s p a c e , t h e n e v e r y c o n t i n u o u s
-'Y a d m i t s a unique c o n t i n u o u s extension : pEX + Y. ( 3 ) The s p a c e pEX is unique i n t h e f o l l o w i n g s e n s e : I f Y i s a n E-compact s p a c e containX densely E admits g and s u c h t h a t evcontinuous f : X ----c o n t i n u o u s e x t e n s i o n to Y , t h e n Y is homeomorphic t o pEX under a homeomorphism t h a t i s the i d e n t i t y on X. Proof. (I) L e t F = C ( X , E ) and l e t u be t h e p a r a m e t r i c map function
g
* g
: X
-f
of
X
into
f o r every
Em = n [ E f
:
f E F ) , where m = I C ( X , E )
fcF, c o r r e s p o n d i n g t o t h e c l a s s
F.
1
and Ef = E
Because
X
is
E-completely r e g u l a r Theorem 3 . 3 e s t a b l i s h e s t h a t t h e r e q u i r e ments of ( 3 ) i n t h e Embedding Lemma a r e s a t i s f i e d .
Accordingly,
26
IJ
EMBEDDING I N TOPOLOGICAL PRODUCTS
i s a homeomorphism.
i n t h e product space
u(X)
pEx of
Now,
Em.
u.
t h a t extends
t i o n s between
and
X
is
Moreover, t h e t o p o l o g i c a l r e l a a r e i d e n t i c a l t o t h o s e between
PEX
E-compact. f* =
t h e mapping
If
*
u
?rf 0
Since
(2)
is
Y n(Ea
f : X
a
'a
o
g.
+
E
where
: aEG) ?r
pEX
E
a
f =
= E
T
f o r every
For each
acG.
a s s o c i a t e d with t h e product, set
a
Then, f o r each
aEG,
a '
i s continuous w i t h
'a
and t h e r e f o r e e x t e n d s t o
: X -+ E
to
f
f 0 a. T h i s p r o v e s ( 1 ) . E-compact w e c a n embed Y a s a c l o s e d s u b -
p r o j e c t i o n mapping T
pEX. Cleari s continuous, then
is dense i n
i s a n e x t e n s i o n of
because t h e c o n s t r u c t i o n y i e l d s s p a c e of
t h e r e exists a superspace c l u ( X ) by a homeomor-
c l u(X); in particular, X
o ( X ) and REX
c l u ( X ) d e n o t e t h e c l o s u r e of
t h a t i s homeomorphic t o
X
a*
phism
ly
Let
g*,
:
pEx
+
E
by p a r t
( 1 ) . W e i l l u s t r a t e t h e s i t u a t i o n i n t h e f o l l o w i n g commutative
diagram.
*
PEX
*Ea
= E
U
X'
Now,
let
H : pEX
-f
9
ilEa * d e n o t e t h e p a r a m e t r i c mapping a s s o -
c i a t e d with t h e c l a s s (9, Observe t h a t i f o t h e r words, t h e
ath
is precisely the
nEa
H(p) = g ( p ) f o r a l l PEX,
IIE,
Therefore,
: acG).
Then H ( p ) = ( g : ( ~ ) ) , , ~ . * g a ( p ) = g a ( p ) = ?ra O g ( p ) . I n
component of
H(p)
a t h component of pcX.
it follows t h a t
is closed i n H(X).
then
PEX
g(p).
Finally, since
H(X) i s dense i n
i n the product X
Therefore, i s dense i n
H(pEX).
However,
a n d , a s w e have j u s t o b s e r v e d , Y H(%X)
C Y
i s t h e d e s i r e d e x t e n s i o n of
Y
contains
from which i t i s immediate t h a t
g. This proves ( 2 ) . Y be a n E-compact s p a c e s a t i s f y i n g t h e a s s u m p t i o n s of t h e theorem. L e t i d e n o t e t h e i n c l u s i o n mapping of X i n t o Y. Then b y p a r t (2) t h e r e i s a c o n t i n u o u s e x t e n s i o n * i : %X -+ Y . S i m i l a r l y , t h e embedding g of X i n t o PEX * e x t e n d s t o g : Y 3 %X by assumption. We i l l u s t r a t e these H
(3)
Let
E- COMPACT SPACES
27
facts in the following commutative diagram.
X
*
*
i
*
*
Now, i (p) = g (p) = p for every p€X. Hence i 0 g (p) = * * g o i ( p ) = p *for *every p c X so that the density conditions Accordingly, on X yield i * g = 1% and g*o i* - idPEx* * g = (i*)-' and consequently Y is homeomorphic to REX under a homeomorphism that leaves X pointwise fixed. This concludes the proof of the theorem. The following corollary is exactly what one would expect and appears in Mrdwkals 1966 paper. 4.4 COROLLARY (Mrdwka). Let X E-completely reqular. Then X is E-compact if alld only if X = pEX. Proof. If X is E-compact it is an extension of itself that satisfies the condition of part (3). Thus X is homeomorphic The to p,X under the identity and consequently X = P EX. converse is immediate.
-
-
Of course, in the instance where E = I O , l ] , BEX coinV cides with the Stone-Cech compactification. If E = R , then PEX is the Hewitt-Nachbin completion UX of a completely regular Hausdorff space. One might wonder, because of the generalized construction for p E X , whether it would be possible to obtain a paracompactification of a Tychonoff space X for which continuous functions from X into a paracompact space admit continuous extensions to the extension space. One of the main results established by J. Van der Slot (1968) proves that such a paracompactification is not possible because arbitrary products of paracompact spaces may fail to be paracompact. An additional characterization of E-compactness turns
EMBEDDING I N TOPOLOGICAL PRODUCTS
28
o u t t o b e q u i t e u s e f u l i n t h e s t u d y o f Hewitt-Nachbin s p a c e s . W e g i v e t h a t r e s u l t a s t h e n e x t theorem.
4.5
Let
THEOREM (Engelking-Mrdwka).
r e q u l a r Hausdorff s p a c e .
The s p a c e
o n l y i f f o r e v e r y Hausdorff s p a c e such t h a t Y f X, --
-
If
X
be a n
E-completely
E-compact i f and
X
containinq
Y
# p EX ,
to
densely
X
t h e r e i s a continuous f u n c t i o n
t h a t c a n n o t be extended -----Proof.
X
f : X
E
-t
Y.
t h e n t h e e x t e n s i o n c o n d i t i o n would deny
p r o p e r t y (1) o f Theorem 4 . 3 .
X = PEX
Thus
so t h a t
X
is
E-compact e s t a b l i s h i n g t h e s u f f i c i e n c y . Conversely, suppose a homeomorphism
h : X
h(X) i s c l o s e d i n
of
Em
onto
+
Em.
Em
Let
h
is
Then t h e r e e x i s t s
E-compact.
m
f o r Some c a r d i n a l denote t h e
T~
fa = ra o h .
and s e t
E,
c o n t i n u o u s and
X
such t h a t
ath projection
Then
fa
: X
--f
E
is
i s t h e p a r a m e t r i c mapping a s s o c i a t e d w i t h
t h e f a m i l y of a l l f u n c t i o n s
d e f i n e d a s above.
fa
According
t o c o n d i t i o n ( i i )of t h e Embedding Lemma, f o r each p r o p e r ext e n s i o n .Y o f
X (i.e.,
i s d e n s e i n Y and Y # X) a t fa f a i l s t o a d m i t a c o n t i n u o u s
X
l e a s t one of t h e f u n c t i o n s extension t o
Y.
This completes t h e p r o o f .
I n t h e f o l l o w i n g example w e examine some p r o p e r t i e s o f N -compact s p a c e s .
4.6
EXAMPLE.
If
IN-compact, t h e n
X
EJ
consequence of t h e f a c t t h a t
( i t i s a c l o s e d subspace o f an
X
i s a zero-
T h i s r e s u l t is a n immediate
d i m e n s i o n a l Hewitt-Nachbin s p a c e .
i s a Hewitt-Nachbin s p a c e IR-compact s p a c e ) and t h a t
p r o d u c t s and s u b s p a c e s of z e r o - d i m e n s i o n a l s p a c e s a r e z e r o dimensional.
Since
PX
i s n o t n e c e s s a r i l y zero-dimensional
f o r every zero-dimensional
X ( i n f a c t , a compact s p a c e i s
z e r o - d i m e n s i o n a l i f and o n l y i f t h e s i n g l e t o n p o i n t s a r e t h e o n l y connected s u b s e t s , i n which c a s e it i s s a i d t o be t o t a l l y d i s c o n n e c t e d ) it t u r n s o u t t h a t t h e c o n v e r s e s t a t e m e n t i s false:
t h e r e d o e s e x i s t a z e r o - d i m e n s i o n a l Hewitt-Nachbin
space t h a t f a i l s t o be
IN-compact.
Nyikos d e s c r i b e s such a s p a c e
A.
I n h i s 1970 p a p e r , P. The s p a c e
A
is a h i g h l y
c o m p l i c a t e d example o r i g i n a l l y i n t r o d u c e d by P r a b i r R o y i n
29
E- COMPACT SPACES
A weaker s t a t e m e n t t h a n t h e c o n v e r s e i s , how-
1 9 6 2 and 1968.
Namely, if p X i s z e r o - d i m e n s i o n a l , t h e n X & a Hewitt-Nachbin s p a c e i f and o n l y if X lN-compact. The -Supp r o o f o f t h i s is d u e t o Mro’wka and p r o c e e d s a s f o l l o w s : ever,true.
pose
i s a Hewitt-Nachbin s p a c e .
X
s i o n a l t h e same h o l d s t r u e o f D
p l e t e l y r e g u l a r , where Moreover, t h e f a c t t h a t that
Let
Y
with
Y
#
X.
E
Let
Let because
Z\x
#
D-com-
IN-completely r e g u l a r i m p l i e s
is
D
IN-compact u t i l i z i n g
is
X
denote i t s Stone e x t e n s i o n . po
E
i s t h e i d e n t i t y on
i*lX
densely
...
PEX,
f
PX
:
and
-+
IR
Z =
Set -1
(pol.
Then
Hence t h e r e
X.
such t h a t
f
f(qo) =
t a k e s on o n l y t h e v a l u e s 1 0,1,3,-$, Define f l : X IN by f l ( p ) = Then f l E C ( X , I N ) b u t c l e a r l y f a i l s t o e x t e n d t o Y . Thus X i s 0
for a l l
*
qo E ( i )
and choose
Y\x
X
d e n o t e t h e i n c l u s i o n mapping and
i : X -,Y
e x i s t s a continuous f u n c t i o n 0,
f(p)
is
d e n o t e a Tychonoff s p a c e c o n t a i n i n g
i * : BX + pY
(i*)-’(Y). q,
i s zero-dimenX
is t h e two-point d i s c r e t e space.
N e x t we w i l l e s t a b l i s h t h a t 4.5.
let
pX
IN-completely r e g u l a r ( 3 . 2 ( 4 ) ) .
is
X
Since
X; a c c o r d i n g l y
fo
--f
.
IN-compact c o m p l e t i n g t h e argument. We remark t h a t s p a c e s
X
w i t h zero-dimensional
now commonly c a l l e d s t r o n q l y z e r o - d i m e n s i o n a l .
PX
are
There a r e a
number o f o u t s t a n d i n g problems t h a t have been p o s e d by Mro)wka i n c o n n e c t i o n w i t h t h e above r e s u l t s .
W e mention a few of
these here. PROBLEM A.
Is e v e r y
IN-compact s p a c e
X
s t r o n g l y zero-
dimensional? PROBLEM B.
Is it t r u e t h a t i n a n
IN-compact s p a c e t h e z e r o -
s e t s b e l o n g t o t h e c l a s s of a l l Bore1 sets g e n e r a t e d by c l o p e n
sets ( i . e . , t h e s m a l l e s t c l a s s c o nt a i n i n g a l l clopen sets t h a t i s c l o s e d under complements and c o u n t a b l e u n i o n s ) ? I n c i d e n t a l l y , Mro/wka h a s shown i n h i s 1972A p a p e r t h a t s t r o n g z e r o - d i m e n s i o n a l i t y i s p r e s e r v e d under p r o d u c t s , b u t w e omit t h a t argument h e r e . I n h i s 1970 p a p e r , K. P. Chew p r o v i d e s a n u l t r a f i l t e r c h a r a c t e r i z a t i o n i n o r d e r t h a t a zero-dimensional
space b e
30
EMBEDDING I N TOPOLOGICAL PRODUCTS
IN-compact
(we w i l l s t a t e h i s r e s u l t precisely i n Section 7)
He a l s o p r o v i d e s a c h a r a c t e r i z a t i o n f o r a z e r o - d i m e n s i o n a l s p a c e t o be
IN-compact i n t e r m s o f t h e " c o m p l e t e n e s s " of
c l o p e n c o v e r i n g s of t h e s p a c e .
( T h i s c o n c e p t of "complete-
n e s s " o r i g i n a t e d i n Z. F r o l i k l s 1961 paper and i s a s s o c i a t e d w i t h t h e n o t i o n of an " a l m o s t r e a l c o m p a c t s p a c e . "
We w i l l
i n v e s t i g a t e t h o s e i d e a s i n Chapter 3 . ) H e r r l i c h ' s (1967B) g e n e r a l i -
W e s h a l l now comment on H.
z a t i o n of c l a s s e s of complete r e g u l a r i t y and c l a s s e s o f compactness.
Herrlich considers a c l a s s
s p a c e s and examines s p a c e s
X
B
of t o p o l o g i c a l
t h a t a r e homeomorphic t o a sub-
s p a c e ( r e s p e c t i v e l y , c l o s e d s u b s p a c e ) of a p r o d u c t o f s p a c e s each of which i s i n
H e r r l i c h d e m o n s t r a t e s t h a t some of
B.
t h e c o n s i d e r a t i o n s on
E-completely r e g u l a r and
E-compact
s p a c e s t h a t w e have been d i s c u s s i n g remain v a l i d i n t h i s more general s e t t i n g , p a r t i c u l a r l y with regard t o the r e s u l t s a s s o c i a t e d w i t h t h e I d e n t i f i c a t i o n Theorem 3.9 and t h e c o m p a c t i f i c a t i o n Theorem 4 . 3 .
E-
Moreover, H e r r l i c h d i s c u s s e s
t h e s e r e s u l t s w i t h i n t h e framework of c a t e g o r y t h e o r y .
(We
w i l l examine t h e c a t e g o r i c a l p o i n t o f view i n t h e n e x t s e c tion.)
Mro/wka p o i n t s o u t i n h i s 1968 p a p e r ( p . 180) t h a t
H e r r l i c h ' s approach d o e s l e a d sometimes t o a s t r o n g e r formulat i o n of some q u e s t i o n s and r e s u l t s , p a r t i c u l a r l y w i t h r e g a r d t o c l a s s e s o f complete r e g u l a r i t y and c l a s s e s of compactness. The r e s u l t s p r e s e n t e d above s u g g e s t t h e f o l l o w i n g u s e f u l definitions i n the case t h a t
E = R.
These d e f i n i t i o n s and
immediate r e s u l t s can b e found i n t h e Gillman and J e r i s o n text. 4.7
be
Let
DEFINITION.
and l e t
S
X
be a n a r b i t r a r y t o p o l o g i c a l s p a c e ,
be a non-empty s u b s e t of
C-embedded i n
X
X.
Then
provided t h a t f o r each
is s a i d t o
S
f
in
there exists a
g
in
C(X) such t h a t t h e r e s t r i c t i o n
f.
S
is
C*-embedded i n
Similarly,
X
bounded c o n t i n u o u s r e a l - v a l u e d f u n c t i o n on bounded c o n t i n u o u s e x t e n s i o n t o
C(S)
glS
i n case e v e r y S
admits a
X.
Using t h e above t e r m i n o l o g y , e v e r y Tychonoff s p a c e i s
is
E- COMPACT SPACES
*
31
V
C -embedded i n i t s Stone-Cech c o m p a c t i f i c a t i o n .
d e n s e and
The f o l l o w i n g r e s u l t s c h a r a c t e r i z e c o n c e p t s of
C-embedding and
*
C -embedding,
C*-embedding,
relate the
and s t r e s s t h e
importance of compact s u b s e t s i n a Tychonoff s p a c e .
We o m i t
t h e p r o o f s t o t h e s e r e s u l t s a s t h e y c a n b e found i n t h e G i l l man and J e r i s o n t e x t ( s e e 1 . 1 7 ,
1.18, and 3 . 1 1 ( c ) of t h a t t e x t
for the d e t a i l s ) . THEOREM (Gillman and J e r i s o n ) .
4.8
s p a c e and
If
i s a non-empty s u b s e t of
S
-is& topolosical
X X,
then t h e followinq
statements are true: (1) The s u b s e t
in
C*-embedded
S
X
any two c o m p l e t e l v s e p a r a t e d sets i n p l e t e l y separated sets i n (2)
The s u b s e t
it is
*
S
is
C -embedded
& I
a r e com-
S
X.
in
in
completely s e p a r a t e d
Every compact s u b s e t C-embedded
if
C-embedded X
from e v e r y z e r o - s e t d i s j o i n t _from (3)
i f and o n l y
X
i f and o n l v i f
&.
of a Tychonoff s p a c e
S
&
X
X.
The s t a t e m e n t (1) i s known a s Urvsohn's E x t e n s i o n Theorem If
i s a metric space t h e n every c l o s e d s e t is a z e r o - s e t ;
X
hence any two c l o s e d d i s j o i n t s e t s a r e c o m p l e t e l y s e p a r a t e d . Furthermore,
in S
if
i s a c l o s e d s u b s e t of
S
are closed i n
S
X, then c l o s e d sets X; t h e r e f o r e c o m p l e t e l y s e p a r a t e d s e t s i n
have d i s j o i n t c l o s u r e s i n
t e n s i o n Theorem: embedded,
Thus w e o b t a i n T i e t z e ' s Ex-
every c l o s e d set i n a m e t r i c space i s
C
*
-
P . Urysohn g e n e r a l i z e d t h a t r e s u l t t o normal
spaces:
~s u b s e t of
X.
T -space
1
X
&
X
i s normal i f and o n l y if e v e r v c l o s e d
C-embedded
in
X.
and o n l y i f e v e r y c l o s e d s u b s e t i s and J e r i s o n ,
3D, p a g e 4 8 ) .
weaker s t i l l t h a n
i s normal i f
( s e e Gillman
There i s a n o t h e r k i n d o f embedding,
* C -embedding,
t i o n p r o p e r t y of n o r m a l i t y .
In fact, X
*
C -embedded
t h a t c h a r a c t e r i z e s t h e separa-
T h i s i s t h e c o n c e p t of " z -
embedding" which w i l l b e i n v e s t i g a t e d i n t h e n e x t c h a p t e r . I n h i s 1 9 7 2 p a p e r J. Green p r o v i d e s f i l t e r c h a r a c t e r i z a t i o n s of
C-
and
*
C -embedding.
embedded i n t h e Tychonoff s p a c e
He proves t h a t X
S
is
*
C
-
i f and o n l y i f t h e " t r a c e
32
on
EMBEDDING I N TOPOLOGICAL PRODUCTS
S"
of e v e r y maximal " c o m p l e t e l y r e g u l a r f i l t e r " on
intersecting f i l t e r " on
S.
X
i s i t s e l f a maximal " c o m p l e t e l y r e g u l a r
S
A similar c h a r a c t e r i z a t i o n i s obtained f o r
embedded s u b s e t s of a Tychonoff s p a c e .
C-
Parallel r e s u l t s for
"z-embedding" w i l l b e p r e s e n t e d i n t h e n e x t c h a p t e r . The f o l l o w i n g r e s u l t i s u s e f u l and o r i g i n a l l y a p p e a r s i n Hewitt' s 1948 p a p e r . 4.9
THEOREM ( H e w i t t ) .
t o p o l o q i c a l space
If then
X,
i s d e n s e and
S
C(S)
is
C-embedded i n t h e
isomorphic
to
C(X).
Proof.
Each f u n c t i o n i n
of
S i n c e each such e x t e n s i o n must a g r e e on t h e d e n s e sub-
X.
s,
C ( S ) h a s continuous extensions t o a l l
IR i n s u r e s t h a t i t must a l s o a g r e e on X. T h e r e f o r e , e a c h f u n c t i o n f E C(S) h a s a D e f i n e t h e mapping 16 from unique e x t e n s i o n f* i n C ( X ) C(S) i n t o C ( X ) by q ( f ) = f*. I t w i l l be shown t h a t (D i s a r i n g isomorphism. I f (o(fl) = P ( f 2 ) , t h e n f l and f 2 must a g r e e on S. Hence, tp i s i n j e c t i v e . I f g E C ( X ) , then t h e r e s t r i c t i o n space
t h e Hausdorff p r o p e r t y o f
.
g/S
in
belongs t o X,
C(S).
Therefore, s i n c e
t h e r e e x i s t s an e x t e n s i o n
(glS)*/S = g/S.
* (gjS) *
S
in
I t f o l l o w s t h a t (91s) = g
of each e x t e n s i o n .
Hence, cp
is
C-embedded
C(X)
satisfying
by t h e u n i q u e n e s s
is surjective.
I n an e n t i r e l y
s i m i l a r manner, t h e u n i q u e n e s s of t h e e x t e n s i o n t o
yields
X
p ( f l + f 2 ) = d f , ) + d f 2 )and a l s o q ( f l f 2 ) = q ( f l ) q ( f 2 ) . Therefore, cp i s a n isomorphism. This completes t h e p r o o f . From t h e l a s t r e s u l t it i s immediate t h a t f o r e v e r y X , t h e r i n g C ( X ) i s isomorphic t o d e n o t e s t h e Hewitt-Nachbin c o m p l e t i o n o f
Tychonoff s p a c e
C(uX),
where
X.
uX
I n t h e n e x t s e c t i o n we w i l l examine t h e n o t i o n s o f compactness and t h e
E-
E - c o m p a c t i f i c a t i o n i n the c o n t e x t of c a t e -
gory theory. Section 5 :
A Cateqorical Perspective
A s we have s e e n i n o u r development t h r o u g h o u t t h i s chap-
t e r , e x t e n s i o n s of t o p o l o g i c a l s t r u c t u r e s abound, of
E-compact s p a c e s w e observed t h a t t h e
I n our study
E-compactification
33
A CATEGORICAL PERSPECTIVE
s a t i s f i e s a f a c t o r i z a t i o n p r o p e r t y r e q u i r i n g t h a t e a c h mapping f
of
t o an
X
E-compact s p a c e
Y
extend t o
pEX:
(The symbol " # ' i n t h e diagram means t h a t it i s commutative: f * o i = 5.1
Although much o f t h e language of t h i s s e c t i o n w i l l be c a t e g o r i c a l i n n a t u r e , most of t h e p r o o f s w i l l be t o p o l o g i c a l . Much o f t h e s e c t i o n i s b a s e d on t h e 1970 n o t e s of S. P. lin.
Frank-
A good s u r v e y of t h e a r e a i s a l s o p r o v i d e d i n t h e 1 9 7 1
p a p e r o f H. H e r r l i c h . We w i l l assume -----
t h a t a l l s p a c e s under c o n s i d e r a t i o n
t h r o u q h o u t t h i s s e c t i o n w i l l b e Hausdorff u n l e s s e x p l i c i t l y
_ s t a t_ ed that
they a r e a r b i t r a r v spaces.
T h i s s e c t i o n may be
o m i t t e d w i t h o u t d e s t r o y i n g t h e c o n t i n u i t y of t h e remainder o f t h i s book.
All o f t h e c a t e g o r i e s t o be c o n s i d e r e d h e r e w i l l be made up o f a c l a s s of s p a c e s t o g e t h e r w i t h t h e mappings between For i n s t a n c e , w e w i l l be d i s -
spaces belonging t o t h e c l a s s .
c u s s i n g t h e c a t e g o r i e s of c o m p l e t e l y r e g u l a r s p a c e s , p l e t e l y r e g u l a r s p a c e s ( f o r some f i x e d
E-com-
E ) , compact s p a c e s ,
I t i s h e l p f u l t o keep s u c h
Hewitt-Nachbin s p a c e s , e t c e t e r a .
examples i n mind when c o n s i d e r i n g t h e f o l l o w i n g d e f i n i t i o n . 5.1
A catesorv
DEFINITION.
objects
Co,
@
c o n s i s t s of a c l a s s of
a c l a s s of morphisms o r maps ( a l s o d e s i g n a t e d by
@), and a n o p e r a t i o n of composition f o r t h e morphisms,
to-
g e t h e r w i t h t h e f o l l o w i n g axioms:
of t h r e e morphisms i s d e f i n e d whenever t h e c o m p o s i t i o n s h o g and g o f a r e defined.
(1) The composition
(2)
h o go f
Composition of morphisms i s a s s o c i a t i v e : h (hog)
0
f
o
(gof) =
and b o t h c o m p o s i t i o n s a r e d e f i n e d i f
EMBEDDING I N TOPOLOGICAL PRODUCTS
34
e i t h e r is defined. (3)
For each o b j e c t morphism g : A g
*
to
A
in
C0
t h e r e i s an i d e n t i t y
lA such t h a t whenever
a r e morphisms t h e n
C
f : B + A
lAC f = f
and
and
lA= g . @ ( A , B ) w i l l d e s i g n a t e t h e c l a s s of morphisms
The n o t a t i o n from
+
A
B , where
A
and
B
belong t o
C0.
The f o l l o w i n g examples i l l u s t r a t e some i m p o r t a n t c a t e gories.
EXAMPLES.
C0
(1) L e t
s p a c e s and l e t
C
d e n o t e any c l a s s of t o p o l o g i c a l
d e n o t e t h e c o n t i n u o u s mappings between
them. T h i s forms a c a t e g o r y of t o p o l o g i c a l s p a c e s . Most of o u r d i s c u s s i o n w i l l c e n t e r around t h e c a t e g o r y where t h e s p a c e s a r e Hausdorf f . (2)
The classes o f a l l s e t s and mappings form a c a t e g o r y .
(3)
Any of t h e c l a s s e s of a l g e b r a i c o b j e c t s , such a s
g r o u p s , r i n g s , o r v e c t o r s p a c e s w i l l form a c a t e g o r y when t h e c l a s s of morphisms i s t a k e n t o b e t h e a p p r o p r i a t e c l a s s of homomorphisms. 5.2
DEFINITION.
A morphism
b e a monomorphism i f whenever
f
in a c a t e g o r y f Cg
and
f oh
is said t o are defined
f o r all morphisms
g and h . The morphism f i s s a i d t o be a n epimorphism i f g o f = h o f i m p l i e s g = h f o r a l l morphisms g and h. I f f is both a monomorphism and an epimorphism, t h e n it i s s a i d t o be a bimorphism. A morphism f : A + B i s s a i d t o b e an i dmorphism i f and o n l y i f i t h a s a r i q h t - i n v e r s e r : B -+ A such t h a t f 0 r = lB and a l e f t - i n v e r s e 4, : B + A s u c h t h a t t , O f = lA. I f f i s a n isomorphism t h e n t h e f o l l o w i n g e q u a l i t i e s show t h a t t h e i n v e r s e s r and .C must b e e q u a l : and e q u a l t h e n
t,=
g = h
t,0lB
= t , o ( f o r ) = ( & o f ) o r= 1A
01: =
r.
Thus, i f f i s a n isomorphism, we w i l l w r i t e f-l = r = t, and r e f e r t o f - l a s t h e i n v e r s e o f f . I f f h a s a r i g h t -
35
A CATEGORICAL PERSPECTIVE
inverse then
i s c a l l e d a r e t r a c t i o n : i f it has a left-
f
i n v e r s e it i s c a l l e d a c o r e t r a c t i o n . I t i s e a s y t o e s t a b l i s h t h a t e v e r y r e t r a c t i o n i s an epi-
morphism and e v e r y c o r e t r a c t i o n i s a monomorphism, b u t t h e converse i s f a l s e . f o r morphisms
g
To s e e t h e f i r s t s t a t e m e n t , o b s e r v e t h a t and
g ' , w i t h t h e i n d i c a t e d compositions
d e f i n e d , w e have g = g o ( f o r ) = glo ( f o r ) = g'. EXAMPLES.
(1)
I n t h e c a t e g o r y o f s e t s and mappings t h e mono-
morphisms a r e t h e i n j e c t i v e mappings, and t h e epimorphisms a r e t h e s u r j e c t i v e mappings. (2)
We w i l l mainly be c o n s i d e r i n g t h e f o u r c a t e g o r i e s o f
Hausdorf f , c o m p l e t e l y r e g u l a r , compact, and Hewitt-Nachbin Since t h e s e p r o p e r t i e s are t o p o l o g i c a l i n v a r i a n t s ,
spaces.
t h e isomorphisms i n each c a t e g o r y a r e p r e c i s e l y t h e homeomorphisms (3)
I n t h e c a t e g o r y o f g r o u p s , t h e isomorphisms a r e t h e
b i j e c t i v e homomorphisms.
5.3
DEFINITION.
gory
C
63
A category
i s a s u b c a t e q o r v of a c a t e -
i f e v e r y o b j e c t and morphism o f
and morphism of
@.
s u b c a t e q o r v i f e v e r y morphism i n
i s a l s o a morphism i n
63.
C
C
is s a i d t o be a f u l l 63
of
C
is said t o
i f every object
A
of
which i s i s o m o r p h i c t o an o b j e c t 63
i s a l s o an o b j e c t
between two o b j e c t s i n
A subcategory
be a r e p l e t e s u b c a t e q o r y of
to
63
The s u b c a t e g o r y
63
B
of
63 63
C
does i t s e l f belong
a s w e l l a s t h e isomorphism,
------We w i l l assume t h a t
be d i s c u s s e d
are full.
a l l of t h e s u b c a t e q o r i e s which w i l l S i n c e any s u b c a t e g o r y o f t h e c a t e g o r y
of t o p o l o g i c a l spaces which i s o b t a i n e d by s p e c i f y i n g t h a t t h e o b j e c t s p o s s e s s a t o p o l o g i c a l i n v a r i a n t i s r e p l e t e , a l l of t h e s u b c a t e q o r i e s which w i l l be d i s c u s s e d w i l l be presumed t o be replete. There i s one i m p o r t a n t example t h a t w e wish t o mention i n c o n n e c t i o n w i t h t h e above d e f i n i t i o n s .
Consider t h e c a t e g o r y
o f m e t r i c s p a c e s and u n i f o r m l y c o n t i n u o u s mappings.
The i s o -
36
EMBEDDING I N TOPOLOGICAL PRODUCTS
rnorphisms a r e t h e i s o m e t r i c s ( d i s t a n c e p r e s e r v i n g maps) s o t h a t t h i s i s a n example of a n o n f u l l s u b c a t e g o r y of t h e c a t e g o r y o f m e t r i c s p a c e s on a l l mappings between m e t r i c s p a c e s . The r e a s o n f o r t h i s s i t u a t i o n i s b e c a u s e t h e p r o p e r t y o f b e i n g a m e t r i c s p a c e f a i l s t o be a t o p o l o g i c a l i n v a r i a n t . (However, t h e p r o p e r t y of b e i n g m e t r i z a b l e i s a t o p o l o g i c a l i n v a r i a n t and i n t h e c a t e g o r y of m e t r i z a b l e s p a c e s , t h e isomorphisms a r e Moreover, t h e s u b c a t e g o r y of m e t r i c
t h e homeomorphisms.)
spaces and u n i f o r m l y c o n t i n u o u s maps i s n o t r e p l e t e b e c a u s e
two metric s p a c e s c a n be t o p o l o g i c a l l y isomorphic w i t h o u t t h e homeomorphism b e i n g a n i s o m e t r y . 5.4
A f u n c t o r from a c a t e g o r y
DEFINITION.
f.2
is a r u l e
f
of
which a s s i g n s t o e a c h o b j e c t
F
an o b j e c t
G
and morphism
FA
t o a category
G
and morphism
A
F ( f ) such t h a t :
(1)
F
p r e s e r v e s i d e n t i t i e s ; 1 . e . , F ( lA)= lFA,
(2)
F
p r e s e r v e s composition: i . e . , i f
in
F ( f ) OF(g) i s defined i n
G , then
equal t o
is defined
f Og C
and i s
F(f o 9).
I f t h e statement ( 2 ) i s a l t e r e d by r e q u i r i n g t h a t compositions
i.e., that
b e reversed:
F(f
0
9) = F ( g ) 0 F ( f ) , t h e n
F
is
called a contravariant functor. The f o l l o w i n g examples i l l u s t r a t e some i m p o r t a n t f u n c t o r s . EXAMPLES.
(1)
For e v e r y s e t
S, l e t
DS
be t h e t o p o l o g i c a l
s p a c e o b t a i n e d by p u t t i n g t h e d i s c r e t e t o p o l o g y on if
f
:
S
*
T
i s a mapping between s e t s , and i f
same mapping r e g a r d e d a s a f u n c t i o n from D
DS
S.
Then
D(f) is the
into
DT, t h e n
i s a f u n c t o r from t h e c a t e g o r y of s e t s t o t h e c a t e g o r y o f
t o p o l o g i c a l spaces. (2) V
For any v e c t o r space
V,
let
FV
t o g e t h e r w i t h v e c t o r a d d i t i o n , and l e t
scalar multiplication i n morphisms.
Then
F
V
be t h e p o i n t s o f F "forget" the
r e l a t i v e t o v e c t o r s p a c e homo-
i s a f u n c t o r from t h e c a t e g o r y of v e c t o r
s p a c e s t o t h e c a t e g o r y of A b e l i a n g r o u p s .
It is called a
f o r a e t f u l f u n c t o r b e c a u s e it d i s r e g a r d s p a r t of t h e u n d e r l y i n g structure.
A CATEGORICAL PERSPECTIVE
(3) f : X
Let
Y
+
P E ( f ) of
The
and
X
Y
be
E-completely r e g u l a r s p a c e s w i t h
a c o n t i n u o u s mapping. f
37
Then t h e r e i s a n e x t e n s i o n
such t h a t t h e f o l l o w i n g diagram commutes:
i s f u n c t o r i a l i n t h e s e n s e t h a t i t can
E-compactification
b e used t o d e f i n e a f u n c t o r
pE
from t h e c a t e g o r y o f
p l e t e l y r e g u l a r s p a c e s t o t h e c a t e g o r y of
E-com-
E-compact s p a c e s .
I n a n e n t i r e l y a n a l o g o u s manner t h e I d e n t i f i c a t i o n Theorem o f Mrdwka c a n be used t o d e f i n e a f u n c t o r from t h e c a t e g o r y o f a l l t o p o l o g i c a l s p a c e s t o t h e c a t e g o r y of
E-completely r e g u l a r
These f u n c t o r s w i l l b e t h e p r i n c i p a l o b j e c t s o f i n -
spaces.
vestigation i n t h i s section.
(4)
Let
power s e t of
be any s e t and r e c a l l t h a t
X
Then f o r any mapping
X.
P(X) denotes the
f : X + Y
between two
s e t s d e f i n e t h e mapping P ( f ) t o be t h e f u n c t i o n f - l : P(Y) + P(X). Then P i s a c o n t r a v a r i a n t f u n c t o r from t h e c a t e g o r y of I t i s c a l l e d t h e power s e t f u n c t o r .
sets t o i t s e l f . and
Observe
are topological spaces, then t h e statement
that i f
X
that
i s c o n t i n u o u s i s a s t a t e m e n t a b o u t t h e image o f
f
Y
f
under t h e power s e t f u n c t o r . 5.5
THEOREM.
& functor
or
contravariant functor preserves
isomorphisms.
Proof.
Let
category
(2
f u n c t o r from
and
f : A
+
with
f Og = 1 B
G
B
g : B
t o a category
F ( f ) o F ( g ) = lFB and
+
A
b e morphisms i n a
go f = 1 and l e t A’ @. Then we must have
and
F
be a
F ( g ) 0 F ( f ) = lFA. A s i m i l a r argument
holds f o r a contravariant functor.
C be a c a t e g o r y . An o b j e c t R E Co i s s a i d t o be u n i v e r s a l r e p e l l i n q i f and o n l y i f f o r e a c h o b j e c t 5.6
DEFINITION.
A E Co
Let
t h e r e i s a unique morphism
f
from
R
to
A.
EMBEDDING I N TOPOLOGICAL PRODUCTS
38
The above d e f i n i t i o n i s u s e f u l i n d e s c r i b i n g t h e s i g n i f i cance of t h e
E-compactification
l y r e g u l a r space
&x
let
of a fixed
PEX
discussed i n the preceding section.
X
X;
i . e . , Hausdorff
The morphisms
X.
a r e simply t h e c o n t i n u o u s mappings.
ex
Hence
is a
f u l l s u b c a t e g o r y of t h e c a t e g o r y o f t o p o l o g i c a l s p a c e s . P X
is a u n i v e r s a l r e p e l l i n g o b j e c t i n
any
E - c o m p a c t i f i c a t i o n of
E
h
: X
-t
Y
E-
E-compact s p a c e s
t h a t c o n t a i n a d e n s e homeomorphic copy of
ex
For
d e n o t e t h e c a t e g o r y whose o b j e c t s c o n s i s t of a l l
c o m p a c t i f i c a t i o n s of of
E-complete-
Ex.
For i f
Y
Then
is
X, t h e n t h e homeomorphism
extends uniquely t o
h* : P E X
-t
Y
a s demanded by
V
the definition.
T h e r e f o r e t h e Stone-Cech c o m p a c t i f i c a t i o n and
t h e Hewitt-Nachbin c o m p l e t i o n a r e u n i v e r s a l r e p e l l i n g o b j e c t s i n t h e c a t e g o r y of a l l Hausdorff c o m p a c t i f i c a t i o n s , respect i v e l y c o m p l e t i o n s , of a c o m p l e t e l y r e g u l a r s p a c e
X.
Next w e should l i k e t o c o n s i d e r a p a r t i c u l a r t y p e of f u n c t o r which i n c l u d e s t h e
p,
E-compactification
a s an
example. 5.7
DEFINITION.
category
R
of
A functor
from a c a t e g o r y
F
C
t o a sub-
is a r e f l e c t i v e f u n c t o r i f f o r e v e r y
(2
of C0 t h e r e i s a morphism h A : A -t FA s u c h t h a t e v e r y morphism from A t o a n o b j e c t R of Ro factors u n i q u e l y t h r o u g h FA v i a h A so t h a t t h e f o l l o w i n g diagram commutes :
object
A
A
If
F : C + R
hA
c
i s a r e f l e c t i v e f u n c t o r , t h e subcategory
c a l l e d a r e f l e c t i v e subcatesorv. t h e r e f l e c t i o n of
A
in
R.
The o b j e c t
The symbol
diagram i n d i c a t e s t h a t t h e morphism unique.
FA
g
g!
FA
R
is called
i n t h e above
i s r e q u i r e d t o be
is
A CATEGORICAL PERSPECTIVE
39
L e t u s examine t h e r e s u l t s of t h e p r e c e d i n g s e c t i o n i n c o n n e c t i o n w i t h t h e above d e f i n i t i o n . c a t e g o r y of
t h e subcategory of r e g u l a r space t i o n of
X
CE
Let
denote t h e
E-completely r e g u l a r s p a c e s and l e t E-compact s p a c e s . the
X
RE
in
For each
E-compactification
by Theorem 4 . 3 ( 2 )
.
RE
denote
E-completely
is the reflec-
"Ex
is a
Therefore, B E V
r e f l e c t i v e functor.
I n p a r t i c u l a r , t h e Stone-Cech compacti-
f i c a t i o n and t h e Hewitt-Nachbin c o m p l e t i o n can be viewed a s images o f o b j e c t s from t h e c a t e g o r y of c o m p l e t e l y r e g u l a r s p a c e s under r e f l e c t i v e f u n c t o r s .
Hence, t h e compact and
Hewitt-Nachbin s p a c e s form r e f l e c t i v e s u b c a t e g o r i e s of t h e c a t e g o r y of c o m p l e t e l y r e g u l a r s p a c e s .
A c t u a l l y we s h a l l s e e
t h a t we can s a y more. The f o l l o w i n g r e s u l t h a s a l r e a d y b e e n v e r i f i e d i n t h e case of 5.8
BE
and t h e c a t e g o r y o f
The r e f l e c t i o n
THEOREM.
E-compact s p a c e s . u n i q u e up
of an o b j e c t
isomorphism. Proof.
63
Let
b e t h e r e f l e c t i o n of a n o b j e c t R
and l e t
from f.
A
@. L e t
be a r e f l e c t i v e s u b c a t e g o r y of
f : A
+
A.
Let
B
FA
b e a n y o b j e c t of
b e a morphism s u c h t h a t a n y morphism
B
t o a n o b j e c t of
R
f a c t o r s uniquely through
B
via
Then w e have t h e f o l l o w i n g commutative diagram i n which
t h e morphisms
h
and
t i v e p r o p e r t i e s of
e x i s t and a r e unique b y t h e r e f l e c -
g and
B
FA, r e s p e c t i v e l y :
n
A
FA
.t
nA
Because
FA
belongs t o
u n i q u e l y through diagram y i e l d s
hog
t h e morphism
as another.
changing t h e r o l e s o f and o b t a i n
R0
hA
must f a c t o r
One s u c h f a c t o r i z a t i o n i s
hA.
g o h = lB.
FA
and
B
Hence, FA
lFA and t h e
h O g = 1FA.
Hence
Inter-
we c a n r e p e a t t h e argument and
B
a r e isomorphic con-
40
EMBEDDING I N TOPOLOGICAL PRODUCTS
cluding t h e proof. Since f o r every o b j e c t f a c t o r through t h e i d e n t i t y
in
R
lR
R
any morphism w i l l
w e immediately o b t a i n t h e
next result.
Any o b j e c t i n a r e f l e c t i v e s u b c a t e q o r y isomorphic t o i t s r e f l e c t i o n .
5.9
COROLLARY.
A r e f l e c t i v e functor F i f t h e morphism hA : A + FA morphism f o r each o b j e c t A.
is s a i d t o be a - r e f l e c t i v e of D e f i n i t i o n 5 . 7 i s a n e p i -
I n t h e p r e c e d i n g s e c t i o n it was observed t h a t t h e f a c t o r i z a t i o n o f a mapping of a s p a c e X t h r o u g h t h e mapping nx : X + BEX i s unique b e c a u s e o f t h e d e n s i t y of hx(X) i n pEX and t h e Hausdorff p r o p e r t y . In t h i s c o n t e x t w e s e e t h a t mappings h a v i n g d e n s e images a r e
pE a n e p i - r e f l e c t i v e f u n c t o r and t h e c a t e q o r y of E-compact spaces i s a n e p i - r e f l e c t i v e s u b c a t e q o r y o f t h e c a t e q o r y of morphisms i n t h e c a t e g o r y o f Hausdorff s p a c e s .
Hence,
E-
completely r e q u l a r s p a c e s . There a r e many a d d i t i o n a l c a t e g o r i c a l r e s u l t s t h a t a r e important t o t h e study of t o p o l o g i c a l spaces.
For i n s t a n c e i t
c a n be shown t h a t t h e monomorphisms i n t h e c a t e g o r y o f Hausd o r f f s p a c e s a r e the i n j e c t i v e mappings and t h e epimorphisms a r e t h e mappings w i t h dense r a n g e .
I t can a l s o be shown t h a t
t h e e p i - r e f l e c t i v e s u b c a t e g o r i e s o f t h e c a t e g o r y o f Hausdorff s p a c e s are c l o s e d h e r e d i t a r y .
I n f a c t , t h e e p i - r e f l e c t i v e sub-
c a t e g o r i e s o f t h a t c a t e g o r y c a n be c h a r a c t e r i z e d a s t h e prod u c t i v e and c l o s e d h e r e d i t a r y s u b c a t e g o r i e s . R e f l e c t i o n s i n t o p o l o g y a r e d i s c u s s e d i n d e t a i l i n t h e 1968 L e c t u r e N o t e s of H.
Herrlich. T h i s c o n c l u d e s our b r i e f e x c u r s i o n i n c a t e g o r i c a l t o p o l o -
gy. I n the next chapter we w i l l begin a d e t a i l e d investigation of t h e p r o p e r t i e s and c h a r a c t e r i s t i c s of Hewitt-Nachbin spaces.
Chapter 2 H E W I T T - N A C H B I N SPACES AND CONVERGENCE
I n t h e p r e v i o u s c h a p t e r w e observed t h a t t h e n o t i o n of a Hewitt-Nachbin s p a c e a r o s e n a t u r a l l y i n c o n n e c t i o n w i t h emb e d d i n g s i n t o p r o d u c t s of t o p o l o g i c a l s p a c c s . However t h a t p o i n t of view d o e s n o t f a c i l i t a t e t h e " i n t e r n a l " s t u d y o f Hewitt-Nachbin s p a c e s ( f o r example, i d e a s l i k e n o r m a l i t y , p a r a compactness, and s o f o r t h ) . T h e r e f o r e w e should l i k e t o c h a r a c t e r i z e t h e p r o p e r t y o f Hewitt-Nachbin c o m p l e t e n e s s i n t e r m s o f convergence f o r c e r t a i n c l a s s e s of f i l t e r s d e f i n e d on t h e
given space. There a r e s e v e r a l p r i n c i p a l o b j e c t i v e s of t h i s c h a p t e r : F i r s t w e want t o t r a n s l a t e t h e embedding d e f i n i t i o n o f H e w i t t -
Nachbin c o m p l e t e n e s s i n t o convergence o f f i l t e r s , and t h a t w i l l r e q u i r e a s u r v e y of t h e s a l i e n t f e a t u r e s of an a p p r o p r i a t e f i l t e r theory. That p o i n t o f view w i l l f a c i l i t a t e t h e d i s c o v e r y of many a d d i t i o n a l c h a r a c t e r i z a t i o n s of H e w i t t Nachbin s p a c e s .
Second w e want t o d e v e l o p t h e Hewitt-Nachbin
uX i n t h e c o n t e x t of t h a t f i l t e r t h e o r y and s t u d y i n d e t a i l t h e p r o p e r t i e s and c h a r a c t e r i s t i c s of v X . Finally w e want t o s t u d y t h e g e n e r a l t o p o l o g i c a l p r o p e r t i e s o f H e w i t t Nachbin s p a c e s . For example, w e a l r e a d y know from Theorem 4 . 2 t h a t t h e t o p o l o g i c a l p r o d u c t of Hewitt-Nachbin s p a c e s i s a Hewitt-Nachbin s p a c e , t h a t any c l o s e d subspace o f a H e w i t t Nachbin s p a c e i s a Hewitt-Nachbin s p a c e , and t h a t a r b i t r a r y i n t e r s e c t i o n s o f Hewitt-Nachbin s p a c e s a r e a g a i n Hewitt-Nachbin s p a c e s . However, a s w e s h a l l see i n t h e development, a g r e a t d e a l more can b e s a i d . A l s o , an i m p o r t a n t f a c e t o f H e w i t t Nachbin s p a c e s a r i s e s i n c o n n e c t i o n w i t h t h e i n t e r p l a y between t h e a l g e b r a i c s t r u c t u r e of t h e r i n g C ( X ) and t h e t o p o l o g i c a l s t r u c t u r e of t h e s p a c e X. The i n v e s t i g a t i o n of t h a t i n t e r p l a y w i l l b e i n i t i a t e d i n t h i s c h a p t e r and t h e n b r o u g h t i n t o f u l l completion
view i n Chapter 3 . Section 6 :
& F i l t e r s and Converqence
The t h e o r y o f f i l t e r s p l a y s a c e n t r a l r o l e i n t h e i n v e s t i g a t i o n o f Hewitt-Nachbin s p a c e s .
Our approach t o t h e H e w i t t -
HEWITT-NACHBIN SPACES AND CONVERGENCE
42
Nachbin completion w i l l follow an approach s i m i l a r t o t h a t i n Gillman and J e r i s o n , b u t i t i s i n t e r e s t i n g t h a t t h i s approach f i t s e a s i l y i n t o a more g e n e r a l s e t t i n g .
Moreover, t h i s gen-
e r a l s e t t i n g provides c o n s i d e r a b l e i n s i g h t i n t o t h e n a t u r e of t h i s p a r t i c u l a r completion a s w e l l a s i n t o t h e n a t u r e of the V
a s s o c i a t e d Stone-Cech c o m p a c t i f i c a t i o n . There a r e s e v e r a l good r e f e r e n c e s f o r t h e u s u a l concept of a f i l t e r such a s t h e 1966 English v e r s i o n of N . Bourbaki and t h e 1970 t e x t by S . W i l l a r d .
The Gillman and J e r i s o n t e x t
t r e a t s the p a r t i c u l a r theory of z e r o - s e t f i l t e r s .
We need t o
make some s l i g h t m o d i f i c a t i o n s t o these t h e o r i e s .
The concept
of a f i l t e r on a non-empty s e t
s u b s e t s of
i s simply a c o l l e c t i o n of
X
I t i s not i n t r i n s i c a l l y r e l a t e d t o any topo-
X.
l o g i c a l s t r u c t u r e u n t i l we i n v e s t i g a t e t h e notion of convergence f o r these f i l t e r s .
However, i n o r d e r t h a t the i n v e s t i -
g a t i o n come t o f r u i t i a n i t i s important t h a t the s u b s e t s belonging t o t h e c o l l e c t i o n s a t i s f y a p p r o p r i a t e axioms.
Our
development i s based on the 1967 n o t e s of R . Alo. A good s u r vey of t h i s a r e a i s a l s o provided i n the 1970 t h e s i s by F . Lynn. Let
X
8
be a non-empty s e t , and l e t
be a non-empty
c o l l e c t i o n of sets i n P ( X ) . W e w i l l say t h a t 9 i s a r i n g i n c a s e i t i s closed under the formation of f i n i t e
-of s e t s
unions and f i n i t e i n t e r s e c t i o n s . If
X
i s a t o p o l o g i c a l space, then the c o l l e c t i o n Z ( X )
of a l l z e r o - s e t s i n
X
i s an example of a r i n g of s e t s , a s i s
t h a c o l l e c t i o n of a l l closed s u b s e t s of a t o p o l o g i c a l space. P ( X ) i s a r i n g of s e t s .
T r i v i a l l y t h e power s e t For a r b i t r a r y
n
(2 :
to
8
8
C
S(X), l e t
~ € 8 ) .observe t h a t even i f 9 happens
fl
n8
denote t h e s e t
8 does not n e c e s s a r i l y belong
t o be a r i n g of sets.
The follow-
ing d e f i n i t i o n i s fundamental. 6.1
DEFINITION.
If
X
i s a non-empty s e t and i f
8
c P(X)
i s closed under f i n i t e i n t e r s e c t i o n s , then a c o l l e c t i o n
elements i n
8
i s s a i d t o be a & f i l t e r on following c o n d i t i o n s a r e s a t i s f i e d : (1) I f
A E ~and i f
BE^
satisfies
X
5
of
i n case the
A c B,
then
BE^.
3- F I L T E R S and
belong t o
5, then
(2)
If
(3)
The empty s e t does not belong t o
If
8
A
B
43
AND CONVERGENCE
i s t h e power s e t
A
5
n
B E 3.
and
5 # @.
P ( X ) w e w i l l say, t h a t t h e
8-
f i l t e r i s a Bourbaki f i l t e r (sometimes simply r e f e r r e d t o a s a f i l t e r , see Bourbaki). z e r o - s e t s on
If
8
i s the c o l l e c t i o n
we w i l l say t h a t the
X
Z ( X ) of a l l
> - f i l t e r is a
2-filter
o r z e r o - s e t f i l t e r ( s e e Gillman and J e r i s o n ) . I t i s immediate from the d e f i n i t i o n t h a t an a r b i t r a r y
i n t e r s e c t i o n of Now l e t X.
2 - f i l t e r s on
X
i s again a
& f i l t e r when-
xc8.
ever
Then
over, i f
-8
denote t h e c o l l e c t i o n of a l l
:(X)
3(X)
+filters
i s p a r t i a l l y ordered by s e t i n c l u s i o n .
on
More-
i s c l o s e d under the formation of f i n i t e i n t e r -
E(X), then U @ i s a $-filter L in E(x) i s a 8 - u l t r a f i l t e r i n c a s e i t i s maximal i n 3-(X); i n o t h e r words, i f 3 i s a & f i l t e r with I r C 3, then L = 5. Thus from our preceding remarks i t i s a consequence of Z o r n ’ s Lemma t h a t if 2 i s closed under f i n i t e i n t e r s e c t i o n s , then f o r each 8 - f i l t e r 5 t h e r e e x i s t s a & u l t r a f i l t e r L s a t i s f y i n q 5 c 11. Observe t h a t i t i s not p o s s i b l e t o d i s c u s s 3 - f i l t e r s i f s e c t i o n s and i f
member of
@
i s a chain i n
I t i s s a i d t h a t a proper
z(X).
-8
f a i l s t o be c l o s e d under f i n i t e i n t e r s e c t i o n s because of
the c o n d i t i o n ( 2 ) i n 6 . 1 .
I n the s t a t e m e n t s of many of the
theorems t h a t a r e t o follow i t w i l l u s u a l l y be r e q u i r e d t h a t the c o l l e c t i o n
8
a l s o be closed under f i n i t e unions.
t h e concern h e r e i s not with the most g e n e r a l theory of
Since
8-
f i l t e r s p o s s i b l e , b u t r a t h e r f o r a theory t h a t i s s u i t a b l e t o
i t w i l l henceforth & assumed that -the distinguished c o l l e c t i o n 8 ------i s always a r i n s o f s e t s unour purposes i n l a t e r work,
-
less e x p r e s s l y s t a t e d o t h e r w i s e . A s u b c o l l e c t i o n $3 of 8 i s s a i d t o be a base f o r a 9- f i l t e r i n c a s e the following c o n d i t i o n s hold: (1) I f B1 and B2 belong t o $3, then t h e r e e x i s t s B E @ s a t i s f y i n g B c B~ n B ~ . ( 2 ) The c o l l e c t i o n b i s non-empty and t h e empty s e t does n o t belong t o 93.
44
HEWITT-NACHBIN
If
i s a base f o r a
b
3 = { A C ~: B erated
a
C
f o r some
A
3;
If
b.
3
i s a base f o r
Bcb
SPACES AND CONVERGENCE
satisfying
& f i l t e r , then t h e c o l l e c t i o n BE@) i s called
is a
8 - f i l t e r and i f
i s case f o r each B
C
the
Fc3
& f i l t e r qen-
b c 3, then
F.
G
I t i s easy t o show from t h e d e f i n i t i o n s t h a t i f
s u b c o l l e c t i o n of
8,
then t h e r e i s a
G
I n t h a t case
Q
has non-empty
3; i s s a i d t o be f i x e d i n c a s e i s s a i d t o be free.
8-filter
A
3
For example, i f the collection ( B
E
A
b(X)
i s a non-empty s u b s e t of : A
G
has the f i n i t e i n t e r s e c t i o n
property . otherwise
is a
& f i l t e r containing
i f and only i f every f i n i t e s u b c o l l e c t i o n of
intersection.
93
t h e r e e x i s t s some
C
n
3
#
a;
then
X,
B ) i s a fixed Bourbaki f i l t e r
on X. o n t h e o t h e r hand, i f X i s an i n f i n i t e s e t , then t h e c o l l e c t i o n ( A E f f ( X ) : X b i s f i n i t e ) i s a f r e e f i l t e r on X . Recall t h a t a space
X
i s compact i f and only i f every
family of closed s e t s w i t h t h e f i n i t e i n t e r s e c t i o n p r o p e r t y has a non-empty i n t e r s e c t i o n .
U t i l i z i n g t h e f a c t coupled with
t h e r e s u l t s s t a t e d i n 3.6 i t i s an easy e x e r c i s e t o show t h a t
-a Tychonoff space X f i l t e r on X i s f i x e d . --
compact i f and only i f every I n fact, X
Z-ultra-
compact i f and only i f
i s fixed. This observation i s h e l p f u l i n d e s c r i b i n g the F r i n k - tvpe o r Wallman-type c o m p a c t i f i c a t i o n
every
Z - f i l t e r on
X
of an a r b i t r a r y Tychonoff space.
B r i e f l y , t h e idea i s t o con-
s i d e r the c o l l e c t i o n w ( 8 ) of a l l 8 - u l t r a f i l t e r s on X , where 8 has c e r t a i n " s u i t a b l e p r o p e r t i e s " (which w i l l be d e f i n e d l a t e r i n t h i s s e c t i o n a s the "normal base" p r o p e r t i e s ) .
Then
w ( 8 ) can be made i n t o a t o p o l o g i c a l space i n such a way t h a t t h e fixed
2 - u l t r a f i l t e r s a r e n a t u r a l l y i d e n t i f i e d with the
p o i n t s of
X.
Moreover, i t can be shown t h a t
~ ( 2 i)s
a
compact Hausdorff space c o n t a i n i n g a dense homeomorphic copy of
X.
on
X
Intuitively,
t h e idea i s t o " f i x " t h e f r e e
and o b t a i n a r e s u l t i n g c o m p a c t i f i c a t i o n of
&filters X
i n the
process.
This procedure w i l l be d i s c u s s e d a t l e n g t h i n Sec-
tion 9.
We introduce i t a t t h i s time because i t provides t h e
motivation f o r many of t h e d e f i n i t i o n s and r e l a t e d r e s u l t s t h a t a r e t o follow i n t h i s s e c t i o n .
8- FILTERS 6.2
Let
DEFINITION.
+ f i l t e r base on
be a t o p o l o g i c a l space, l e t
X
and l e t
X,
45
AND CONVERGENCE
pcx.
I t i s said t h a t
94 be a p is a
i n c a s e f o r each neighborhood N(p) of p t h e r e e x i s t s a B E @ s a t i s f y i n g B C N(p) . I t i s s a i d t h a t p i s a c l u s t e r p o i n t of 94 i n c a s e f o r each neighborhood N(p) l i m i t p o i n t of
of
p,
94
n
N(p)
B
# @
Observe t h a t
94
p o i n t ) of
for a l l p
BE@.
i s a l i m i t point (respectively, c l u s t e r
i f and only i f
l y , c l u s t e r p o i n t ) of t h e
p
is a l i m i t point (respective-
8 - f i l t e r generated by
Moreover,
94.
5 i s a & f i l t e r and i f 94 c 3 i s a base f o r 3 , then p i s a l i m i t p o i n t ( r e s p e c t i v e l y , c l u s t e r p o i n t ) of 3 i f and only i f p i s a l i m i t p o i n t ( r e s p e c t i v e l y , c l u s t e r p o i n t ) if
of
94. I t i s easy t o show t h a t whenever
3 c Q, then
f i l t e r s satisfying only i f of
G
p
i s a l i m i t p o i n t of
only i f
p
0
are
3-
i s a l i m i t p o i n t of
Q; and
3
is a cluster point
p
5.
i s a c l u s t e r p o i n t of
p
and
3
Informally, w e
say t h a t " l i m i t p o i n t s go up" whereas " c l u s t e r p o i n t s go down." 3 - f i l t e r has a c l u s t e r point.
Note a l s o t h a t every fixed
8 - f i l t e r s o n a topologi-
I n studying t h e convergence of
c a l space First,
s e v e r a l important q u e s t i o n s p r e s e n t themselves.
X,
i s it the case t h a t
8 - f i l t e r s have unique l i m i t s when-
ever a l i m i t point e x i s t s ? point i n p?
X,
Second, i f
does t h e r e e x i s t a
& f i l t e r t h a t converges t o
Finally, i s it possible for d i s t i n c t
have a common c l u s t e r p o i n t ?
i s an a r b i t r a r y
p
8-ultrafilters to
A l l of t h e s e q u e s t i o n s a r e some-
t i m e s answered i n t h e a f f i r m a t i v e .
However, t h e following pre-
s e n t a t i o n w i l l provide p r e c i s e answers t o t h e s e q u e s t i o n s .
We
w i l l a l s o g i v e some examples of t h e p a t h o l o g i c a l c a s e s . 6.3
Let
DEFINITION.
8
collection
C
be a non-empty s e t .
k ( X ) i s s a i d t o be
each
Z E ~ and
x,&
xcZ1
and
n
Z1 =
Z
X
(X\C1)
n
C1,C2
(x\C2)
Z1
E
8
= @. I f
with
x
Z1,Z2 Z1
E
E
8
The c o l l e c t i o n
normal i n case f o r each p a i r there e x i s t
& d i s j u n c t i v e i n case f o r
there e x i s t s a
a.
A non-empty
8
c X\cl,
8
satisfying i s s a i d t o be with Z1 n Z 2 = @ Z 2 c X\C2
and
i s a t o p o l o g i c a l space, then
8
HEWITT-NACHBIN SPACES AND CONVERGENCE
46
i s s a i d t o be d i s j u n c t i v e i n case f o r each closed s e t and
xifF
n
Z1 =
F
base i n of
there exists a
a.
Finally,
E
8
satisfying
8
the c o l l e c t i o n
c a s e f o r each p o i n t there exists a
p
Z1
PEX
C
X
and
i s s a i d t o be a l o c a l
and each neighborhood
satisfying
Zc8
xtZl
F
p
E
(int
Z) C
N(p)
Z c N(p).
Note t h a t t h e above d e f i n i t i o n i s meaningful even i f the c o l l e c t i o n 8 f a i l s t o be a r i n g of s e t s . Also observe t h a t t h e p r o p e r t i e s of
9 being d i s j u n c t i v e , normal, o r a l o c a l
base a r e analogous t o those of complete r e g u l a r i t y , normality, and a base f o r a t o p o l o g i c a l space, r e s p e c t i v e l y , except t h a t the collection topology
8
may possess s e t s t h a t f a i l t o belong t o t h e
7.
The following remarks a r e e a s y t o v e r i f y . 6.4
REMARKS.
if -
8
(1) I f
closed s u b s e t s of
X,
i s a d i s j u n c t i v e c o l l e c t i o n of
8
then
The c o l l e c t i o n
&disjunctive.
Moreover,
8
9 i s a l o c a l b a s e , then (2)
is
disjunctive. S(X) p o s s e s s e s a l l of t h e proper-
t i e s of being d i s j u n c t i v e , normal, and a l o c a l b a s e . ( 3 ) I f X i s a Tychonoff space, then t h e c o l l e c t i o n Z ( X ) of a l l z e r o - s e t s i s a l o c a l b a s e by 3 . 6 ( 3 ) and a l s o nor-
mal.
Moreover, i f
closed s u b s e t s i n Now, i f
8
i s normal, then the c o l l e c t i o n of a l l
X
X
-
i s normal.
i s a l o c a l base on an a r b i t r a r y t o p o l o g i c a l
X, then f o r each ptX t h e c o l l e c t i o n Ir ( p ) = ( i s a neighborhood of p ] i s a 8 - f i l t e r on X t h a t v e r g e s t o p. I t i s c a l l e d 8-neighborhood f i l t e r a t e d with the p o i n t p. The following r e s u l t s t r e s s e s
space
Z
---
Z E :~ conassocithe
importance of t h e concept of a l o c a l b a s e . 6.5
8
THEOREM. J &
c b(X)
let
pcx.
X
5l o c a l base
3
-3
The
if
2
8 - f i l t e r on
X,
and
followinq s t a t e m e n t s a r e t r u e :
(1) The p o i n t (2)
be an a r b i t r a r y t o p o l o q i c a l space, l e t
p
b ( p ) c 3. The p o i n t p
i s a l i m i t p o i n t of
is 2
c l u s t e r p o i n t of
3
i f and only
3
i f and only
9- F I L T E R S
a
i f there ---
3‘
exists 8-filter and s a t i s f y i n q 5 c a t .
-
If
(3)
Lc
is g
of
p.
satisfying
p
al. (2)
is
p
Let
3
C
E
b(p)
int
Z C 2
5
n z
c N.
F C ~ and
Since
Ir
p
is a
in
and l e t
X.
N
be any neigh-
p.
3
z
E
Z
Z E U(p)
E
~
so
The converse i s t r i v i and c o n s i d e r
b(p)).
B =
The c o l l e c t i o n
B
3 ’ . I t follows
&filter
a t . Also, by (1) above the p o i n t p 3 ’ . conversely, l e t 5 c a r and l e t p 3 ’ . Then p i s . a c l u s t e r p o i n t of 5 ’ a.
and hence a c l u s t e r p o i n t of there e x i s t s a
if 8-
b(p) c
and
Suppose
is a l i m i t point
p
I t follows t h a t
converges t o
for
i s a l i m i t p o i n t of be a l i m i t p o i n t of (3)
5
C
be a c l u s t e r p o i n t of
3l
p
is a cluster
Hausdorff i f and o n l y
i s a f i l t e r base and g e n e r a t e s a
that
-if
p
By the l o c a l base p r o p e r t y t h e r e i s a
Z E ~ . Hence
{BE8 : B = F
then
unique l i m i t points
(1) suppose t h a t
borhood of that
X
have
filters
Proof.
i f and only
converginq t o
LA.
The space
(4)
3-ultrafilter,
11
p o i n t of
47
AND CONVERGENCE
I&. Then by ( 2 ) above & f i l t e r 3 ’ converging t o p with Ir c $ I 8-ultrafilter, Ir = 5 ’ . The converse i s t r i v i s a c l u s t e r p o i n t of
.
ial.
x and y i n X s a t i s f y i n g N ( x ) fi N ( y ) # f o r all neighborhoods of x and y. I t follows t h a t x i s a c l u s t e r p o i n t of t h e a n e i g h b o r hood f i l t e r b ( y ) . Hence by statement (2) above t h e r e i s a (4)
Suppose t h e r e a r e two d i s t i n c t p o i n t s
&filter
3
converging t o
x
statement (1) above t h e p o i n t
5. However, from i s also a l i m i t of 3 so
with y
k(y)
C
3 - f i l t e r s cannot have unique l i m i t p o i n t s .
that
The proof o f
t h e converse i s t r i v i a l and does not r e l y on the l o c a l b a s e p r o p e r t y completing the p r o o f . Observe t h a t f o r a Hausdorff space converges t o a p o i n t of
3.
pcx, then
Otherwise some
p
X,
if a
&filter
3
i s t h e only c l u s t e r p o i n t
B f i l t e r which c o n t a i n s
3
would have
two d i s t i n c t l i m i t p o i n t s . Answers t o t h e f i r s t two q u e s t i o n s posed e a r l i e r have
HEW1 TT- NACHBIN SPACES AND CONVERGENCE
48
The remaining q u e s t i o n a s t o whether
now been e s t a b l i s h e d .
d i s t i n c t & u l t r a f i l t e r s can have a common c l u s t e r p o i n t ( i . e . ,
fj
l i m i t point i f
i s a l o c a l b a s e ) may be answered a f f i r m a -
t i v e l y a s t h e f o l l o w i n g examples i l l u s t r a t e . x
Let
U ( m i denote t h e one-point compactifica-
= N
W
*
IN, and l e t i n t e g e r s and l e t m
fj = P ( N ) .
t i o n of
y l = [ A F ~: ( N b) fl E i s f i n i t e ] , and ( N \B) I1 ID is f i n i t e ] . I t i s easy t o v e r i f y
a2
and
S1
D e f i n e t h e follow-
*
= (BE8 :
that
d e n o t e t h e even
E
*
ing collections:
z2
Let
d e n o t e t h e odd o n e s .
b o t h c o n t a i n t h e neighborhood f i l t e r 6.5(1) t h a t
m
*
IN , and t h a t
a r e Bourbaki u l t r a f i l t e r s on ~ ( C O ) .
I t f o l l o w s from
i s a l i m i t p o i n t , and hence a c l u s t e r p o i n t ,
of each of Sl and example. I n f a c t
z2.
Note t h a t 8 i s a l o c a l b a s e i n t h i s P ( X ) i s a l o c a l b a s e f o r any X. IR
A s a n o t h e r example, c o n s i d e r t h e r e a l numbers
8
t h e i n d i s c r e t e topology and l e t a r b i t r a r y r e a l number. hence h a s
y
Then f o r e a c h r e a l number
Sx
Bourbaki u l t r a f i l t e r
Let
= P(IR).
: xcA) c o n v e r g e s t o
= (A
y
under
be an
x
the
y
and
as a cluster point.
The n e x t r e s u l t r e v e a l s t h a t t h e u s e of $ f i l t e r s f o r a s u i t a b l e c o l l e c t i o n 8 e l i m i n a t e s t h e above s i t u a t i o n and stresses t h e importance of t h e c o n c e p t of a d i s j u n c t i v e collection. 6.6
Let
THEOREM.
-a d i s j u n c t i v e each p o i n t
pcX
ultrafilter &filter
3
2 t o p o l o q i c a l s p a c e , and l e t
X
c o l l e c t i o n of c l o s e d subsets
on
the X.
5
collection
Furthermore,
i f and o n l y
if
3
C
P p
3
of
X.
= [ Z c g : ~ E Z )i s a
&a
8 &
Then f o r
8-
c l u s t e r p o i n t of a
Therefore, d i s t i n c t
P' u l t r a f i l t e r s c a n n o t have a common c l u s t e r p o i n t .
8-
5 i s a 3 - f i l t e r i s an immediate conP sequence of t h e d i s j u n c t i v e p r o p e r t y . Next suppose t h a t G
Proof.
is a
The f a c t t h a t
I f Bcq\zp, t h e n p{B. Ztg such t h a t p c Z and Z fl B = gi. Thus Z E 3 and hence i n G. T h i s c o n t r a d i c t s t h e f i n i t e i n t e r P s e c t i o n p r o p e r t y of q . H e n c e G = 3 P' Note t r i v i a l l y t h a t p i s a c l u s t e r p o i n t o f 8 To P' prove t h e second s t a t e m e n t , l e t p be a c l u s t e r p o i n t o f 3 8 - f i l t e r which c o n t a i n s
Hence there is a
8- FILTERS and suppose t h e r e i s an that
SO
X\F
Therefore, 8
5.
C
.
3
5
i s a c l u s t e r p o i n t of
P'
p.
But
p
sp. F
n
Then (X\F)
pj!F =
fl
i s a c l u s t e r point
if 3 C 3 then p P i s a c l u s t e r p o i n t of
Conversely, because
is a
Finally, i f 11 aP then c l e a r l y k. = 3
F j!
such t h a t
FEZ
i s a neighborhood of
which c o n t r a d i c t s t h e h y p o t h e s i s t h a t of
49
AND CONVERGENCE
p
8 - u l t r a f i l t e r with c l u s t e r p o i n t
p
This completes t h e p r o o f .
P'
S i n c e every l o c a l base i s d i s j u n c t i v e w e o b t a i n t h e following c o r o l l a r y . 6.7
If
COROLLARY.
if
8
X,
then
i s an a r b i t r a r y t o p o l o q i c a l space and
X
c P ( X ) i s a l o c a l base c o n s i s t i n g of c l o s e d s u b s e t s
point
ap is t h e unique
of
& u l t r a f i l t e r converqinq t o t h e
p.
I n t h e p r e v i o u s d i s c u s s i o n of t h e Wallman-Frink compactiX, i t was s t a t e d t h a t t h e
f i c a t i o n of a Tychonoff space p o i n t s of
X
would be n a t u r a l l y i d e n t i f i e d w i t h c e r t a i n
~ ( 8 ) .I n
u l t r a f i l t e r s of that
ptX
8 is
the case t h a t
and a b a s e f o r t h e c l o s e d s e t s i n
disjunctive
i t now becomes c l e a r
X,
i s t o be i d e n t i f i e d with t h e
However, t h e c o n d i t i o n t h a t
8-
8-ultrafilter
8
P' be a normal c o l l e c t i o n i s
8
needed i n o r d e r t h a t t h e c o m p a c t i f i c a t i o n
~ ( 8be )
Hausdorff.
Moreover, t h e f o l l o w i n g r e s u l t d e m o n s t r a t e s how normal c o l l e c t i o n s p r o v i d e u s e f u l c h a r a c t e r i z a t i o n s of 6.8
8C
THEOREM.
-
is 2
8 - f i l t e r on
X,
8-ultrafilters.
i s a normal c o l l e c t i o n and i f
P(X)
5
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 equiv-
alent:
(1) The f i l t e r (2)
For each that -
(3)
Zt5
For each
z1 E 8
3
# fl
Z E ~ e,i t h e r
Z&?
.
satisfyinq
f i l t e r @ , Z ) g e n e r a t e d by that
a
is a
follows t h a t
8-ultrafilter.
n
F
(1) i m p l i e s ( 2 ) :
Proof.
is a
Z E ~ , Z
and
F
Z Z
implies
z1 E 3. # @
f o r each
contains
9 - u l t r a f i l t e r implies t h a t Z E ~ .
FEZ
or there e x i s t s a
z1 c X\Z
Since
3
for a l l
3 =
FEZ, the
3. The f a c t ($,Z).
It
50
HEWITT-NACHBIN
( 2 ) implies ( 3 ) :
SPACES AND CONVERGENCE
Z E ~ and ZL3 . Then b y ( 2 ) t h e r e F E ~such t h a t Z n F = @. S i n c e 8 i s
Let
e x i s t s an element
normal, t h e r e e x i s t e l e m e n t s
z
C
C1
and
C2
in
8
such t h a t
F C X\C2 and ( X \ C l ) n ( X \ C 2 ) = @ . Then from which i t f o l l o w s t h a t C1 t 31 and c1 c
x\Cl,
c C1
F c X\C2
x\Z.
( 3 ) i m p l i e s (1): Suppose t h a t 31' is a + f i l t e r with 5 c 3'. I f 2 ~ 3 ' and Z i a , t h e n by ( 3 ) t h e r e e x i s t s a Z1 F 8 such t h a t Z1 c X V and Z1 t 3. I t f o l l o w s t h a t Z1 F 3' and Z1 ri Z = This c o n t r a d i c t s t h e f i l t e r propert y of 3'. H e n c e , 3 i s a 3 - u l t r a f i l t e r .
a.
T h i s completes t h e d i s c u s s i o n c o n c e r n i n g s p e c i f i c prop e r t i e s of t h e c o l l e c t i o n
8.
Next w e t u r n o u r a t t e n t i o n t o
a d i s c u s s i o n of c e r t a i n k i n d s of
& f i l t e r s o f which t h e
8-
u l t r a f i l t e r s are a special case, I n h i s 1969 p a p e r , S . Ciampa i n t r o d u c e s t h e n o t i o n o f a
maximal
net which t u r n s o u t t o b e u s e f u l i n t h e s t u d y o f con-
v e r g e n c e a s r e l a t e d t o compactness and c o m p l e t e n e s s i n uniform spaces.
W e w i l l d e f i n e t h e a n a l o g u e of t h e "maximal n e t " con-
c e p t i n t h e c a s e of
9 - f i l t e r s on
X.
However w e w i l l avoid
t h e word "maximaltg i n o u r d e f i n i t i o n i n o r d e r t o p r e v e n t con-
fusion since a
8 - u l t r a f i l t e r i s maximal i n t h e c l a s s
z(x)
with respect t o t h e p a r t i a l ordering of set inclusion.
3 on a t o p o l o g i c a l s p a c e X i s s a i d t o b e c l u s t e r a b l e i n c a s e 3 converges t o each of i t s
6.9
DEFINITION.
3-filter
A
c l u s t e r p o i n t s i n X. I n the case t h a t 6 . 5 ( 3 ) t h a t every
3
and
Q
are
8
i s a l o c a l b a s e i t f o l l o w s from
8-ultrafilter is clusterable.
&filters
Moreover,
with 3 c G, then 3 is
if
clusterable
o n l y i f G is c l u s t e r a b l e . The p r o o f o f t h e l a s t s t a t e m e n t i s immediate from o u r e a r l i e r o b s e r v a t i o n t h a t " c l u s t e r p o i n t s g o down" and " l i m i t p o i n t s g o up." t h a t e v e r y Cauchy
Finally,
& f i l t e r i n a uniform s p a c e c o n v e r g e s t o
i t s c l u s t e r p o i n t s , whence e v e r y Cauchy
able. filter
i t i s w e l l known
S i n c e whenever
8 is
8 - f i l t e r is cluster-
a l o c a l base the
b ( p ) i s Cauchy, i t f o l l o w s t h a t
&neighborhood
b ( p ) is c l u s t e r a b l e .
However, i n g e n e r a l b ( p ) i s n o t a 8 - u l t r a f i l t e r . I n particu l a r , b ( p ) i s n o t a z e r o - s e t u l t r a f i l t e r when 8 i s
8- F I L T E R S the c o l l e c t i o n
Z(X)
51
AND CONVERGENCE
.
The f o l l o w i n g p r o v i d e s a n o t h e r i m p o r t a n t example of
3- f i l t e r s .
clusterable 6.10
DEFINITION,
and
A
6.11
8
in
B
A U B E
with
8
Since
REMARK.
from 6 . 8 ,
9-filter
A
3
i s s a i d t o be prime i n c a s e
3
implies
BE^.
A E ~o r
i s a r i n g of sets, i t i s e a s y t o show
(1) i m p l i e s ( Z ) , t h a t every
prime.
3-ultrafilter
The f o l l o w i n g r e s u l t w i l l be u s e f u l throughout t h i s book.
-5 If
Let
space and l e t
9
l o c a l base t h a t i s a l s o a base f o r t h e c l o s e d s e t s of
X.
6.12 be
THEOREM.
i s a p o i n t of
-then t h e
The
(2)
proof.
3
p
and l e t
G
(int Z)
8,
9 - f i l t e r on
prime
to
Suppose t h a t
p
t h e r e e x i s t s a set
X,
5.
c l u s t e r p o i n t of p.
i s a cluster point
be an open s e t c o n t a i n i n g
c G.
C Z
&a
converqes
5
(3) n 3 = (PI. (1) i m p l i e s ( 2 ) :
$
are e q u i v a l e n t :
is 2
&filter
base p r o p e r t y of E
and i f
X
following statements
(1) The p o i n t
p
T1-topoloqical
------------
p
of
be a
X
p.
By t h e l o c a l
Z E ~s a t i s f y i n g
does n o t belong t o t h e c l o s e d
Hence, p
8
i s a base f o r t h e c l o s e d s e t s t h e r e * * Z c Z and piZ Hence, p E X\Z* C i n t Z C Z . Now, Z* U Z = X belongs t o 3. I t * f o l l o w s from t h e primeness of 3 t h a t Z E 5 or Z E ’ ~ . How* e v e r , p,dZ and p t n 3 s i n c e p i s a c l u s t e r p o i n t of 3 and 8 i s a b a s e f o r t h e c l o s e d s e t s . I t follows t h a t Z E ~ set
X\int Z.
exists
Z
*
and hence
€8
3
Since
such t h a t
converges t o
( 2 ) implies (3) :
y # p.
Then
Again
p
p. E
p ,d ( y ) so t h a t
t h e r e e x i s t s an element converges t o
p.
.
X\int
n
3.
suppose t h a t
p
X\(y].
E
Z E ~w i t h
p
E
y
n
E
3
and
I t follows t h a t
Z c X\(y] s i n c e
This c o n t r a d i c t s t h e f a c t t h a t
The i m p l i c a t i o n ( 3 ) i m p l i e s (1) i s immediate.
y
E
3
fl 3.
T h i s concludes
the proof. Observe t h a t
if 8
i s a collection s a t i s f y i n q the
c o n d i t i o n s s t a t e d i n t h e p r e v i o u s theorem,
then every prime
HEWITT-NACHBIN SPACES AND CONVERGENCE
52
is
I n particular, the collection Z ( X ) of a l l z e r o - s e t s i n a Tychonoff space s a t i s f i e s 6 . 1 2 .
3 - f i l t e r on
6.13
X
DEFINITION.
clusterable.
8-filter
A
5
on
i s s a i d t o have t h e
X
countable i n t e r s e c t i o n p r o p e r t y i n c a s e t h e i n t e r s e c t i o n of
5
every countable c o l l e c t i o n of members of
i s non-empty.
3 i s s a i d t o be closed under countable i n t e r s e c t i o n s i n c a s e t h e i n t e r s e c t i o n o f every countable c o l l e c -
The
2-filter
t i o n of members of
6.14
belongs t o
5
If 8 c
THEOREM.
5.
9-
P ( X ) i s a normal c o l l e c t i o n then a
u l t r a f i l t e r has t h e countable i n t e r s e c t i o n p r o p e r t y i f and only i f i t i s closed under countable i n t e r s e c t i o n s . Proof.
L be a
Let
& u l t r a f i l t e r with the countable i n t e r -
s e c t i o n property, and l e t Li.
members of
For any
countable s o t h a t
n
C'
C
be a countable c o l l e c t i o n of
Z ~ l r the c o l l e c t i o n
# 6.
C'
Thus t h e c o l l e c t i o n
= C U (Z] is
L U (fl @)
g e n e r a t e s a 8 - f i l t e r 3. I t follows t h a t La = 3 and t h a t l7 C E b. H e n c e L i s c l o s e d under countable i n t e r s e c t i o n s . The converse i s t r i v i a l .
8
If
i s a l o c a l base, then t h e
b ( p ) i s an example of a section property.
&neighborhood f i l t e r
& f i l t e r t h a t h a s t h e countable i n t e r b ( p ) w i l l not be
However, i n g e n e r a l
c l a s e d under countable i n t e r s e c t i o n s . X
i s an i n f i n i t e s e t and
(A E
P(X)
: X\F)
3
A s another example,
if
i s t h e Bourbaki f i l t e r
i s f i n i t e ) , then
5
f a i l s t o have t h e counta-
b l e intersection property. One might wonder whether o r n o t i t i s p o s s i b l e t o embed an a r b i t r a r y into a
9- f i l t e r with t h e countable i n t e r s e c t i o n p r o p e r t y
& u l t r a f i l t e r t h a t a l s o has t h e countable i n t e r s e c t i o n
property.
However, such an embedding i s g e n e r a l l y impossible.
I n f a c t , such i s not t h e c a s e even i f the
with the property of being c l u s t e r a b l e .
9 - f i l t e r i s endowed This phenomenon w i l l
be i l l u s t r a t e d i n S e c t i o n 7 when w e c o n s i d e r an example of a
I n that example i t w i l l be shown t h a t such an embedding f a i l s even when 8 i s t h e d i s t i n g u i s h e d c o l l e c t i o n Z ( X ) of a l l z e r o - s e t s i n a Tychonoff space. However, i f we impose a d d i t i o n a l c o n d i t i o n s
Hewitt-Nachbin space t h a t f a i l s t o be L i n d e l o f .
8- FILTERS
,8
on t h e d i s t i n g u i s h e d c o l l e c t i o n p o s s i b l e f o r prime
53
AND CONVERGENCE
8-filters.
then such an embedding i s
T h i s motivates t h e following
d e f i n i t i o n which w i l l a l s o be u s e f u l i n p r e s e n t i n g the WallmanFrink completion i n Section 9 . 6.15
8
A non-empty c o l l e c t i o n
DEFINITION.
8
be a d e l t a r i n q of s e t s i n c a s e
C
P ( X ) i s said t o
i s a r i n g of s e t s t h a t i s
closed under countable i n t e r s e c t i o n s .
s a i d t o be complement qenerated i n c a s e f o r each
(cn : n E m ) n {cn : nElN 1 .
2 =
Zc8
is
there
of complements of members of
e x i s t s a sequence such t h a t
9
The c o l l e c t i o n
I t i s easy t o v e r i f y t h a t t h e c o l l e c t i o n
9
Z ( X ) of a l l
z e r o - s e t s i n a Tychonoff space i s a d e l t a r i n g of s e t s t h a t i s complement g e n e r a t e d . 6.16
If
THEOREM.
-normal and
-with t h e unique Proof.
We have the following r e s u l t .
8
i s a d e l t a r i n q of s e t s on
countable i n t e r s e c t i o n p r o p e r t y
8-filter
embeddable i n a
4 u l t r a f i l t e r with the countable i n t e r s e c t i o n property. Let 3 be a prime 8 - f i l t e r on X with t h e countable Then, by Z o r n ' s Lemma, 5
i n t e r s e c t i o n property.
in a
is
that i s
X
complement qenerated, then every prime
L.
a-ultrafilter
countable i n t e r s e c t i o n p r o p e r t y .
i s contained
LI h a s t h e
I t w i l l be shown t h a t
Suppose, t o t h e c o n t r a r y ,
Lc
t h a t t h e r e e x i s t s a countable s u b c o l l e c t i o n ( Z n : n c n ] of with empty i n t e r s e c t i o n . countable cover of
x
Then the family ( X \ Z n
by complements of
complement generated, f o r each [Cn,i
:
fl ( C n , i U
ncm
U
icm ).
Zh,
1t follows t h a t
where (X\Cn, i )
nclN] is a
Since
8
is
such t h a t
X = U (x\Zn : nclN
= Zn ' , i
8
t h e r e e x i s t s a sequence
i c m ) of complements of members of :
8.
:
c (X\Zn).
1
Zn -
=
I n o t h e r words,
ncm icm the family ( X \ Z n : n E N ] h a s a countable refinement ( 2 ; : n c m ) ( a f t e r re-labeling the Z h , i n o r d e r t o s i m p l i f y t h e notat i o n ) which a l s o covers
in
such t h a t
X.
Zh c X\Zi
For each
n c m , s e l e c t an index
and a f i n i t e cover
n
gn
of m e m b e r s
of 9 such t h a t each element of an meets a t most one of 2; and Z (which i s p o s s i b l e because t h e normality of 8 i n in
HEWITT-NACHBIN SPACES AND CONVERGENCE
54
-
-
s u r e s t h e e x i s t e n c e o f Z1 and Z 2 i n 8 w i t h Z:, c ( X \ z l ) , Zi c ( X \ z 2 ) and Z1 U Z 2 = X ) . S i n c e 5 i s a prime 8n f i l t e r , f o r e a c h nclN t h e r e e x i s t s a s e t E n E 6n such t h a t
-
belongs t o
En
&filter.
n
C l e a r l y , En
3.
I t follows t h a t
En
an.
a c t e r i z a t i o n of t h e cover s i n c e [ Z h : n ~ m c] o v e r s t h a t t h e prime & f i l t e r
X.
5
n
# @
Zi
Ir.
because
is a
n
Zh = @
because o f t h e c h a r -
Therefore,
n
(En
: ncN] =
@
T h i s c o n t r a d i c t s t h e assumption has the countable intersection
property . F i n a l l y w e e s t a b l i s h t h a t \I i s u n i q u e . Suppose t o t h e contrary t h a t there is a & u l t r a f i l t e r Irl containing 3 w i t h L1 # L . Then t h e r e e x i s t A t L and B t L l w i t h A n B =
0,
A c x\cl,
X
8 there e x i s t x\c2, and (x\c,) n (x\cz)
By t h e n o r m a l i t y of B c
so t h a t
U C2 E 3.
C1
a,
C1,C2 =
8
satisfying
Thus
c1 u c 2
E
0.
=
Without loss of g e n e r a l i t y , by t h e
Ir. and since A c X\C1 i t f o l l o w s t h a t t h e empty s e t b e l o n g s t o L4. This i s a c o n t r a d i c t i o n and completes t h e p r o o f . p r i m e n e s s of
assume
C1
E 3.
Then
C1
E
I t f o l l o w s from t h e p r e v i o u s theorem t h a t i f
X
is a
Tychonoff s p a c e , then e v e r y prime 2 - f i l t e r w i t h t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y i s embeddable i n a u n i q u e Z - u l t r a f i l t e r with the countable i n t e r s e c t i o n p r o p e r t y .
However, more t h a n
t h i s is t r u e as the next r e s u l t demonstrates.
3
on a Tychonoff space X i s 5 z e r o - s e t u l t r a f i l t e r with the countable i n t e r s e c t i o n p r o p
6.17
THEOREM.
A zero-set f i l t e r
- --
e r t y i f and o n l y i f intersections. p r o o f . Assume t h a t intersection. member o f
3.
Let
3 , i s prime and c l o s e d under c o u n t a b l e
3 f
E
i s prime and c l o s e d under c o u n t a b l e C ( X ) and suppose
For each p o s i t i v e i n t e g e r
Z ( f ) meets e v e r y
n, let
and
>*I.
= (X€X : ( f ( x ) f
2;
Since
Zn U Z A
E
3
but
Z,:
/ 3, w e have
Zn
E
3
f o r every
n,
%FILTERS AND CONVERGENCE
n
Z(f) =
and hence
iZn
:
nelN
1
55
3.
belongs t o
3
Thus
is a
z e r o - s e t u l t r a f i l t e r t h a t h a s the countable i n t e r s e c t i o n p r o p erty. The c o n v e r s e f o l l o w s from 6 . 1 1 and 6 . 1 4 which c o n c l u d e s the proof. The n e x t r e s u l t p r o v i d e s a f o r m u l a t i o n f o r
2-filters
w i t h t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y i n terms o f f u n c t i o n s belonging t o the r i n g
I t i s proved i n d i r e c t l y i n G i l l -
C(X).
man and J e r i s o n by u s i n g r e s u l t s i n 5 . 6 , 5 . 7 ,
and 5 . 1 4 of t h a t
text.
6.18
THEOREM.
let 5 & a are true : --
If
(1)
Let
b e an a r b i t r a r y t o p o l o g i c a l s p a c e , and
X
2 - f i l t e r on
5
is a
X.
Then t h e f o l l o w i n q s t a t e m e n t s
Z-ultrafilter
with the
s e c t i o n property, then every on some z e r o - s e t -----
If
(2)
8
f
countable i n t e r -
C(X)
E
&
bounded
3.
in
f a i l s t o have t h e c o u n t a b l e i n t e r s e c t i o n
p r o p e r t y , then t h e r e e x i s t s a c o n t i n u o u s f u n c t i o n f
E
C ( X ) t h a t i s n o t bounded on any z e r o - s e t be-
% 3.
longinq
(1
Proof.
zn
f
Let
= (X€X : / f ( x )
I 2
exists a zero-set bounded on
~s a t i s f y i n g
po
Zn
/ 3 Z
n , then t h e r e
f o r some
n
Zn = f6.
belongs t o
n c m ] belongs t o
3
3.
Hence
Therefore,
is
f
f o r every
such t h a t I f ( p o ) ) 2 n
Z'
E
Zn
nEJN
there
f o r every f
is a subset
IR.
of
L e t (Fn : nelN ] be a sequence i n
(2)
section.
Choose
fn
I f n 5 1. D e f i n e
that
E
3
C ( X ) such t h a t
w i t h empty i n t e r Fn = Z ( f n ) and
OD
the function
g =
I: 2-"fn
and o b s e r v e
n=l
is continuous because t h e series converges uniformly.
g
x
from
n
(Fi
: 1
n ) , then g ( x ) 2-". Observe t h a t 1 i s d e f i n e d . Also, - 2 2" f o r every 9 n ) . I f Z i s a z e r o - s e t b e l o n g i n g t o 5, i
Z (9) i s empty so t h a t
x
E
If
d e f i n e t h e set
nElN
This i s impossible s i n c e the range of
ncN.
If
.
ll [Zn :
e x i s t s a point
0
Z
n)
Otherwise
Z.
Z' =
so t h a t
C ( X ) and f o r e a c h
E
E fl IFi
:
li i
-9
56
SPACES AND CONVERGENCE
HEWITT-NACHBIN
then f o r every
n
m u s t i n t e r s e c t the s e t
Z
nEIN,
IFi
: 1
because 3 has t h e f i n i t e i n t e r s e c t i o n p r o p e r t y . 1 Therefore, - cannot be bounded on any z e r o - s e t of 3. This g concludes t h e p r o o f . i
5 n)
I t i s i n t e r e s t i n g and u s e f u l t o r e l a t e z e r o - s e t f i l t e r s
between d i f f e r e n t t o p o l o g i c a l s p a c e s .
Thus l e t
tinuous mapping from t h e t o p o l o g i c a l space Y.
l o g i c a l space
If
3
(5)
=
is a
X
f
be a con-
i n t o the topo-
X, d e f i n e the
Z - f i l t e r on
collection f
(The mapping
#
iz
F
8(y)
: f+Z)
E
5).
i s introduced i n 4 . 1 2 of t h e Gillman and I t is immediate t h a t f # (5) i s a 2 - f i l t e r on
fx
Jerison t e x t . )
Y because f - l p r e s e r v e s unions and i n t e r s e c t i o n s . However, if 3 i s a 2 - u l t r a f i l t e r on X i t w i l l n o t n e c e s s a r i l y be true that
f
# (3) i s a
2 - u l t r a f i l t e r on
Y
( s e e Gillman and
Nevertheless the following r e s u l t i s easy t o
Jerison, 4 H . 2 ) . verify. 6.19
THEOREM (Gillman and J e r i s o n )
s i v e n a s i n the d e f i n i t i o n (1)
If
(2)
prime If 3
3
of
Y,
and
f
&
f # (3)
X, then
&a
h a s t h e countable i n t e r s e c t i o n property 01: under countable i n t e r s e c t i o n s , then t h e
same holds t r u e pf f#
X,
above.
i s a prime Z - f i l t e r on Z - f i l t e r on Y .
--i s closed
The mapping
fn
. Let
f'(3).
i s sometimes r e f e r r e d t o a s t h e " s h a r p
mapping" induced by
f.
This concludes our survey of t h e theory o f for arbitrary collections t h a t f o r a Tychonoff space
8 X
of
P(X).
g-€ilters
I t h a s been observed
the distinguished collection
Z ( X ) possesses all of t h e d e s i r a b l e p r o p e r t i e s of being a r i n g
of sets ( i n f a c t , a d e l t a r i n g of s e t s ) , a l o c a l b a s e , d i s j u n c t i v e , normal, and a base f o r t h e closed s e t s i n
X.
In
f a c t , Z ( X ) provided t h e motivation which lead t o many of t h e more g e n e r a l concepts and r e s u l t s presented above.
A major
57
R- FILTERS AND CONVERGENCE
v
p o r t i o n of t h e s t u d y of Hewitt-Nachbin s p a c e s w i l l concern i t s e l f solely with zero-set f i l t e r s . a l t h e o r y of
However,
t h e more g e n e r -
9 - f i l t e r s w i l l be n e c e s s a r y d u r i n g t h e p r e s e n t a -
t i o n of t h e Wallman-Frink completion i n S e c t i o n 9 .
L e t us
pause f o r a moment and examine some of t h e r e s u l t s and quest i o n s i n c o n n e c t i o n w i t h t h e Wallman-Frink c o m p a c t i f i c a t i o n and c o m p l e t i o n . I t i s w e l l known t h a t H . Wallman
(1938) used a p r o p e r t y
of n o r m a l i t y o f t h e c l a s s of c l o s e d s e t s i n a normal Hausdorff t o p o l o g i c a l space i n o r d e r t o c o n s t r u c t t h e Wallman compactif i c a t i o n ( s e e a l s o t h e 1966 paper by 0 . N j i s t a d ) .
I n 1964 0 .
F r i n k g e n e r a l i z e d Wallman's method i n c o n s t r u c t i n g Hausdorff c o m p a c t i f i c a t i o n s o f Tychonoff s p a c e s b y i n t r o d u c i n g t h e following concept. 6.20
DEFINITION.
b a s e on
X
Let
be a t o p o l o g i c a l s p a c e .
X
is a distinguished collection
8
A normal
c P(X) that is a
r i n g o f sets, d i s j u n c t i v e , normal, and a b a s e f o r t h e c l o s e d
sets of
X.
As was p r e v i o u s l y p o i n t e d o u t , t h e c o l l e c t i o n normal b a s e on a Tychonoff s p a c e .
Z ( X ) is a
I t i s e a s y t o show t h a t
e v e r y normal b a s e i s a l o c a l b a s e . For a normal b a s e s t r u c t e d t h e space tification. collection
8 on a Tychonoff s p a c e , F r i n k con-
w ( 8 ) of a l l
f j - u l t r a f i l t e r s f o r h i s compac-
H e t h e n proceeded t o show t h a t f o r t h e p a r t i c u l a r Z ( X ) of a l l z e r o - s e t s i n
p r e c i s e l y t h e Stone-&ch
X
t h e space
w(8) is
c o m p a c t i f i c a t i o n (meaning t o w i t h i n a
homeomorphism a s d i s c u s s e d p r e v i o u s l y )
.
The Alexandrof f one-
p o i n t c o m p a c t i f i c a t i o n of a l o c a l l y compact Hausdorff s p a c e can a l s o b e o b t a i n e d a s a Wallman-Frink c o m p a c t i f i c a t i o n :
a
s u i t a b l e normal b a s e i s g i v e n by t h e c o l l e c t i o n of z e r o - s e t s of t h o s e c o n t i n u o u s f u n c t i o n s on
X
complement of some compact s u b s e t of by R. Alo and H .
Shapiro).
t h a t a r e c o n s t a n t on t h e X (see t h e 1968A p a p e r
Alo and S h a p i r o have a l s o shown
t h a t t h e Fan-Gottesman and F r e u d e n t h a l (1952) c o m p a c t i f i c a t i o n s
I n f a c t , t h e y observed t h a t a l l of t h e normal b a s e s which t h e y used w e r e s u b c o l l e c t i o n s of t h e
a r e of t h e Wallman-Frink t y p e .
SPACES AND CONVERGENCE
58
HEWITT-NACHBIN
collection
Z ( X ) of a l l z e r o - s e t s .
A q u e s t i o n posed by F r i n k
was whether or n o t e v e r y c o m p a c t i f i c a t i o n of a Tychonoff s p a c e could b e obtained a s a space base
8. Alo
w ( 8 ) f o r some s u i t a b l e normal
and S h a p i r o r a i s e d t h e a d d i t i o n a l q u e s t i o n t h a t ,
8 always b e t a k e n a s some
i f such i s indeed t h e c a s e , c o u l d a p p r o p r i a t e s u b c o l l e c t i o n of
Z(X)?
The former q u e s t i o n h a s
been answered a f f i r m a t i v e l y i n t h e c a s e o f m e t r i c s p a c e s by E . S t e i n e r i n 1968B.
However, t h e q u e s t i o n remains open f o r t h e
general case. The c o n c e p t of a normal b a s e p l a y s a n o t h e r i m p o r t a n t r o l e i n t h e s t u d y of t o p o l o g i c a l s p a c e s b e c a u s e i t p r o v i d e s an i n t e r n a l c h a r a c t e r i z a t i o n o f completely r e g u l a r
T1-s p a c e s .
S p e c i f i c a l l y , 2 t o p o l o g i c a l space i s a completely r e q u l a r s p a c e i f and o n l y i f i t h a s a normal b a s e .
TO see t h i s ,
s e r v e t h a t i f a space is a completely r e g u l a r the collection
T1-space,
Z ( X ) of a l l z e r o - s e t s i s a normal b a s e .
T1ob-
then
on
t h e o t h e r hand, i f a T1-space h a s a normal b a s e t h e n i t h a s a F r i n k c o m p a c t i f i c a t i o n and hence i s c o m p l e t e l y r e g u l a r . We w i l l s e e i n S e c t i o n 9 how Alo and S h a p i r o u s e a v a r i a t i o n o f F r i n k ' s n o t i o n of a normal b a s e , by demanding t h a t i t a l s o be a complement g e n e r a t e d d e l t a r i n g o f s e t s , i n cons t r u c t i n g t h e Wallman-Frink c o m p l e t i o n of a Tychonoff s p a c e . I t w i l l be shown t h a t t h s Hewitt-Nachbin c o m p l e t i o n i s j u s t a
s p e c i a l c a s e o b t a i n e d by t h e i r t e c h n i q u e .
Analogous t o F r i n k ' s
q u e s t i o n posed above, ona might a s k whether o r n o t e v e r y comp l e t i o n o f a Tychonoff s p a c e man-Frink method.
We
X
can be o b t a i n e d by t h e Wall-
w i l l address t h a t question during our
presentation i n Section 9. Section 7 :
Hewitt-Nachbin Completeness v i a I d e a l s . F i l t e r s , and N e t s
W e now f o c u s o u r a t t e n t i o n on t h e s t u d y o f H e w i t t -
Nachbin completeness from t h e p o i n t o f view of maximal i d e a l s
i n t h e r i n g C(X) of 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 h e X, i n t e r m s o f z e r o - s e t u l t r a f i l t e r s on X, and i n t e r m s of n e t s . I n o r d e r t o f a c i l i t a t e o u r s t u d y w e b e g i n by i n c o r p o -
space
r a t i n g t h e n e c e s s a r y r e s u l t s c o n c e r n i n g t h e t h e o r y of i d e a l s
IDEALS, FILTERS, AND NETS
i n the ring space
59
of 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 h e
C(X)
F u r t h e r d e t a i l s concerning t h e s e r e s u l t s , t o g e t h e r
X.
w i t h t h e i r p r o o f s , may b e found i n C h a p t e r s 2 and 5 o f t h e Gillman and J e r i s o n t e x t . Let
and
Ir
that
b e an a l g e b r a i c r i n g w i t h i d e n t i t y .
R
an i d e a l
of
I C
acP
An ideal
or
I
C
implies
J
An i d e a l I = J
i s s a i d t o b e prime i n c a s e
P
rI
C I
W e w i l l adopt the convention
when r e f e r r i n g t o i d e a l s .
t o be maximal i n c a s e
J.
rcR.
f o r every
I
# R
I
Recall t h a t
i s an a d d i t i v e subgroup such t h a t
R
i s said
I
f o r any i d e a l ab
E
implies
P
bEP.
I t i s c l e a r t h a t t h e i n t e r s e c t i o n of any f a m i l y o f
ideals i n
i s a g a i n an i d e a l i n
R
Moreover, an a p p e a l t o
R.
Z o r n ’ s Lemma e s t a b l i s h e s t h e r e s u l t t h a t e v e r y i d e a l i s con-
-tained
i n 2 maximal i d e a l .
F i n a l l y , i t i s an e a s y e x e r c i s e t o
show t h a t e v e r y maximal i d e a l i s prime. The f o l l o w i n g lemma i s e a s y t o v e r i f y .
If
7.1
LEMMA.
PEX,
then t h e s e t
ideal i n section
M = ( f E C ( X ) : f ( p ) = 0 ) i s a maximal P Moreover t h e p o i n t p b e l o n q s t o t h e i n t e r -
C(X).
n
i s an a r b i t r a r y t o p o l o q i c a l s p a c e and i f
X
iz(f)
E
z(x)
: f
M ~ ) .
E
The n e x t r e s u l t e s t a b l i s h e s t h e fundamental r e l a t i o n s h i p between
2 - f i l t e r s on a s p a c e
and t h e i d e a l s of
X
C(X)
.
omit t h e p r o o f s which may b e found i n Gillman and J e r i s o n
We
(2.3
and 2 . 5 ) . 7.2
THEOREM (Gillman and J e r i s o n )
If
M
X
b e an a r b i t r a r y
Then t h e f o l l o w i n q s t a t e m e n t s a r e t r u e :
topoloqical space. (1)
. Let
i s an i d e a l i n
Z[M] = [ Z ( f ) Moreover,
if
Z(X)
E
M
:
C(X),
then t h e c o l l e c t i o n
EM] i s a
maximal,
then
2 - f i l t e r on
X.
a
Z-
Z [MI
ultraf ilter. (2)
If
+
Z
3:
[a]
is 2
Moreover, Zt[3]
Z - f i l t e r on
= ( f E C(X)
if
3
: Z(f) E
X,
a)
then t h e c o l l e c t i o n
i s an i d e a l i n
Z-ultrafilter
i s a maximal i d e a l .
on
X,
C(X)
then
60
HEWITT-NACHBIN
SPACES AND CONVERGENCE
Because of t h e above p r o p o s i t i o n , an i d e a l
is s a i d t o be f i x e d i n c a s e the otherwise
Z- f i l t e r
in
M
C(X)
is fixed;
Z [MI
i s s a i d t o be f r e e .
M
I n S e c t i o n 6 i t was observed t h a t a Tychonoff s p a c e i s compact i f and o n l y i f e v e r y
Z - f i l t e r on
f o l l o w s from 7 . 2 t h a t 2 Tychonoff s p a c e
only i f e v e r y i d e a l i n
C(X)
is fixed.
X
2
X
X
compact
It
if and
i s f i x e d (Gillman and J e r i s o n ,
4.11). If
i s a Tychonoff s p a c e and i f
X
,
C (X)
follows t h a t
f (p) = 0
f o r every
On t h e o t h e r hand,
then
Z ( g ) m e e t s e v e r y member of t h e
Therefore,
gcM
if
If
THEOREM.
maximal i d e a l s i n
7.1, (pcX).
X C(X)
Moreover,
n
E
f o r some
Z [MI
g
. P
It
by
C(X),
E
2-ultrafilter M c M. P
which i m p l i e s t h a t
r e s u l t has been established.
7.3
p
Hence, M t M
fEM.
g(p) = 0
7.1.
i s a f i x e d maxi-
M
then t h e r e i s a p o i n t
mal i d e a l i n
Z[M]
.
The f o l l o w i n g
2 Tychonoff s p a c e , then t h e f i x e d
are p r e c i s e l y th2 c o l l e c t i o n s they a r e d i s t i n c t
for
M in P d i s t i n c t points
P. Now,consider t h e mapping
p
from
p ) f o r each pcx. -p ips (af )r =i n gf ( homomorphism with
f i n e d by
into
C(X)
IR
de-
I t i s easy t o v e r i f y
that kernel M Therefore, P' by t h e Fundamental Homomorphism Theorem f o r r i n g s , t h e quotient ring
C(X)/Mp
IR f o r each ptX. C(X)/Mp o n t o I€? i s g i v e n
i s isomorphic t o
I n f a c t t h e isomorphism
p"
from
by F ( f + MP) = p ( f ) . I t f o l l o w s from 7 . 3 t h a t f o r e a c h f i x e d maximal i d e a l M C ( X ) the q u o t i e n t C(X)/M is isomorphic t o t h e r e a l f i e l d
m.
One might n o w wonder what
o c c u r s i n t h e c a s e t h a t t h e maximal i d e a l
is free.
M
This
prompts t h e f o l l o w i n g d e f i n i t i o n . 7.4
A maximal i d e a l M i n C ( X ) is s a i d to be c a s e t h e q u o t i e n t r i n g C(X)/M is isomorphic t o IR;
DEFINITION.
real i n
otherwise
M
is s a i d t o be h y p e r - r e a l .
mal i d e a l , then i t i s s a i d t h a t
Z[M]
If
M
is a real
W e remark t h a t f o r e a c h maximal i d e a l
M
i s a r e a l maxi2-ultrafilter.
in
C(X) the
IDEALS, FILTERS, AND NETS quotient ring
C(X)/M
61
always c o n t a i n s an isomorphic copy o f
m. The f o l l o w i n g p r o p o s i t i o n s a r e found i n Gillman and J e r i son ( 5 . 8 , 5.14, and 2 . 4 ,
respectively).
W e s t a t e them h e r e
f o r emphasis and p u r p o s e s of r e f e r e n c e a l t h o u g h w e o m i t t h e proofs 7.5
. If
THEOREM (Gillman and J e r i s o n ) .
2 Tychonoff
X
space, then the followinq s t a t e m e n t s a r e t r u e :
*
(1) Every maximal i d e a l i n Every maximal i d e a l i n
(2)
is r e a l .
C (X)
i s r e a l i f and o n l y i f
C(X)
is pseudocompact.
X
7 . 6 THEOREM (Gillman and J e r i s o n ) . If X & a Tvchonoff space and i f M i s a maximal i d e a l i n C ( X ) , t h e n t h e follow-
%
statements
(1) (2)
The The
=
equivalent:
maximal i d e a l 2-ultrafilter
is real.
M
Z[ M]
i s c l o s e d under c o u n t a b l e
Z[M]
has t h e countable i n t e r -
intersections. (3)
The
Z-ultrafilter
s e c t ion p r o p e r t y 7.7
(Gillman and J e r i s o n )
COROLLARY
5
s p a c e and i f Moreover, 3
.
is a
2-ultrafilter
i s r e a l i f and o n l y i f
.
If
on 3
X
is g then
X,
Tychonoff 5 = Z[Zc[3]].
has the countable
intersection property. I n S e c t i o n 4 w e c o n s t r u c t e d t h e Hewitt-Nachbin completion
vX
of a Tychonoff s p a c e
Theorem 4 . 3 when X
E = IR.
X
v i a the
E-Compactification
I n t e r p r e t i n g 4 . 4 i t was s e e n t h a t
X = uX.
i s a Hewitt-Nachbin s p a c e i f and o n l y i f
the r i n g
C(X)
i s isomorphic t o t h e r i n g
the r e s u l t s t a t e d i n 4 . 9 .
Moreover
C ( u X ) according t o
These f a c t s w i l l b e u s e f u l i n
e s t a b l i s h i n g t h e f o l l o w i n g fundamental r e s u l t which o r i g i n a l l y appeared i n E . H e w i t t ' s 1948 p a p e r 7.8
THEOREM ( H e w i t t ) .
(Theorem 5 9 ) .
& Tvchonoff s p a c e
X
is a H e w i t t -
Nachbin s p a c e i f and o n l y i f e v e r y r e a l maximal i d e a l i n
is fixed. --
C(X)
62
SPACES AND CONVERGENCE
HEWITT-NACHBIN
Proof.
If
Necessity:
i s a Hewitt-Nachbin s p a c e , then t h e
X
i d e a l s t r u c t u r e s of
C ( X ) and
vious observations.
Hence, l e t
ideal i n M(f) i n Since
C(sX)
.
F
f
C(-;X)
E
the e l e m e n t
i s a r e a l number by 7 . 4 .
C('JX)/M
C ( L I X ) a r e isomorphic i t f o l l o w s t h a t w e can
C ( X ) and
c(x)
with a p o i n t i n the product
Moreover, s i n c e
C(X)).
d e n o t e any r e a l maximal
M
For each f u n c t i o n
the q u o t i e n t r i n g
identify ( M ( f ) ) f f
C ( L X ) a r e e q u i v a l e n t by o u r pre-
n[lRf : Z - u l t r a f i l t e r on X
is a
Z[M]
( 7 . 2 ( 1 ) ) ( a g a i n w e make u s e of t h e isomorphism) i t h a s t h e
f i n i t e intersection property.
.,,fk
i n C(X) there exists a point f i ( p ) = M(fi) for a l l i = 1, . . . , k: namely, p
t i o n of f u n c t i o n s satisfying
PEX
Hence, f o r any f i n i t e c o l l e c -
fl, f 2 , .
k Ti Z ( f i - M ( f i ) ) b e c a u s e i=l T h e r e f o r e , an a r b i t r a r y neighbor-
i s contained i n t h e i n t e r s e c t i o n f i - M(fi) belongs t o hood
...,fk)
U(f,,
space n[lRf : f
c(x)
( f (p)) for to
E
in
into
C(vX)
.
CJ ( X )
,
u
where
i s t h e p a r a m e t r i c mapping
f
E
((M(f))f
c(x)
Now, r e c a l l t h a t t h e isomorphism
i n 4 . 9 was g i v e n by
i s t h e p r o j e c t i o n mapping from f o r each
i n t h e product
of t h e p o i n t ( M ( f ) )
f E C(X) C ( X ) } w i l l c o n t a i n a p o i n t o f t h e form
I t follows t h a t t h e p o i n t
C(X).
~JX= c l O(X)
C(X)
M.
belongs from
cp
cp(f) = T ~ I L J X where
nlRf
into
C ( X ) i t i s the case that
f
Tf
Therefore,
IRf.
vanishes a t the
c ( x ) i f and o n l y i f M(f) = 0 . However, M ( f ) = pcint (M(f) 1 0 i f and o n l y i f f b e l o n g s t o t h e i d e a l M. Hence, the ideal
c o n s i s t s p r e c i s e l y of t h o s e f u n c t i o n s i n
M
vanish a t the point (M(f)) f maximal i d e a l by 7 . 3 .
E
C(X)'
Therefore, M
By 7 . 3 t h e f i x e d maximal i d e a l s i n
Sufficiency:
C(vX) t h a t
is a fixed a r e pre-
C(X)
c i s e l y of t h e form M = I f E C ( X ) : f ( p ) = 0 ) where PEX. By P h y p o t h e s i s , t h e s e i d e a l s a r e p r e c i s e l y t h e r e a l maximal i d e a l s
in
C(X),
i . e . , a n i d e a l i s r e a l i f and o n l y i f i t i s f i x e d .
T h e r e f o r e , t h e mapping which a s s o c i a t e s t o e a c h mal i d e a l
M
P
i s i n j e c t i v e from
a l l r e a l maximal i d e a l s i n
C(X).
X
pcX
t h e maxi-
onto the collection The c o l l e c t i o n
h
m
i s made
i n t o a t o p o l o g i c a l s p a c e by t a k i n g , a s a b a s e f o r t h e c l o s e d
s e t s , a l l s e t s of t h e form h ( f ) = (MP E h : f E M ) where P f E C(X) T h e f a c t t h a t t h i s i s a b a s e f o l l o w s from
.
of
IDEALS, FILTERS, AND NETS
M
the observation t h a t Since
M
P
belongs t o
L [ t n ( f ) u m ( g ) ] o n l y i f M~ ,4 m ( f g ) . P h ( f ) i f and o n l y i f f ( p ) = 0, t h e
correspondence between
p
M
and
P
c a r r i e s the z e r o - s e t s of
o n t o t h e f a m i l y of a l l s e t s of t h e form
X
more, s i n c e
63
h(f).
Further-
i s a Tychonoff s p a c e , t h e c o l l e c t i o n
X
Z(X) of
i s a base f o r the closed sets i n X (3.6 (1)) which shows t h a t t h e t o p o l o g y on X can be r e c o v e r e d from C ( X ) . H e n c e , X i s homeomorphic t o h . Moreover, s i n c e C ( u X ) i s isomorphic t o C ( X ) t h e same argument can b e used t o e s t a b l i s h t h a t UX i s homeomorphic t o h. T h e r e f o r e , X is a l l zero-sets i n
sX
homeomorphic t o space.
X
and, a c c o r d i n g l y , i s a Hewitt-Nachbin
This concludes t h e p r o o f , I f w e s u b s t i t u t e t h e Hewitt-Nachbin s p a c e
Y
for
VX
i n t h e above s u f f i c i e n c y proof w e o b t a i n immediately t h e f o l l o w i n g r e s u l t due t o H e w i t t (1948, Theorem 5 7 ) . 7.9
COROLLARY
(Hewitt)
a r e homeomorphic C(Y)
are
,
The Hewitt-Nachbin s p a c e s
i f and o n l y i f t h e f u n c t i o n r i n g s
and Y C ( X ) and X
a l q e b r a i c a l l y isomorphic.
The p r e c e d i n g r e s u l t p a r a l l e l s t h e i m p o r t a n t f a c t t h a t two compact Hausdorff s p a c e s X and Y a r e homeomorphic i f and only i f t h e f u n c t i o n r i n g s
C
*
Y
( X ) and
C
(Y)a r e a l g e b r a i c a l l y
isomorphic (see, f o r example, Gillman and J e r i s o n , 4 . 9 ) .
A
few a d d i t i o n a l remarks a r e i n o r d e r c o n c e r n i n g t h e c o n s t r u c t i o n u t i l i z e d i n t h e proof o f t h e s u f f i c i e n c y c o n d i t i o n of 7 . 8 .
h
If
denotes the c o l l e c t i o n o f
h
then
all
maximal i d e a l s i n
C(X),
can be made i n t o a t o p o l o g i c a l s p a c e by t a k i n g , a s a
b a s e f o r t h e c l o s e d s e t s , a l l s e t s of t h e form ( M E m : f c M ) , f
E
C(X).
The topology t h u s d e f i n e d i s c a l l e d t h e S t o n e
topoloqy and t h e r e s u l t a n t t o p o l o g i c a l s p a c e S t r u c t u r e space of t h e r i n g
C(X)
compact Hausdorff s p a c e and t h a t
. X
g i v e n i n 7 . 8 above.
is called the
In
is a
i s homeomorphic t o t h e
c o l l e c t i o n of a l l f i x e d maximal i d e a l s i n pwMp
m
It turns out t h a t
Ih. v i a t h e mapping
A d d i t i o n a l information concerning
t h e S t r u c t u r e s p a c e can b e found i n G i l l m a n and J e r i s o n ( 4 . 9 ,
7M, and 7 N ) . With t h e a i d of 7 . 8 t o g e t h e r w i t h 7 . 6 w e can now g i v e
64
SPACES AND CONVERGENCE
HEWITT-NACHBIN
t h e f o l l o w i n g c h a r a c t e r i z a t i o n o f Hewitt-Nachbin c o m p l e t e n e s s i n terms of z e r o - s e t u l t r a f i l t e r s on t h e s p a c e . 7.10
THEOREM (Gillman and J e r i s o n )
. A
Tychonoff s p a c e
Hewitt-Nachbin complete i f and o n l y i f e v e r y
X
Z-ultrafilter
on
X with t h e countable i n t e r s e c t i o n property is f i x e d . proof. I f 5 i s a 2 - u l t r a f i l t e r o n X w i t h the countable i n t e r s e c t i o n p r o p e r t y , t h e n 5 = Z [ Z c [ 3 ] ] by 7 . 7 and Z c [ 3 ] i s a maximal i d e a l by 7 . 2 ( 2 ) .
Since
3
h a s t h e countable
c
i n t e r s e c t i o n p r o p e r t y , Z [ a ] i s r e a l by 7 . 6 . Nachbin complete, then
If
is H e w i t t -
X
i s f i x e d by 7 . 8 and hence
Zc[3]
i s f i x e d by d e f i n i t i o n . iT i s a r e a l maximal i d e a l i n
Z[Zc[3]]
Conversely, suppose
I t f o l l o w s from 7 . 6 ( 3 ) t h a t
Z[M]
countable i n t e r s e c t i o n property.
h
assumption which means t h a t X
C(X).
is a
2 - u l t r a f i l t e r with the
Then
Z[M]
i s f i x e d by
i s f i x e d by d e f i n i t i o n .
Thus
i s Hewitt-Nachbin complete by 7 . 8 which c o n c l u d e s t h e p r o o f .
I n h i s 1 9 7 0 p a p e r , K . P . Chew p r o v i d e s a c h a r a c t e r i z a t i o n f o r a z e r o - d i m e n s i o n a l s p a c e t o b e I"-compact t h a t i s a n a l o gous t o t h e p r e c e d i n g r e s u l t f o r Hewitt-Nachbin s p a c e s ( i . e . ,
IR-compact s p a c e s ) .
Namely, a z e r o - d i m e n s i o n a l s p a c e
X
on
X
IN-compact i f and o n l y i f e v e r y c l o p e n u l t r a f i l t e r
the countable
with
intersection property is fixed.
W e have a l r e a d y o b s e r v e d i n t h e p r e v i o u s c h a p t e r t h a t
e v e r y compact Hausdorff s p a c e i s a
Hewitt-Nachbin s p a c e .
The
following r e s u l t w i l l a s s i s t u s i n providing s e v e r a l a d d i t i o n a l i n t e r e s t i n g and i m p o r t a n t examples of Hewitt-Nachbin s p a c e s . 7.11
THEOREM.
statements
are
If
equivalent:
(1) The s p a c e
(2)
i s a Tychonoff s p a c e , t h e n t h e f o l l o w i n q
X
Every
X
Lindelzf.
Z - f i l t e r on
X
with the countable inter-
section property is fixed. (3)
Every c l u s t e r a b l e
Z - f i l t e r on
X
w i t h t h e count-
-
able intersection property is fixed.
Proof.
(1) i m p l i e s ( 2 ) :
I t i s e a s y t o show t h a t
X
is
L i n d e l o f if and o n l y i f e v e r y f a m i l y o f c l o s e d s u b s e t s w i t h
FILTERS, AND NETS
IDmLS,
65
t h e countable i n t e r s e c t i o n p r o p e r t y i s f i x e d .
I n particular,
Z- f i l t e r with t h e countable i n t e r s e c t i o n p r o p e r t y i s
every
such a family.
Clearly,
( 3 ) i m p l i e s (1):
( 2 ) implies
Suppose t h a t
(3).
i s n o t L i n d e l o f . Then X with no
X
Q = (Oa : ~ E G of ] I f w e d e f i n e 5 = (X\Oa
t h e r e e x i s t s an open cover countable subcover.
:
a&],
3
then
i s a family of c l o s e d s e t s with the countable i n t e r s e c t i o n
property.
As
X
i s a Tychonoff space, t h e c o l l e c t i o n
i s a base f o r t h e closed s e t s i n
X\Oa
t h a t each closed s e t set
I t follows
i s contained i n some zero-
The c o l l e c t i o n of a l l z e r o - s e t s t h a t c o n t a i n a t
Z.
3 has the f i n i t e i n t e r s e c t i o n property
l e a s t one member of
G
since
by 3.6(1).
X
5
in
Z(X)
has no countable subcover, and hence g e n e r a t e s a
5*
Z-filter
with the p r o p e r t y t h a t each member of
a*
con-
Furthermore, 3 has the countable i n t e r s e c t i o n p r o p e r t y because 5 h a s the 3;.
t a i n s a f i n i t e i n t e r s e c t i o n of members of
that
a*
then
p
Z
E
x\Z
*
5 f a i l s t o have a c l u s t e r p o i n t i n
countable i n t e r s e c t i o n p r o p e r t y , and E
Oa
f o r some
Z ( X ) such t h a t
acG.
pkZ,
p
X\Oa
C
and
Z,
p
We claim
For i f
X.
2 t
a*.
PEX,
Moreover,
f o r which (X\Z)
fl Z =
a.
3,; hence 5* conI t follows t h a t 3" i s
cannot be a c l u s t e r p o i n t of
verges t o each of i t s c l u s t e r p o i n t s . clusterable ( 6 . 9 ) . i t is a f r e e
3.
= fl
Hence, t h e r e e x i s t s some
i s an open neighborhood of
Therefore
fl
*
Moreover, s i n c e
Z-filter
(every f i x e d
5* h a s no c l u s t e r p o i n t Z - f i l t e r has a c l u s t e r
p o i n t ) and t h e proof i s complete. A n immediate consequence of t h e previous r e s u l t i s t h a t
every Lindelof space i s Hewitt-Nachbin Lindelof space every
complete s i n c e i n a
Z- f i l t e r (and hence every
Z- u l t r a f i l t e r )
with the countable i n t e r s e c t i o n p r o p e r t y i s f i x e d . more, s i n c e every
241), every
Further-
o-compact space i s Lindelof (Dugundji, page
a-compact space i s Hewitt-Nachbin complete.
In
p a r t i c u l a r , every countable space i s Hewitt-Nachbin complete. Moreover, a s every second countable space i s Lindelof i t follows t h a t every second countable space i s a Hewitt-Nachbin space.
H e n c e every s e p a r a b l e metric space is Hewitt-Nachbin
complete so t h a t every subspace of a Euclidean space i s Hewitt-
66
SPACES AND CONVERGENCE
HEWITT-NACHBIN
I n t h e next c h a p t e r we s h a l l e s t a b l i s h t h e
Nachbin complete.
s t r o n g e r r e s u l t t h a t every m e t r i c space of ‘Inonmeasurable c a r d i n a l “ i s a Hewitt-Nachbin space. b l e m e t r i c space we s e e t h a t Hewitt-Nachbin complete.
IR
IR
Since
i s a separa-
and a l l of i t s subspaces a r e
T h e r e f o r e , u n l i k e t h e compact Haus-
d o r f f s p a c e s , Hewitt-Nachbin subspaces of a Hewitt-Nachbin space need n o t be c l o s e d .
F i n a l l y , we p o i n t o u t t h a t Hewitt
i n 1948 f i r s t discovered t h a t Lindelof spaces a r e H e w i t t Nachbin complete.
On t h e o t h e r hand t h e r e do e x i s t Hewitt-
Nachbin spaces t h a t f a i l t o be Lindelof a s t h e f o l l o w i n g example illustrates. 7.12
A Hewitt-Nachbin
EXAMPLE.
space t h a t f a i l s t o b e
Lindelof and f a i l s t o be paracompact. The following space a p p e a r s i n t h e 1947 paper by R . denote t h e s e t of r e a l numbers with a P base f o r t h e open s e t s given by i n t e r v a l s of t h e form ( a , b ] = Sorgenfrey. : a
[xEIR
Lindelof
<
Let
x
E
5 b).
I t i s w e l l known t h a t
(Dugundji, Chapter V I I I ,
---i t i s n o t second E
3, page
i s reqular
E x . 3 , page 1 7 4 ) a l t h o u g h
E
146).
P
It
i s completely normal (Dugundji, Moreover,
t h e p r o d u c t space
i s n o t normal ( l o c . c i t . , Ex. 3 , page 144) and hence
X E
iL P--is n o t paracompact
mal)
CI
c o u n t a b l e ( l o c . c i t . , Ex. 2 , page 1 7 3 ) .
h a s a l s o been shown t h a t Chapter V I I ,
6,
E
nor
Lindelof
( s i n c e r e g u l a r paracompact spaces a r e nor( s i n c e i n Lindel6f spaces t h a t a r e Hausdorff
t h e concepts of r e g u l a r i t y and paracompactness a r e e q u i v a l e n t (Dugundji, Chapter V I I I ,
6 . 5 , page 174)).
However, s i n c e
i s Lindelof it i s Hewitt-Nachbin complete by 7 . 1 1 , fore
E
P
X E
iL
i s a Hewitt-Nachbin
i s the r e a l l i n e
E )I
and t h e r e -
space by 4 . 2 ( 3 ) (where
E
IR) .
The preceding example i s important i n connection w i t h t h e r e s u l t 7 . 1 1 because i t e s t a b l i s h e s t h a t t h e r e a r e
Z-
f i l t e r s with t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y t h a t cannot be embedded in a 2 - u l t r a f i l t e r w i t h t h e c o u n t a b l e i n t e r s e c tion property. I n f a c t , there a r e c l u s t e r a b l e 2 - f i l t e r s with t h e countable i n t e r s e c t i o n p r o p e r t y which cannot be embedded
in a
Z - u l t r a f i l t e r with t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y .
67
IDEALS, F I L T E R S , AND NETS
For, i f i n a Hewitt-Nachbin space every c l u s t e r a b l e Z - f i l t e r w i t h t h e countable i n t e r s e c t i o n p r o p e r t y could be embedded i n a
Z - u l t r a f i l t e r with the countable i n t e r s e c t i o n p r o p e r t y ,
then every such c l u s t e r a b l e 2 - f i l t e r would be f i x e d which i n t u r n would y i e l d by 7 . 1 1 ( 3 ) t h a t the space i s Lindelof conOn t h e o t h e r hand, r e c a l l t h a t i t was
t r a r y t o example 7 . 1 2 .
shown i n 6.17 t h a t every prime
2 - f i l t e r t h a t i s closed under
countable i n t e r s e c t i o n s i s a z e r o - s e t u l t r a f i l t e r with t h e countable i n t e r s e c t i o n p r o p e r t y .
This o b s e r v a t i o n s u g g e s t s
the next r e s u l t which i s found i n Gillman and J e r i s o n . THEOREM (Gillman and J e r i s o n ) .
7.13
space then t h e following s t a t e m e n t s
(1) The space (2)
If
i s a Tychonoff
X
are e q u i v a l e n t :
a Hewitt-Nachbin space.
X
Every p r i m e
Z - f i l t e r on
w i t h the countable
X
intersection property is f i x e d , (3)
Every p r i m e
2 - f i l t e r on
x
t h a t i s c l o s e d under
countable i n t e r s e c t i o n s i s f i x e d . Proof.
3
If
(1) i m p l i e s ( 2 ) :
i s a prime
with t h e countable i n t e r s e c t i o n p r o p e r t y , then by 6.16 contained i n a unique
2-ultrafilter
intersection property.
X,
L
L
x
Z - f i l t e r on
3
is
with t h e countable
By t h e Hewitt-Nachbin completeness of
i s f i x e d : whence
3
i s fixed.
The i m p l i c a t i o n ( 2 ) i m p l i e s ( 3 ) i s immediate. ( 3 ) i m p l i e s (1): I f
3
is a
i n t e r s e c t i o n p r o p e r t y , then
2 - u l t r a f i l t e r with the countable
3
i s a prime
Z-filter that is
c l o s e d under countable i n t e r s e c t i o n s by 6 . 1 7 .
The r e s u l t i s
now immed i a t e . The next r e s u l t , o r i g i n a l l y due t o H e w i t t (1948, Theorem 54),
r e l a t e s a n o t h e r i n t e r e s t i n g c l a s s of t o p o l o g i c a l spaces
t o the Hewitt-Nachbin spaces.
I n f a c t , i t y i e l d s t h e important
r e s u l t t h a t t h e property of Hewitt-Nachbin completeness coupled w i t h pseudocompactness y i e l d s compactness. I t a l s o prov i d e s a u s e f u l t o o l f o r sometimes a s c e r t a i n i n g whether o r n o t a given space i s Hewitt-Nachbin complete.
We w i l l utilize
t h e r e s u l t t o p r e s e n t an e s p e c i a l l y important example of a space t h a t f a i l s t o be a Hewitt-Nachbin
space.
68
SPACES AND CONVERGENCE
HEWITT-NACHBIN
7.14
.
THEOREM ( H e w i t t )
pseudocompact Tychonoff s p a c e
X
- -
i s 2 Hewitt-Nachbin s p a c e i f and o n l y i f i t i s compact.
proof. C(X)
Since
X
i s r e a l by 7 . 5 ( 2 ) .
maximal i d e a l i n fore
i s pseudocompact, e v e r y maximal i d e a l i n
X
C(X)
X
If
i s n o t compact, then some
i s f r e e a s a consequence of 7 . 2 .
There-
c a n n o t b e a Hewitt-Nachbin s p a c e by t h e r e s u l t 7 . 8 .
The s u f f i c i e n c y w a s observed p r e v i o u s l y . 7.15
EXAMPLE.
The o r d i n a l
s p a c e [O,n]
f a i l s t o be a H e w i t t -
N a c h b i n space. Let
l e t [O,n]
n
d e n o t e t h e f i r s t u n c o u n t a b l e o r d i n a l number, and
denote t h e set of a l l o r d i n a l s less than o r e q u a l
0. A b a s e f o r t h e open sets i n [o,n] i s g i v e n by t h e > a) n ( x : x < p + 11. T h i s topology i s u s u a l l y r e f e r r e d t o a s
to
c o l l e c t i o n of a l l s e t s o f t h e form ( a , p ) = ( x : x
the
i n t r i n s i c topoloqy f o r a c h a i n and i s s t u d i e d e x t e n s i v e l y by R . Alo and 0 . F r i n k i n t h e i r 1967 p a p e r .
The s p a c e [ O , n ]
with
t h e r e s u l t a n t topology i s c a l l e d t h e o r d i n a l s p a c e and h a s t h e following p r o p e r t i e s : The s p a c e -
[0, n]
2 compact Hausdorff s p a c e
(Dugundji, Chapter V I I I , The subspace -
[O,n)
=
2, Ex.
[O,n]\[n] i s
(Dugundji, Chapter V I I ,
2, Ex.
I n f a c t , b o t h [0,hl] and [O,n) (see Alo and F r i n k , 1 9 6 7 ) . Every c o n t i n u o u s f u n c t i o n constant
on 2
Chapter X I , [O,
n)
a normal
[O,hl)
2, E x .
2 , page 1 4 4 ) .
from
[O,hl)
2, Ex.
into x
<
IR
is
0)
7 , page 8 1 ) .
c o u n t a b l y compact (Dugundji,
1, page 2 2 8 ) .
i s pseudocompact.
It follows t h a t
Moreover,
t h a t [O,n) i s s e q u e n t i a l l y compact: l e m E ( e ) , page 1 6 3 ) . The s p a c e [ O , n )
T -space 1
a r e c o m p l e t e l y normal
"tailvv [p,n) = {x : p
(Dugundji, Chapter 111,
The subspace
2 , page 1 6 2 ) .
i t can be shown ( K e l l e y , Prob-
i s not compact b e c a u s e i t f a i l s t o
be a c l o s e d s u b s e t of t h e compact Hausdorff s p a c e [O,n].
T h e r e f o r e , by 7 . 1 4 and ( 4 ) above,
n o t a Hewitt-Nachbin space.
Therefore,
[O,n)
is
qeneral,
69
NETS AND HEWITT-NACHBIN COMPLETENESS
s e q u e n t i a l compactness does not imply H e w i t t Nachbin completeness. NETS AND HEWITT-NACHBIN COMPLETENESS
For our f i n a l development i n t h i s s e c t i o n we would l i k e t o c o n s i d e r the notion of Hewitt-Nachbin completeness from t h e p o i n t of view of n e t s .
I n o r d e r t o b r i n g t h i s development i n -
t o focus i t w i l l f i r s t be necessary t o p r e s e n t an i n v e s t i g a t i o n of t h e a p p r o p r i a t e c l a s s e s of n e t s t h a t provide the corr e c t connection with zero- s e t f i l t e r s p o s s e s s i n g the countable intersection property.
The main r e s u l t 7 . 2 4 then r e p h r a s e s
Theorem 7 . 1 0 i n t e r m s o f t h e s e c l a s s e s of n e t s .
The r e a d e r
who s o d e s i r e s may omit t h i s m a t e r i a l without d e s t r o y i n g the c o n t i n u i t y of the development i n t h i s book and proceed d i r e c t l y t o Section 8 . I t is w e l l known t h a t the theory of n e t s on a topological space i s e q u i v a l e n t t o the theory o f Bourbaki f i l t e r s on t h a t space ( s e e , f o r example R . B a r t l e ' s 1955 p a p e r ) .
I n the i n t e r -
e s t of completeness we w i l l include a s h o r t summary of t h a t r e l a t i o n s h i p i n t h e p r e s e n t s e c t i o n and we w i l l i n c l u d e p r o o f s of some a d d i t i o n a l r e s u l t s which do not appear i n B a r t l e ' s 1955 paper. 7.16
2.
order
ED
A
is a s e t
and
2
p.
a t o p o l o g i c a l space
Furthermore, 1 (a') v
of a n e t a function
x
in 1
in cp
is a directed set
D
If
X.
aED
X
and i f
set
[PED :
i s a mapping from a p i s a n e t from p
(a) by
pa.
w i l l denote t h e s e t ( 1P : B E D and P 2 a]. Dv i s s a i d t o be a subnet with d i r e c t e d s e t D i n case there e x i s t s
with directed s e t
x
from
condition s : (1) For a l l
into
Dv
a
( i i ) For each
that
thera e x i s t s a
E D
If
with a p a r t i a l
D
w i l l denote the r e s i d u a l
into
D
y
a,P
X, then i t i s customary t o denote
into
A net
2 a a+
net i n
directed s e t D
y
then
aED,
2 a).
set
such t h a t f o r each p a i r
satisfying
and i f P
A directed
DEFINITION.
cp(P')
Dv,
E
* 2 P E
D
D
w
P
s a t i s f y i n g t h e following
va - F ~ ( ~ and ) ,
t h e r e e x i s t s an a whenever P ' 2 a.
E Dv
such
70
SPACES AND CONVERGENCE
HEWITT-NACHBIN
Now,
let
be a n e t i n
P
X
and l e t
IB(b) d e n o t e t h e
f a m i l y I b ( a + ) : a E D ) . I t is e a s y t o v e r i f y from t h e d e f i n i t i o n s t h a t a ( & ) i s a b a s e f o r a (Bourbaki) f i l t e r on X. We
a ( ~ ) On .
w i l l denote t h a t f i l t e r by
B
that
~ ( a =)
[ ( x , ~ ):
B c BI.
case
set.
XCB
Then
and
D('A)
into
The above correspondence b e t w e e n n e t s i n b a k i f i l t e r s on
is a filter ----cf(p(63))
7.17
is
and
b a s e on
then
X,
!l3(2(63))= B,
defined
X
and Bour-
X
If
Let X,
X
A c X.
a r e s a i d t o be e q u i v a l e n t i n c a s e
i s s a i d t o be e v e n t u a l l y
63.
be any t o p o l o g i c a l s p a c e , l e t
and l e t
& I
The n e t s
p
and
v
The n e t
p
p
a ( ~= )a ( v ) .
i n case
A
i3
and moreover
p r e c i s e l y t h e f i l t e r q e n e r a t e d b~
be n e t s i n
v
(x' ,B' ) i n
i s one-one i n t h e f o l l o w i n g sense:
X
DEFINITION.
2
B E B ] , and d e f i n e ( x , B )
2 ( d ) from i s a n e t i n X.
= x
Define
X.
D(B) is e a s i l y verified t o be a directed
Moreover, t h e mapping
%(a)(x,B)
by
t h e o t h e r hand, suppose
i s a b a s e f o r a (Bourbaki) f i l t e r on
p(a+) c A
f o r some
i n the d i r e c t e d set D. The n e t b i s s a i d t o b e freq u e n t l y in A i n c a s e f o r each a E D t h e r e e x i s t s a B E D
a
satisfying
p
2
a
and
wLp
u n i v e r s a l i n c a s e f o r each B
or
is eventually i n
p
p.
The n e t
A.
B
C
X
X\B.
EX
if p I n such a c a s e
verqe t o the point borhood of
E
i s s a i d t o be
y
either
w
is eventually i n
The n e t
p
is s a i d to
i s e v e n t u a l l y i n e v e r y neiqhp
i s c a l l e d a l i m i t p o i n t of
i s f r e q u e n t l y i n e v e r y neighborhood o f i s c a l l e d a c l u s t e r p o i n t of p .
P.
If
con-
p
p
then
p
I t i s c l e a r t h a t every l i m i t p o i n t of a n e t is a l s o a c l u s t e r p o i n t . Easy examples show t h a t t h e c o n v e r s e i s n o t
The n e t 1 i s s a i d t o b e c l u s t e r a b l e i f each of i t s cluster points is also a l i m i t point. ( T h i s i s S. Ciampa's true.
n o t i o n o f "maximal n e t " g i v e n i n h i s 1969 p a p e r . )
(1) I t can now be shown t h a t f o r any n e t 14 % ( P e ( k ) ) and I I ( ~ ( P a r )e )e q u i v a l e n t . I n f a c t , each i s e q u i v a l e n t t o t h e n e t P . 7.18 in
REMARKS.
X,
the n e t s
(2)
If
v
i s a subnet of
w , then
a(v)
3 iJ((4).
NETS AND HEWITT-NACHBIN COMPLETENESS
71
The n e x t r e s u l t i s fundamental i n e s t a b l i s h i n g t h e r e l a t i o n s h i p between t h e convergence p r o p e r t i e s of t h e n e t s and The r e s u l t and i t s proof may be found
t h o s e of t h e f i l t e r s .
i n t h e B a r t l e 1955 paper. 7.19 p
L2t
THEOREM ( B a r t l e ) ,
be a n e t i n
and l e t
X,
--
X
3
be any t o p o l o q i c a l s p a c e , l e t Bourbaki f i l t e r on X .
&a
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 t r u e : The n e t --
F
is
u n i v e r s a l i f and o n l y i f
3
i s an u l t r a f i l t e r i f and o n l y i f
a(p)
i s an
u ltrafilter. The f i l t e r --
% ( a ) is 5
The n e t -A
universal
p
is
frequently
meets e v e r y member o f
A subset -~
only i f The n e t
A c X
&I A f a ( ~ ,.)
~r,
-
i s eventually
&I
REMARKS.
i f and --
3
only i f i f and
A.
A c X
if and -
only i f
cf(b).
The s u b s e t A c X b e l o n g s -% ( a ) is e v e n t u a l l y in A . 7.20
c X
m e e t s e v e r y member of
T(S) is f r e q u e n t l y &
belongs
A
a.
5
i f and o n l y i f
(1) Because of t h e p r e v i o u s theorem i t i s
e v i d e n t t h a t t h e t h e o r y of convergence of n e t s i n a t o p o l o g i c a l space i s e q u i v a l e n t t o t h e t h e o r y o f convergence of
I n p a r t i c u l a r , a n e t p is c l u s t e r a b l e i f and o n l y i f 3 ( p ! i s c l u s t e r a b l e : i n o t h e r words, i n c a s e F converges t o each of i t s c l u s t e r p o i n t s . ( 2 ) I t i s e a s y t o v e r i f y t h a t e q u i v a l e n t n e t s have t h e same c l u s t e r p o i n t s and t h e same l i m i t p o i n t s . More p r e c i s e l y , i f F and v a r e e q u i v a l e n t n e t s , then p c X i s a c l u s t e r p o i n t ( l i m i t p o i n t ) of p i f and o n l y i f i t i s a c l u s t e r p o i n t ( l i m i t p o i n t ) of v. f i l t e r s i n t h a t space.
-
W e should l i k e t o conclude o u r b r i e f summary o f n e t s by
e s t a b l i s h i n g t h e r e l a t i o n s h i p which e x i s t s between c e r t a i n
nets in
X
and z e r o - s e t u l t r a f i l t e r s on
X
t h a t have t h e
countable i n t e r s e c t i o n property. 7.21
DEFINITION.
A net
F = (b,
: atD)
i n a topological
72
SPACES AND CONVERGENCE
HEWITT-NACHBIN
space
is said t o be
X
Z - u n i v e r s a l i f f o r each
w i t h non-empty i n t e r i o r , e i t h e r
Z
there e x i s t s a Z
C
f o r the n e t
2
ai
7.22
&
h a s non-empty i n t e r i o r , The d i r e c t e d s e t
2.
t h e r e e x i s t s some
D
D
with
aED
Z - u l t r a f i l t e r s on a Tycho-
The f o l l o w i n g r e s u l t r e l a t e s
x
or
E
icm.
for a l l
n o f f space
Z(X)
E
i s s a i d t o b e s e q u e n t i a l l y bounded i f f o r each
k
sequence ( a i : i c l N ) i n
a
Z
Z ( X ) such t h a t
E
is eventually i n
and
X\E,
E
is eventually i n
p
Z-universal n e t s on
to
Let
THEOREM.
X.
2 Tychonoff s p a c e .
X
Then t h e follow-
statements a r e t r u e :
(1)
If
L
&a
Z-ultrafilter
on
w i t h t h e count-
X
9(L)
a b l e i n t e r s e c t i o n p r o p e r t y , then
is 2
Z-
u n i v e r s a l n e t whose u n d e r l y i n q d i r e c t e d s e t i s s e q u e n t i a l l y bounded.
If
(2)
is a
y
Z-universal n e t i n
whose under-
X
l y i n g d i r e c t e d s e t i s s e q u e n t i a l l y bounded, t h e n there e x i s t s 2 --
Z-ultrafilter
Lb
a b l e i n t e r s e c t i o n p r o p e r t y such
-lent to Proof.
is a
(1) L e t
Cn(LcI). E
E
Z ( X ) have non-empty i n t e r i o r .
Z-ultrafilter, either
EEL
If
E E L , then
(x,E)
(y,U) E D ( 8 ) and if (y,U)
E
2
and
D(b) f o r some
XEX.
Z c
( x , E ) , then
T h e r e f o r e , 2(L) i s e v e n t u a l l y i n
E.
Since Z(X)
Z E
ZcL
L
by 6 . 8
Hence,
if
%(L) (y,U) = y ~ c u E.
On t h e o t h e r hand, i f
P(L) i s e v e n t u a l l y i n Z by a s i m i l a r argument. %(L) i s Z - u n i v e r s a l .
then
ZEL
or there e x i s t s
x’$
w i t h non-empty i n t e r i o r such t h a t (3).
w i t h t h e count--i s equivathat y
CI
This proves t h a t
Now, suppose t h a t ( (xi,Ui) D(d) .
By assumption
e r t y so t h a t
6.14. (y,
i s a sequence i n
L& h a s t h e c o u n t a b l e i n t e r s e c t i o n p r o p
i s c l o s e d under c o u n t a b l e i n t e r s e c t i o n s by
I t follows t h a t t h e r e e x i s t s a p o i n t
n
E D(%), and c l e a r l y
Ui)
( y , fI
y E
ui) 2 (xi,ui)
n
Then
Ui.
for a l l
T h e r e f o r e , D ( b ) i s s e q u e n t i a l l y bounded.
iEN.
(2)
L
: iElN )
3 ( ~b)e t h e f i l t e r g e n e r a t e d by t h e g i v e n n e t
Let
Since
X
E
a ( y ) , the collection
Lw = ( F E ;4(k)
: F E
1.
Z(X)) is
NETS AND HEWITT-NACHBIN COMPLETENESS
a
Z-filter.
p
is eventually i n
E c
Now l e t
E
Z ( X ) have non-empty
E
i s eventually i n
Z
is
14
Therefore, a s
E
L&
F
k
then
X\E,
C
p (a+) C E
then
E,
3 ( ~which ) implies t h a t
.
73
interior.
f o r some
Hence
acD.
On t h e o t h e r hand, i f
L
Z F
IL
by t h e same argument.
2-universal i t follows t h a t
It
Z - u l t r a f i l t e r by 6 . 8 ( 3 ) . Once i t i s shown t h a t
If
is a
w
c f ( k ) i s c l o s e d under c o u n t a b l e
i n t e r s e c t i o n s i t i s e a s y t o e s t a b l i s h t h a t t h e same h o l d s t r u e
Ik
for
since
w
i s c l o s e d under c o u n t a b l e i n t e r s e c t i o n s .
Z(X)
11 LI
By 6 . 1 4 i t w i l l f o l l o w t h a t tion property.
h a s the countable intersec-
To t h i s end, suppose t h a t { p ( n i + ) : ai
i c I N 1 is a countable c o l l e c t i o n i n
e x i s t s some
. a
. a
f o r which
D
t
(ao+) c n (@(ai+): i c I N completes t h e p r o o f .
u
:, and
ai
D,
E
By assumption t h e r e
B(p).
for a l l
.
itN
the r e s u l t follows.
Thus
This
W e now f o c u s our a t t e n t i o n on t h e c h a r a c t e r i z a t i o n of
Hewitt-Nachbin c o m p l e t e n e s s b y way of
Z-universal n e t s .
The
f o l l o w i n g lemma w i l l b e needed. 7.23
Z - u n i v e r s a l n e t i n a Tychonoff s p a c e
Every
LEMMA.
X
is clusterable. Proof.
If
p oi nt of N(p) of
p.
2 - u n i v e r s a l n e t and i f
is a
)I
w,
then
Moreover, a s
is a cluster
i s a Tychonoff s p a c e , t h e r e
X
e x i s t s a z e r o - s e t neighborhood N(p) by 3 . 6 ( 3 ) .
p
i s f r e q u e n t l y i n e v e r y neighborhood
p
Z
of
I t follows t h a t the
p
satisfying
2-ultrafilter
p E Z c Li
as
P
c o n s t r u c t e d i n t h e proof o f 7 . 2 2 ( 2 ) h a s t h e p r o p e r t y t h a t e a c h
U
E
that
L
meets Z
k
L
k
by 7 . 1 9 ( 3 ) .
converges t o
7 . 2 0 ( 2 ) and 7 . 2 2 ( 2 ) .
p.
Hence, Z
E
Therefore, k
I t follows t h a t
p
L
k
which i m p l i e s
converges t o
p
by
is clusterable.
The f o l l o w i n g r e s u l t i s a r e p h r a s i n g of t h e c h a r a c t e r i -
-
z a t i o n of Hewitt-Nachbin c o m p l e t e n e s s g i v e n i n 7 . 1 0 i n terms of n e t s .
*
7.24
THEOREM.
statements
If
Tychonoff s p a c e , t h e n t h e follow-
X
equivalent:
(1) The s p a c e
X
i s Hewitt-Nachbin complete.
HEWITT-NACHBIN SPACES AND CONVERGENCE
74
Z-universal n e t i n
Every
(2)
whose u n d e r l y i n g
X
d i r e c t e d s e t i s s e q u e n t i a l l y bounded c o n v e r g e s . (3)
Z-universal n e t i n
Every
set
directed
whose u n d e r l y i n q
X
i s s e q u e n t i a l l y bounded h a s a c l u s t e r -
-
a b l e converqent s u b n e t .
(1) i m p l i e s ( 2 ) :
Proof.
suppose t h a t
n e t a s given i n s t a t e m e n t ( 2 ) ultrafilter
L
IA
i s equivalent t o
complete, L
able.
Z-
Since
9(Lkp).
X
i s Hewitt-Nachbin Hence, fl(LCI)
c o n v e r g e s by 7 . 2 0 ( 1 ) and 7 . 2 0 ( 2 ) .
p
implies (3) :
(2)
By 7 . 2 2 ( 2 ) t h e r e e x i s t s a
,
i s f i x e d and t h e r e f o r e c o n v e r g e s .
CI
and t h e r e f o r e
Z-universal
w i t h t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y such )I
that
is a
p
Z-universal n e t i s c l u s t e r -
By 7 . 2 3 e v e r y
Hence ( 2 ) y i e l d s t h a t i t h a s a c l u s t e r a b l e c o n v e r g e n t
s u b n e t , namely t h e n e t i t s e l f .
5
( 3 ) implies (1): I f
is a
2 - u l t r a f i l t e r with t h e count-
%(a) is
a b l e i n t e r s e c t i o n p r o p e r t y , then
a
Z-universal n e t
whose u n d e r l y i n g d i r e c t e d s e t i s s e q u e n t i a l l y bounded by 7 . 2 2 ( 1 ) . Hence, by ( 3 )
Now,
w.
S(a(3))=
the case that
a ( 3 ) h a s a c l u s t e r a b l e convergent subnet 5
and s i n c e
v
3 c a ( v ) by 7 . 1 8 ( 2 ) .
i s a subnet of
F i n a l l y , by 7 . 2 0 ( 1 )
3 ( v ) is a c l u s t e r a b l e convergent f i l t e r .
Therefore, 3
I t f o l l o w s from 6 . 1 2 ( 2 ) t h a t
cluster point.
a ( 5 ) i t is
5
has a
converges
completing t h e p r o o f . S e c t i o n 8:
C h a r a c t e r i z a t i o n s and P r o p e r t i e s o f Hewitt-Nachbin Spaces
tion cl
I n Chapter 1 w e c o n s t r u c t e d t h e Hewitt-Nachbin compleux o f a Tychonoff s p a c e X a s a c l o s e d subspace
~ ( x of )
a n embedding o f
t h e proof of t h e
i t was e s t a b l i s h e d t h a t
-
a r e isomorphic and t h a t s p a c e i n which
X
i n a p r o d u c t o f r e a l l i n e s (see
E - C o m p a c t i f i c a t i o n Theorem 4 . 3 ) .
X
the uX
alqebraic rinqs
C(X)
Moreover,
and
C(uX)
i s t h e unique Hewitt-Nachbin
i s d e n s e and
C-embedded.
I n the present
s e c t i o n we w i l l d i s c u s s a d d i t i o n a l p r o p e r t i e s of
uX, e s t a b -
l i s h s e v e r a l i m p o r t a n t c h a r a c t e r i z a t i o n s of Hewitt-Nachbin completeness,
and i n v e s t i g a t e numerous t o p o l o g i c a l p r o p e r t i e s
a s s o c i a t e d w i t h Hewitt-Nachbin s p a c e s .
To b e g i n w e o b s e r v e t h a t i t i s u n n e c e s s a r y t o d i s t i n -
SPACES
PROPERTIES OF HEWITT-NACHBIN
g u i s h between homeomorphic c o p i e s o f reason f o r ambiguity. morphic t o
For suppose t h a t t h e s p a c e
homeomorphic t o a d e n s e subspace
i t i s immediate t h a t for
f
E
C ( X ) t h e r e e x i s t s an e x t e n s i o n
*
so t h a t
$(X) of f
X
Y . Moreover,
C(+(X))
*
E
.
Hence,
C ( i r X ) by 4.3(1)
Y. I n
f o h i s an e x t e n s i o n o f f t o t h e space o t h e r words, t h e f o l l o w i n g diagram i s commutative:
h
Y
T h e r e f o r e , w e may c o n s i d e r
X
a s a d e n s e and
C-embedded sub-
Y. o n t h e o t h e r hand, i f
s p a c e of t h e s p a c e
is
i t follows t h a t
UX
i s isomorphic t o
C(X)
i s homeo-
Y
Since
h.
o ( X ) of
i s homeomorphic t o a d e n s e subspace
X
unless there i s a
UX
under t h e homeomorphism
LJX
75
C-embedded i n t h e Hewitt-Nachbin s p a c e
i s d e n s e and
X
Y , then
Y
i s homeo-
uX. Thus w e need n o t d i s t i n g u i s h between homeomorphic c o p i e s o f UX a s c l a i m e d . Moreover, w e w i l l c o n s i d e r morphic t o
X
of
a s a subspace V
and, s i m i l a r l y ,
UX
--
Stone- Cech compactif i c a t i o n
8.1
If
REMARK.
d e n s e and fication
i s a Tychonoff s p a c e i n which
T
C-embedded,
X
follows t h a t s p a c e i n which
*
then
X
C -embedded
is
Since
PT.
PX
*
C -embedded
is
in
X T.
is It
V
i n t h e Stone-Cech compacti-
i s t h e unique compact Hausdorff
i s dense and
X
subspace o f t h e
PX.
*
C -embedded i t f o l l o w s t h a t
PX = P T ( w e a r e i d e n t i f y i n g t h e homeomorphic c o p i e s h e r e ) .
Hence, X c T t i o n UX
-
C
PX.
&
compactification
UX
I n p a r t i c u l a r , t h e Hewitt-Nachbin compleV r e g a r d e d a s a subspace of t h e Stone-cech
PX.
(Again, w e d o n o t d i s t i n g u i s h between
and i t s homeomorphic copy i n
ax.)
e s t a b l i s h e s t h e p r e c i s e manner i n which (up t o homeomorphism). Jerison (8.5) .
The f o l l o w i n g r e s u l t uX
is related t o
T h i s r e s u l t i s proved i n Gillman and
px
76
HEWITT-NACHBIN
SPACES AND CONVERGENCE
THEOREM (Gillman and J e r i s o n )
8.2
.
UX
(1) The Hewitt-Nachbin completion
subspace
of
pX
i n which
C-embedded.
UX
Hewitt-Nachbin subspace between Proof.
(1) Suppose t h a t
is
C-embedded.
the case t h a t that
Since
i s d e n s e and
X
(2)
C-embedded i n
Y
bedded i n
Then
Y = uX.
so t h a t
Y.
X
is the
uX
C-em-
X c Y c UX c pX.
i s a Hewitt-Nachbin
Y X
i s d e n s e and
C-em-
This concludes t h e p r o o f .
Now w e have a l r e a d y d e f i n e d t h e c o n c e p t of a on
it is
I t follows
i s d e n s e and
which i m p l i e s t h a t
X c Y c uX.
i n which
pX
However,
uY. X
PX.
pX
is a subspace o f
Suppose t o t h e c o n t r a r y t h a t
space s a t i s f y i n g
and
i s a d e n s e s u b s e t of
C-embedded i n
H e n c e , UX = uY
X
X
unique Hewitt-Nachbin s p a c e i n which bedded.
is the smallest
Y
i s d e n s e and
X
is the larqest
X
The Hewitt-Nachbin completion
(2)
X
& 2 Tychonoff
X
Then t h e f o l l o w i n q s t a t e m e n t s a r e t r u e :
space.
Z-filter
converging t o a p o i n t o f
s e t of a Tychonoff s p a c e
X. When X i s a d e n s e subw e would l i k e t o b e a b l e t o d i s -
T
X converging t o The m o t i v a t i o n f o r t h i s comes from t h e
c u s s a n a l o g o u s l y t h e n o t i o n o f a f i l t e r on
a point
p
in
T.
following question:
How d o e s one c o n s t r u c t a s p a c e
X
t a i n i n g a given space
T
con-
d e n s e l y such t h a t c e r t a i n c l a s s e s of
f i l t e r s on X which do n o t converge w i l l converge t o p o i n t s added i n the new s p a c e ? 8.3
DEFINITION.
then a
Z-filter
If
3
X
on
i s a subset of a t o p o l o g i c a l s p a c e X
converges t o a p o i n t
pcT ( o r
T,
p
3 ) i f e v e r y open ( i n T ) s e t c o n t a i n i n g p c o n t a i n s a member Z ( f ) E 5. The p o i n t P E T i s a c l u s t e r p o i n t of 3 i f e v e r y open ( i n T ) s e t c o n t a i n i n g p h a s a i s a l i m i t p o i n t of
non-empty i n t e r s e c t i o n w i t h e v e r y m e m b e r of
3.
L e t u s now c o n s i d e r what c o l l e c t i o n s o f c o n t i n u o u s func-
t i o n s may be extended i n a c o n t i n u o u s f a s h i o n from
ux. What i s v e r y h e l p f u l h e r e i s t h e f i l t e r p r e v i o u s l y by t h e s h a r p mapping (see 6 . 1 9 ) . L e t u s suppose t h a t
noff space
T
and t h a t
X
3
into
X
f 8 (3) d e f i n e d
i s a dense subspace o f a Tycho-
i s a prime
Z - f i l t e r on
X
with
PROPERTIES OF HEWITT-NACHBIN SPACES
the countable i n t e r s e c t i o n property. f u n c t i o n from
If
i s a continuous
f
Y, t h e n by 6 . 1 9
i n t o a Hewitt-Nachbin s p a c e
X
77
and 7 . 1 3 , f # (3) h a s a l i m i t p o i n t y f E Y and yf E n f # ( 3 ) . Now i f 3 a l s o happens t o b e a unique such 2 - f i l t e r conprT, then w e may d e f i n e a c o n t i n u o u s
verging t o t he point extension
f*
of
* I f
t o t h e subspace f(x),
f
(x) =
yf,
if
XEX
if
x
=
p.
I n t h i s way w e may show t h a t t h e f u n c t i o n
f
o u s l y extended t o a f u n c t i o n mapping every p o i n t
pcT
U ( p ) by
T* = X
can b e c o n t i n u -
Y
into
T
whenever
i s t h e l i m i t of a unique such
Z-filter
converging t o i t . That i s , w e
W e f o r m u l a t e t h i s r e s u l t more f o r m a l l y .
have shown t h a t s t a t e m e n t ( 5 ) i m p l i e s s t a t e m e n t (1) i n t h e f o l l o w i n g theorem. 8.4
THEOREM (Gillman and J e r i s o n ) .
Tychonoff s p a c e
T.
The
Hewitt-Nachbin s p a c e Y c o n t i n u o u s mapping from (2)
The s p a c e
If 2
X
be d e n s e i n t h e
X
are
followinq statements
(1) Every c o n t i n u o u s mappinq
(3)
Let
is
7
from
equivalent:
i n t o any
X
h a s an e x t e n s i o n T i n t o Y.
C-embedded
T.
countable c o l l e c t i o n of zero-sets i n
&
empty i n t e r s e c t i o n , then t h e i r c l o s u r e s empty i n t e r s e c t i o n . clT (5)
n (zn
: nEm ) =
n (clTzn
:
ncm )
have
T
zn & x,
For a n y c o u n t a b l e f a m i l y o f z e r o - s e t s
(4)
X
. Z-
Every p o i n t of
T
i s t h e l i m i t of a unique
on
X
with t h e countable i n t e r s e c t i o n
ultrafilter property. Proof.
(1) i m p l i e s ( 2 ) :
Nachbin s p a c e ,
( 2 ) implies ( 4 ) :
Z ( f T ) where
T f
Since the real l i n e i s a H e w i t t -
( 2 ) i s j u s t a s p e c i a l case of
If E
X
is
C-embedded i n
C ( T ) i s t h e e x t e n s i o n of
(1).
then
T,
f
E
C (X)
clTZ(f) =
.
If
ncm’ ] i s a c o u n t a b l e c o l l e c t i o n of z e r o - s e t s i n 1 ( r e s p e c t i v e l y , T ) then f ( f n A 1) i s a c o n t i n u o u s
r Z ( f -I,)
:
=c2”
x
HEWITT-NACHBIN SPACES AND CONVERGENCE
78
function i n ncm j .
X ( r e s p e c t i v e l y , T ) f o r which
n
Z(f) =
(Z(fn)
:
Thus, 00
ciT
n
z(fn) = ciTz(f)
00
00
T
z(f ) =
=
n=l
n
z ( f nT ) = n c i z ( f ) n n= 1 n= 1 T
where t h e p e n u l t i m a t e e q u a l i t y h o l d s s i n c e t h e clsumll d e f i n e d above f o r (now t h e ) !Z ( f n T ) ) a g r e e s w i t h f on t h e dense subspace X of T . I t i s obvious t h a t s t a t e m e n t ( 4 ) i m p l i e s statement ( 3 ) . (3) i m p l i e s
(4) :
zero-sets i n
X
a
f
E
:
n c m ) i s a c o u n t a b l e c o l l e c t i o n of
p
p cl n
I f fZn and i f
[Zn
1,
: nElN
then t h e r e i s
co
c(x)
such t h a t
p
F
n zn r l
c l T z ( f ) and
However, s t a t e m e n t ( 3 ) would then imply t h a t
p
0.
z(f) =
n= 1
00
p
fl
clTzn;
E
Z(X)
n= 1 t h a t i s , i t would imply s t a t e m e n t ( 3 ) implies ( 5 ) :
p
i
clTZ] i s a
If
(4).
3 = (2
PET, then t h e f a m i l y
Z - u l t r a f i l t e r on
X
t h a t converges t o
Under t h e assumption of s t a t e m e n t ( 3 ) i t i s a with the countable i n t e r s e c t i o n p r o p e r t y . u l t r a f i l t e r F on a l s o converging t o
z(f) p
E
3
with
:
p.
Z-ultrafilter
If there is a
Z-
with t h e countable i n t e r s e c t i o n property p, t h e n t h e r e must e x i s t Z ( g ) E G and
X
Z(g)
n
Z ( f ) = @.
Statement ( 3 ) implies t h a t
p c l T Z ( g ) c o n t r a d i c t i n g t h e convergence o f
G
to
p.
Upon i n t e r p r e t i n g 8 . 4 w e can add t o t h e r e s u l t s a l r e a d y obtained f o r 8.5
uX
by t h e
E - C o m p a c t i f i c a t i o n Theorem 4 . 3 .
THEOREM (Gillman and J e r i s o n ) .
Every Tychonoff s p a c e
has 2 Hewitt-Nachbin completion uX, c o n t a i n e d t h e followinq e q u i v a l e n t p r o p e r t i e s : (1) Every c o n t i n u o u s mapping
Hewitt-Nachbin s p a c e from
ux
(2)
Every f u n c t i o n
(3)
If 2
function
fv
7
from
pX, X
with
i n t o any
h a s a continuous extension
Y
Y.
into
C ( X ) h a s an e x t e n s i o n
f
to gi
C(UX).
countable c o l l e c t i o n o f z e r o - s e t s i n
empty i n t e r s e c t i o n ,
have empty
X
then t h e i r c l o s u r e s
intersection.
&
X
has UX
PROPERTIES O F HEWITT-NACHBIN
(4)
SPACES
79
zn
For a n y c o u n t a b l e f a m i l y of z e r o - s e t s
clvx
n
i z n : nc I N ) =
ultrafilter
x,
iclbxzn : ncm 2 .
UX i s t h e l i m i t of a unique ZX with the countable i n t e r s e c t i o n
Every p o i n t o f
(5)
n
on
property. Furthermore,
t h e space
i f 2Hewitt-Nachbin
i s unique, -
UX
space
T
i n the following s e n s e :
containing
densely s a t i s f i e s
X
any one of t h e l i s t e d c o n d i t i o n s , t h e n t h e r e e x i s t s a homeomorphism
of
onto
uX
T
t h a t leaves
pointwise fixed.
X
W e remark t h a t Gillman and J e r i s o n prove a n a l o g o u s reY-
s u l t s t o 8 . 4 and 8 . 5 i n t h e c a s e o f
C -embedding and compact
Thus, i n 8 . 4 f o r example, e v e r y o c c u r r e n c e o f " H e w i t t -
spaces.
Nachbin space" would b e r e p l a c e d by Ilcompact s p a c e , I'
em-
IIC-
6
bedding" i s r e p l a c e d by " C -embedding,
"countable c o l l e c t i o n s "
by " f i n i t e c o l l e c t i o n s , " and " Z - u l t r a f i l t e r w i t h t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y " by simply " Z - u l t r a f i l t e r .
I n the
(I
analogous c a s e 8 . 5 , " t h e Hewitt-Nachbin c o m p l e t i o n V
be-
uXrt
pX," and s o f o r t h .
comes t h e IfStone-Cech c o m p a c t i f i c a t i o n
(See 6 . 4 and 6 . 5 of t h e Gillman and J e r i s o n t e x t f o r t h e p r e -
cise s t a t e m e n t s concerning t h e s e r e s u l t s .
W e omit r e p e a t i n g
them h e r e s i n c e t h e y a r e s o l i k e t h e r e s u l t s s t a t e d i n 8 . 4 and 8 . 5 above, and we w i l l simply r e f e r e n c e Gillman and J e r i s o n . )
ux
W e should now l i k e t o employ t h e p r o p e r t i e s o f
in
o r d e r t o e s t a b l i s h s e v e r a l i m p o r t a n t and u s e f u l c h a r a c t e r i z a t i o n s of Hewitt-Nachbin c o m p l e t e n e s s .
The n e x t c o n c e p t o r i g -
i n a t e d i n t h e 1 9 5 7 A paper o f S . 'Mro'wka and t u r n s o u t t o b e v e r y u s e f u l t o o u r development. 8.6
pcG
G6-set
c l o s u r e of t o be
set i n
pcX
G -dense
in
6-X meets
i s s a i d t o be
G -closed
G -set 6
6--
G
in
if
X
such t h a t
6 . The G 6 - c l o s u r e of A i s t h e s e t of s a t i s f y i n g t h e c o n d i t i o n t h a t whenever G
containing A
b e an a r b i t r a r y t o p o l o g i c a l s p a c e .
there e x i s t s a
p#A
A II G =
and
a l l points
X
A c X
subset
f o r each p o i n t
a
Let
DEFINITION.
A non-empty
by
X
in
X A.
p, then if
G
n A # 6.
G -cl2.
6
W e denote the
The subspace
A
is
6is said
X = G - c 1 2 : i . e . , i f every 6
6-
HEWITT-NACHBIN SPACES AND CONVERGENCE
80
The terminology i n t h e above d e f i n i t i o n i s found i n t h e /
Mrowka used t h e t e r m "Q-closedI1 i n -
1972 paper o f R . B l a i r .
s t e a d of " G - c l o s e d . " ( I n t h e i r 1974 book, A l o and S h a p i r o 6 u s e t h e terminology l l r e a l c l o s e d . l l ) I t i s immediate from t h e A, A c G -cl? C cl?. 6 i s any open s e t c o n t a i n i n g p ,
above d e f i n i t i o n t h a t f o r e v e r y s e t For i f then
p r G6-clp
P
G
IR
closed i n 03
n
n= 1
# 6.
A
and
G
For example, t h c open i n t e r v a l ( 0 , l ) i s
because t h e
g i v e n by
G,-set
6-
G =
U
3 (1 - ,;1 7 ) c o n t a i n s 1, b u t
.~ -
n o t belong t o
G
-elm ( 0 , l ) .
n
G
6 I t follows t h a t ( 0 , l ) = G 6 - c l m
( 0 , l ) = @.
Similarly,
0
1 does
Hence
j!
G6-clm
(0,l).
(0,l) is G -closed i n IR. 6 The f o l l o w i n g i n t e r e s t i n g r e s u l t i s found i n t h e 1957A
paper by MroGka.
I t g e n e r a l i z e s t h e p r o p e r t y t h a t c l o s e d sub-
s p a c e s of Hewitt-Nachbin s p a c e s a r e Hewitt-Nachbin complete, and i t w i l l b e u s e f u l i n e s t a b l i s h i n g t h e many c h a r a c t e r i z a t i o n s of Hewitt-Nachbin completeness which a r e t o f o l l o w .
The
proof i s from B l a i r ' s 1964 N o t e s . THEOREM (Mrowka).
8.7
Nachbin s p a c e Proof. i :A
Let +
Every
i s Hewitt-Nachbin c o m p l e t e .
X
d e n o t e a G - c l o s e d s u b s e t of
A
6
d e n o t e t h e i n c l u s i o n mapping.
X
f i l t e r on
Z - f i l t e r on
s e c t i o n p r o p e r t y by 6 . 1 9 . p
n
F
nz.
P E
If
X
and l e t
3
is a
Z-ultra-
with the countable i n t e r s e c t i o n property,
A
i # (a) i s a prime point
G6-closed s u b s e t of a H e w i t t -
then
with the countable i n t e r -
X
H e n c e , by 7 . 1 3 ( 2 ) t h e r e e x i s t s a
i# (3). I t w i l l be shown t h a t
Suppose t o t h e c o n t r a r y t h a t
p#A.
PEA
and t h a t
Then, s i n c e
A
is
G - c l o s e d , t h e r e e x i s t s a G - s e t G = n [Oi : i c I N ] such t h a t 6 6 PEG and G n A = 6 . Moreover, b y t h e complete r e g u l a r i t y o f
and 3.6(3), f o r each
X
borhood
icm, p
E
n
6.12.
Zi 2
€
nA
Z ( X ) with =
6.
i ffi( a ) , i t i s
A)
E
5
E
Zi
C
the case t h a t
Then f o r e a c h
(zi n
p
W e c l a i m t h a t f o r some
Oi.
Then, s i n c e
For suppose o t h e r w i s e .
t h a t i s contained i n that
t h e r e e x i s t s a z e r o - s e t neigh-
icm
itm Zi
i ff (a) c o n v e r g e s t o
p
t h e r e e x i s t s a zero-set i n and hence
f o r each
iEIN
Zi
E
i ff
(a).
i n which c a s e
by
i ff (5)
I t follows
n icm
(zi n
A) =
81
PROPERTIES OF HENITT-NACHBIN SPACES
(
n
iim
Zi)
e r t y of G
n
n
i s non-empty by t h e c o u n t a b l e i n t e r s e c t i o n prop-
A
n
3. On t h e o t h e r hand,
c
Zi
i E 7N
A = @.
This i s a c o n t r a d i c t i o n .
n oi c
and
G
i EN
Therefore,
there e x i s t s
a z e r o - s e t neighborhood Z ' t Z ( X ) s u c h t h a t p E Z ' and 2' 9 A = @. Finally, since 2' i s a neighborhood of p, t h e convergence of i x (a) i m p l i e s t h a t Z ' E i # (3);whence ( Z ' n A ) c 3. T h i s i s i m p o s s i b l e s i n c e 3 i s a Z-filter. H e n c e , PEA. W e now c l a i m t h a t
Zt3
e x i s t s a member clxZ
n
A
p
n 3.
E
such t h a t
For i f n o t , t h e n t h e r e
p{Z.
Since
Z = cl Z =
A
ptA, i t i s t h e c a s e t h a t
and s i n c e
p
T h e r e f o r e , t h e r e e x i s t s a z e r o - s e t neighborhood satisfying Z' Z
F I ~ I
i'(3)
p
and
ZI
E
as before.
a.
(2' C A ) #
I t follows t h a t
n
Z'
Z =
a.
n
A)
Then (2'
j!
2'
clxZ.
Z(X)
t
I t follows t h a t t
3
implies t h a t
This i s a c o n t r a d i c t i o n so t h a t
p t
n
3.
i s a Hewitt-Nachbin s p a c e c o n c l u d i n g t h s
A
proof. I t w i l l b e shown i n 9 . 6 t h a t t h e Hewitt-Nachbin comple-
tion
VX
is the
G - c l o s u r e of t h e s p a c e
6 Cech c o m p a c t i f i c a t i o n V
i n i t s Stone-
X
pX.
The f o l l o w i n g theorem c h a r a c t e r i z e s Hewitt-Nachbin com-
px, i n t e r m s
p l e t e n e s s i n terms o f c o n t i n u o u s f u n c t i o n s on
of t h e G - c l o s u r e c o n c e p t , and i n t e r m s of s u b s p a c e s o f p X . 6 With e a c h c h a r a c t e r i z a t i o n w e i n d i c a t e t o whom i t i s due by an a p p r o p r i a t e r e f e r e n c e t o the b i b l i o g r a p h y . 8.8
THEOREM.
statements
If
X
d a Tychonoff
are e q u i v a l e n t :
space, then t h e followinq
(1) The s p a c e X i s a Hewitt-Nachbin s p a c e . V (2) ( K a t e t o v , 1951B). If Y 2 Tychonoff s p a c e i n which (3)
X
C-embedded,
po
(Mrdwka, 1 9 5 7 A ) . For e a c h p o i n t
-exists
a continuous f u n c t i o n
f(p ) = 0 (4)
i s d e n s e and
O"
and
f(p)
>
0
f
e x i s t s g function po-
E
f t C(X)
X = Y.
PX\x
there
C ( p x ) such t h a t
for a l l points
( K a t e t o v , 1951B). For each p o i n t extendable
E
then
~ E X .
po E pX\X
there
t h a t i s not continuously
82
SPACES AND CONVERGENCE
HEWITT-NACHBIN
(5)
(Mro/wka, 1957A). The s p a c e V
Stone- Cech -(6)
Gg-closed i n i t s
X
pX.
compac t i f i c a t i o n
(Mr&wka, 1957A). The s p a c e
is
X
G6-closed i n
some Hausdorff c o m p a c t i f i c a t i o n . (Wenjen, 1966). The s p a c e
(7)
of
i s a n intersection
X
X and c o n t a i n e d PX. a(Wenjen, 1966). There e x i s t s 2 compact Hausdorff
F -sets containing
(8)
space
that
B
of
intersection (9)
contains
F -sets i n
in
Y,
X
then
i s an i n t e r s e c t i o n
of
X
is an intersection
of
PX.
of
u-compact s u b s p a c e s
(1) i m p l i e s ( 2 ) : i s dense and
If
PX. i s dense and
X
C-embedded i n
vY.
thi! unique Hewitt-Nachbin s p a c e i n which embedded by 4 . 3 (3), i t f o l l o w s t h a t assumption
C-embedded i n Since
X c Y c uY = vX.
Y.
>
f(po)
Define t h e space
0.
f-
=
to
C(X)
.
ded i n
I t w i l l b e shown t h a t
Y.
Hence, l e t
and
by
X = VX
X = Y.
Y = X U (p,)
t a k e s t h e r e l a t i v e topology a s a subspace o f
i s dense i n
C-
By
( 2 ) i m p l i e s ( 3 ) : Suppose t h a t t h e r e e x i s t s a p o i n t Po such t h a t e v e r y f u n c t i o n f E C ( p X ) t h a t i s p o s i t i v e on satisfies
is
uX
i s d e n s e and
X
i s Hewitt-Nachbin complete s o t h a t
X
Hence
4.4.
X.
X
( F r o l f k , 1963). The s p a c e
(10)
i s an
X
containinq
B
a--
( F r o l f k , 1963). The s p a c e cozero-sets ---
Proof.
such t h a t
X
f
f A 0.
E
X
is
PX.
f = f
+ +
x
where
Clearly
Y
X
C-embedded i n
C ( X ) and d e f i n e t h e f u n c t i o n s
Then
PX\X
t
f+ = f V 0
and e a c h summand b e l o n g s
f-
I t s u f f i c e s t o show t h a t each summand i s
C-embed-
TO t h i s end, d e f i n e the f u n c t i o n
Y.
1
g=-
1
+
. f+
*
*
s i n c e X i s C -embedded i n P X , t h e r e e x i s t s a c o n t i n u o u s e x t e n s i o n gP : pX + IR such t h a t g P I X = g . Furthermore, gP i s p o s i t i v e on X so t h a t by o u r i n i t i a l assumption g P (p,) > 0. T h e r e f o r e , t h e func+ 1 tion f l = p - 1 i s a c o n t i n u o u s e x t e n s i o n of f + from Y Then
g
into
IR.
f-
from
belongs t o
C (X) and,
9
S i m i l a r l y , t h e r e exists a continuous extension of Y
into
IR.
However,
X
#
Y
which c o n t r a d i c t s ( 2 ) .
SPACES
PROPERTIES OF HEWITT-NACHBIN
( 3 ) implies ( 5 ) :
Let
>
f(p)
for a l l
0
Then t h e set ing the p o i n t t i v e on
pcX.
G = fl ( G n
po.
:
f
For each
n
Moreover, G
is
such t h a t
t C(pX)
n t m ) is a
Therefore, X
X.
px\x.
denote an a r b i t r a r y p o i n t i n
po
By ( 3 ) t h e r e e x i s t s a function
and
83
define
nEN
G -set i n
6
because
X = @
G -closed
in
6
The i m p l i c a t i o n s ( 2 ) implies ( 4 ) ,
f(po) = 0
pX
contain-
f
i s posi-
by d e f i n i t i o n .
PX
( 7 ) i m p l i e s ( 8 ) , and
( 5 ) implies ( 6 ) a r e t r i v i a l .
( 6 ) implies ( 1 ) : I f t h e space
d o r f f c o m p a c t i f i c a t i o n , then
X X
i n some Hausb i s Hewitt-Nachbin complete is
G -closed
by 8 . 7 . ( 4 ) implies ( 2 ) :
Suppose t h a t
the Tychonoff space by 8.2(1).
Y c uX
and a f u n c t i o n p
to
P
x
pX
n
CPX\G
P
: p
B
on
X.
Moreover,
pX,
f
E
G
Fu-set
pXYG
(5
Since
and moreover
px.
let
NOW,
X
C
po
f(po) = 0
pX
in
F
Z
P
in
n 2 = $5. Hence, x = n {pX\!Z, P s e c t i o n of c o z e r o - s e t s i n pX. If
and
f(p)
be a p o i n t i n
p
Let
X
of c o z e r o - s e t s i n
Fo-set i n
under
B
be an a r b i t r a r y p o i n t
i s a non-empty i n t e r s e c t i o n of
there e x i s t s a zero-set
(9) implies ( 3 ) :
x in-
FU
-sets
such t h a t
I t follows t h a t t h e r e e x i s t s a f u n c t i o n
F.
C ( p X ) such t h a t
( 3 ) implies ( 9 ) :
pX
ip of
i s t h e i d e n t i t y mapping
iplX
t h e r e e x i s t s a closed s e t and
6 by ( 5 ) .
X =
P
the i n v e r s e image of a
x
G -set
denote the i n c l u s i o n mapping from
i
Let
n
such t h a t
px\x).
E
F -set i n
~x\x.
po p( F
p
such t h a t t h e r e s t r i c t i o n
ip i s a
in in
Then t h e r e e x i s t s a
PX\X.
E
Then t h e r e e x i s t s a Stone e x t e n s i o n
B.
to
p
containing
(8) i m p l i e s ( 3 ) :
into
Let
i s a s u b s e t of the
X
=
t h a t i s n o t continuously extendable
by assumption.
in
Then
Then by 8 . 1 , X C Y c pX. Therefore, X # Y , t h e r e e x i s t s a p o i n t p E Y\X
Y.
If
f E C(X)
( 5 ) implies ( 7 ) : G
C-embedded i n
i s dense and
X
X =
n
> o
for a l l
Z ( R X ) such t h a t : p E
PEX.
Then by ( 3 )
pX\X.
p
E
zp
and
p X w ] which i s an i n t e r -
( a x \ z ( f a ) : a&)
pX, then f o r each p o i n t
is a n intersection p
E pX\X
it is
84
SPACES AND CONVERGENCE
HEWITT-NACHBIN
t h e case t h a t
p c z(f,)
Hence, t h e function
and
f = f
a
V
Z(f ) n X = f o r some acG. a i s the r e q u i r e d f u n c t i o n
0
satisfying ( 3 ) .
(lo): Each
equivalent
(7)
F -set i n 5
is
PX
5-compact
s i n c e i t i s a countable union of closed s u b s e t s of argument i s r e v e r s i b l e s i n c e each
is a
F -set. 0
8.9
REMARKS.
PX.
The
o-compact subspace of
pX
This concludes the proof of t h e theorem. (1) Statement ( 6 ) of t h e previous theorem a l s o
p o i n t s up t h e d i f f e r e n c e between Lindelof spaces and H e w i t t Nachbin spaces because i t can be shown t h a t 2 space i s Lindelof i f and only i f i t i s compactification.
G
- c l o s e d i n every Hausdorff
6--
f
This r e s u l t was proved by Mrowka (1958B,
( v i ) , page 8 4 ) . Theorem 8.8(10) a l s o y i e l d s the f a c t t h a t an i n t e r -
(2)
s e c t i o n of Lindelof spaces need n o t be L i n d e l 6 f . For l e t X be a Hewitt-Nachbin space t h a t f a i l s t o be Lindelof (an example of which was given i n 7 . 1 2 ) . Then X i s an i n t e r s e c t i o n of X
that
a-compact subspaces of
pX
by 8.8(10). I t follows
i s an i n t e r s e c t i o n of Lindelof subspaces of
However, i t was shown i n 4 . 2 ( 5 )
PX.
t h a t an a r b i t r a r y i n t e r s e c t i o n
of Hewitt-Nachbin spaces i s Hewitt-Nachbin complete. A number of
i n t e r e s t i n g questions r e l a t e d t o the H e w i t t -
Nachbin completion if
x
and
Y
vX
remain t o be answered.
For i n s t a n c e ,
a r e Tychonoff spaces, then i n what way i s
v ( X x Y) related t o
UX
x uY?
This q u e s t i o n , a s w e l l a s sev-
e r a l o t h e r s , w i l l r e c e i v e c o n s i d e r a b l e a t t e n t i o n i n S e c t i o n 11. We have a l r e a d y e s t a b l i s h e d a number o f t o p o l o g i c a l p r o p e r t i e s a s s o c i a t e d w i t h Hewitt-Nachbin s p a c e s . of these were e s t a b l i s h e d f o r t h e more g e n e r a l
Since many
E-COmpaCt
spaces t r e a t e d i n Chapter 1, w e w i l l c o l l e c t them t o g e t h e r h e r e i n t o a s i n g l e theorem f o r t h e s p e c i a l c a s e of HewittNachbin spaces. 8.10
THEOREM.
X
5 Tychonoff space.
Then the follow-
inq statements a r e t r u e : (1)
(Gillman and J e r i s o n , 1960).
If
empty family of Hewitt-Nachbin
(Ya : aEG) i s a nonsubspaces of X, then
PROPERTIES O F HEWITT-NACHBIN SPACES
85
of
Y = f? (Y : a c G ) i s a Hewitt-Nachbin subspace a (Gillman and J a r i s o n ,
If
1960).
X.
is a Hewitt-
X
Nachbin s p a c e , t h e n e v e r y c o z e r o - s e t i n
is
X
Hewitt-Nachbin c o m p l e t e .
If
(Gillman and J e r i s o n , 1960).
a Hewitt-
X
Nachbin s p a c e and i f each p o i n t o f then e v e r y -
subspace
of
is a
X
G
-set,
6
i s a Hewitt-Nachbin
X
space. (Katztov, 1 9 5 1 B ) .
If
i s Hewitt-Nachbin
X
p l e t e , then e v e r y c l o s e d subspace
of
X
e-
is Hewitt-
Nachbin complete. (Mrdwka, 1957A).
-
then e v e r y
If
i s Hewitt-Nachbin c o m p l e t e ,
X
of
G - c l o s e d subspace
6Nachbin s p a c e .
(Gillman and J e r i s o n , from
X
Nachbin subspace (Hewitt,
f
i n t o t h e space
Nachbin subspace 1948).
be a H e w i t t -
X
b e a c o n t i n u o u s mappinq
If
Y.
Y , then
of of
Tha
Let
1960).
Nachbin s p a c e and l e t
is a H e w i t t -
X
is a H e w i t t -
F
f-l(F) is a Hewitt-
X.
t o p o l o q i c a l p r o d u c t of H e w i t t -
Nachbin s p a c e s i s Hewitt-Nachbin c o m p l e t e . S t a t e m e n t s (l), ( 4 ) , (5), ( 6 ) and ( 7 ) have a l r e a d y
Proof.
been e s t a b l i s h e d .
W e w i l l o f f e r p r o o f s f o r ( 2 ) and
w e l l a s an a d d i t i o n a l proof of
(3) as
( 6 ) due t o R . B l a i r (1965)
because w e t h i n k t h e proof i s i n s t r u c t i v e . (6)
R e c a l l t h e d e f i n i t i o n and p r o p e r t i e s a s s o c i a t e d w i t h t h e
f#
mapping
on t h e c o l l e c t i o n
c o n t i n u o u s (see 6 . 1 9 ) .
Z ( Y ) whenever
Now, l e t
A = f-l(F), let
d e n o t e t h e i n c l u s i o n mapping, and l e t
T
Y
i : A
is
*
X
d e n o t e t h e restric-
f/A
f i l t e r on
A
w i t h t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y , then
i # (3) and
# 7 (3) a r e prime
satisfying
x
shown t h a t
XEA
E
A
into
+
t i o n mapping
section property.
from
: X
f
F.
hence
f (x) Z
n i# (a) and y and f l 3 # 6 .
# y.
and
Zl
Z-ultra-
Z - f i l t e r s w i t h t h e countable i n t e r -
Hence there e x i s t points E
xtX
n T # (3) by 7 . 1 3 .
Suppose t o t h e c o n t r a r y t h a t hoods
is a
If
x,dA.
Then
f
and
ycF
I t w i l l be
(x) # F
and
Therefore, t h e r e e x i s t z e r o - s e t neighbor-
in
Z ( Y ) with
f(x)
E
Z,
y
E
Z ' ,
and
86
HEWITT-NACHBIN
n
Z
z' n
that T
-1
@.
2' = (2')
n
SPACES AND CONVERGENCE
Now, t h e convergence of
6 T (a),
belongs t o
F
3.
belongs t o
Z
T
6 (3)
whence
T
n
f-'(Zl)
i t i s the case t h a t
A
which c a s e
n
Z
# $5.
Z'
implies
Hence, t h e
f-'(Z) Since
n
f-'(Z)
F) =
Z
E
x.
n
convergence of i ( a ) t o x implies t h a t i # (a) ; whence f-'(Z) n A belongs t o 3.
implies
(z'
f (x)
Furthermore,
f - l ( Z ) i s a z e r o - s e t neighborhood of
that
y
to
-1
belongs t o 7-l(Z1)
#
f-'(z')
n
A =
in
@
This c o n t r a d i c t i o n e s t a b l i s h e s t h a t
XEA.
n
x
Now, suppose t h a t
3.
Then t h e r e e x i s t s a
x { clxZ
such t h a t
xgZ.
clxZ fl A .
Hence, t h e r e e x i s t s a z e r o - s e t neighborhood
Z'
t
Z'
Z ( x ) with Z = @.
case t h a t
x.
I t follows t h a t
x
Z'
E
2'
n
A)
# @.
n
n
= Z'
clxZ =
@.
z ~ 3
Z = cl Z = A
I t follows t h a t
n
x
belongs t o
A
because i # ( a ) converges t o belongs t o 3 s o t h a t
i 6 (3)
belongs t o
Therefore, i - ' ( Z ' )
Z f l (Z'
2'
and
Moreover, s i n c e
since
Z'
it i s the
A
This c o n t r a d i c t i o n concludes t h e proof of
statement ( 6 ) .
(2)
Every c o z e r o - s e t
Since both
X
and
X\Z(f) i s of t h e form
f-'(IR\{O]).
a r e Hewitt-Nachbin
IR\[O]
spaces, the
r e s u l t follows from s t a t e m e n t ( 6 ) . (3)
Let [ p ) =
n
a singleton s e t i n 3 . 6 ( 3 ) f o r each that
p E Zn c
a zero-set i n
: nc7N
(Un
X.
i s open, d e n o t e
Un
By t h e complete r e g u l a r i t y of
X.
there e x i s t s a zero-set
nglN
un.
] where each
Hence,
(p] =
n
Zn =
n c IN
n un
nE IN
Zn -.
E
non-empty s u b s e t of follows from (1) t h a t
then
X, F
F =
ptX.
n
If
F
and
Z ( X ) such
so that [p] is
I t f o l l o w s from ( 2 ) t h a t t h e s e t
Hewitt-Nachbin complete f o r every
X
X\(p] i s
i s an a r b i t r a r y
( X \ ( p ) : p€X\F).
It
i s Hewitt-Nachbin complete.
This
concludes t h e proof of t h e theorem. We remark t h a t t h e product theorem f o r Hewitt-Nachbin spaces was a l s o proved i n t h e 1952 paper by T . S h i r o t a . The following r e s u l t i s due t o Gillman and J e r i s o n (1960, 8.lO(a)).
8.11
COROLLARY (Gillman and J e r i s o n ) .
subspace of t h e Tychonoff space
X,
If
then
Y
2
c l u x Y = uY.
C-embedded
87
PROPERTIES OF HEWITT-NACHBIN SPACES
Proof.
If
in
and hence i n
uX
is
Y
C-embedded i n cldXY.
then
X,
Moreover, clSxY
COROLLARY (Gillman and J e r i s o n ) .
Hewitt-Nachbin subspace Proof.
Let
be a
Y
t h e Hewitt-Nachbin
of
Every
a Hewitt-Nachbin
by 4 . 3 ( 3 ) . C-embedded
space i s c l o s e d .
C-embedded Hewitt-Nachbin
space
C-embedded
i s Hewitt-
clJxY = UY
Nachbin complete by 8 . 1 0 ( 4 ) s o t h a t 8.12
is
Y
subspace of cl Y = X
Then, by 8 . 1 1 we have
X.
cluxY = UY = Y .
I n 7 . 1 5 t h e example of t h e o r d i n a l space [0,62] was pres e n t e d . Since [ O , n ] i s compact by 7.15(1), i t i s HewittMoreover, s i n c e by 7 . 1 5 ( 3 ) every c o n t i n u o u s
Nachbin complete.
[o,n)
r e a l - v a l u e d f u n c t i o n on t h e subspace [p,n) = (x : B
"tail"
C-embedded i n [ O , n ] .
x
< n),
i s c o n s t a n t on a
is
i t i s immediate t h a t [ O , n )
Hence, a
C-embedded s u b s e t of a H e w i t t -
Nachbin space need n o t be c l o s e d .
Therefore, the condition
t h a t t h e subspace be Hewitt-Nachbin complete i n 8.11 cannot be dropped.
F u r t h e r on i n t h i s s e c t i o n we w i l l g i v e an ex-
ample demonstrating t h a t c l o s e d Hewitt-Nachbin Hewitt-Nachbin
space need n o t be
subspaces of a
C-embedded.
The n e x t r e s u l t concerns unions of Hewitt-Nachbin 8.13
THEOREM.
(1)
spaces.
(Gillman and J e r i s o n , 1 9 6 0 ) . I n anx
Tychonoff s p a c e , the union of a compact subspace
- -
w i t h 2 Hewitt-Nachbin
subspace i s Hewitt-Nachbin
complete. (2)
If
(Mrdwka, 1 9 5 7 A ) .
that
: n c l m ) where each
X = U (Xn
Hewitt-Nachbin
i s a normal
X
subspace
of
X,
then
T1-space Xn X
such
is a c l o s e d i s Hewitt-
Nachbin complete. (1) L e t
Proof.
not Hewitt-Nachbin
complete.
i s n o t Hewitt-Nachbin p
E
cluxY.
Let
E
is
X
g
E
C(uX)
Since
uX\X.
Y U (p).
Since
C(Y).
u l a r t h e r e e x i s t s a function
p
i t f o l l o w s from
wX
Consider t h e space
an a r b i t r a r y f u n c t i o n i n
i s compact and
K
I t w i l l be e s t a b l i s h e d t h a t
complete.
compact, hence c l o s e d , i n that
where
X = Y U K
ux
Y
is
K
cluxX = UX Now, l e t
f
be
is completely reg-
such t h a t
g(x) = 0
88
SPACES AND CONVERGENCE
HEWITT-NACHBIN
whenever
xtK
and
is
g
(glY)(f)
t h e function
1 on a neighborhood o f
can be extended t o a f u n c t i o n
by s e t t i n g i t e q u a l t o
on
0
c o n t i n u o u s l y extended t o
hv
Furthermore, h
K.
in
Since
C(uX).
p. Therefore, Y Y U [ p ) completing t h e argument by 8 . 8 ( 4 ) . po
Let
be a p o i n t i n
U [clPxXn : n c m
1,
f o r each p o i n t
2-”
If
PX\X.
then f o r each
f n : px
uous f u n c t i o n
po
with
p c c lpxXn.
Let
C(X)
E
can be h”
and
f
f
can
C-embadded
does not belong t o
t h e r e exists a contin-
nc7N
[0,2-”]
--f
h
is
b e c o n t i n u o u s l y extended t o (2)
Hence
p, i t f o l l o w s t h a t
a g r e e on a d e l e t e d neighborhood of
in
p.
fn(po) = 0 f
and
fn(p) =
denote t h e function
x [ f n : n c W ’i which i s c o n t i n u o u s b e c a u s e of uniform convergence.
0
Therefore, X c l PxXno 11
is
n0
f o r some
suppose t h a t
C-embedded i n
I t follows t h a t
does belong t o
.
clPxXn0 = BXn
x
C -embedded
Now, s i n c e
fl X,
>
f (p)
whenever
0
denote t h e r e s t r i c t i o n
flXn
there exists a function
g(p)
and
>
e x t e n s i o n of
g
t^
p
.
xn
F
f
E
) with
C(pX-
I1
0
Let
by 8 . 8 ( 3 ) .
0
Then, by t h e n o r m a l i Y of
ox C ’ (X)
such t h a t
glxn
=
fl
0
0
whenever g
pcX.
Let
gP
Then, g P (p,)
PX.
over
, and moreover
f/Xn
cl
0
s p a c e by assumption, t h e r e e x i s t s a f u n c t i o n and
pX.
i s a Hewitt-Nachbin
Xn
0
in
X,
in
C -embedded
*
i s dense and
Xn
po
and t h e r e f o r e
X
0
f(Po) = 0
pex.
whenever
0
IN, Because o f t h e n o r m a l i t y of
6
0
so t h a t
>
f(p)
i s Hewitt-Nachbin complete by 8 . 8 ( 3 ) .
On t h e o t h e r hand,
x-
and
Moreover, f ( p ) = 0
P
g (p) = g ( p )
>
0
denote t h e Stone = 0 b e c a u s e qP
whenever
pcx.
‘n
0
-
There-
0
f o r e , t h e space
X
i s Hewitt-Nachbin complete by 8 . 8 ( 3 ) .
This
c o n c l u d e s t h e proof of t h e theorem. /
I n h i s 1954 paper Mrowka p r o v i d e s an example demonstrat i n g t h a t t h e assumption o f n o r m a l i t y i n 8 . 1 3 ( 2 ) c a n n o t be dropped.
The example a l s o a p p e a r s i n G i l l m a n and J e r i s o n
(Problem 51) and w e s h a l l p r e s e n t it a t t h e end o f t h i s section. The n e x t r e s u l t i s found i n t h e 1967 p a p e r of P . Kenderov
SPACES
PROPERTIES O F HEWITT-NACHBIN
89
and w i l l c h a r a c t e r i z e Hewitt-Nachbin completeness f o r normal I t w i l l make use
Hausdorff and countably paracompact s p a c e s .
of t h e following c h a r a c t e r i z a t i o n of t h e s e spaces due t o J . Horne ( 1 9 5 9 ) and J . Mack ( 1 9 6 5 ) . LEMMA (Horne-Mack).
8.14
A normal Hausdorff
space
2
X
countably paracompact i f and only i f f o r every d e c r e a s i n q sequence IFn : n c I N } of c l o s e d s e t s t h e r e i s 2 sequence ( G n
tion,
i n t e r s e c t i o n such t h a t
Fn
C
with empty i n t e r s e c -
X
: n E l N ) of open s e t s w i t h empty
f o r every
Gn
THEOREM (Kenderov) .
8.15
&
X
nElN.
be a normal Hausdorff space,
denote t h e c o l l e c t i o n of a l l c l o s e d s u b s e t s
of
followinq s t a t e m e n t s a r e t r u e : (1) If X i s a Hewitt-Nachbin space, then e v e r y
8-
-Then t h e --
fj
and l e t
X.
u l t r a f i l t e r w i t h t h e c o u n t a b l e i n t e r s e c t i o n property i s fixed.
If
(2)
&
X
8-
countably paracompact and i f every
u l t r a f i l t e r with t h e c o u n t a b l e i n t e r s e c t i o n prop e r t y i s f i x e d , then
(1) Let
Proof.
( F A : A c r ) denote a
& u l t r a f i l t e r on
intersection property. X,
zero-sets i n
3.
tion property.
so
and l e t
with the countable
X
d e n o t e t h e c o l l e c t i o n of a l l
zo
Note t h a t
has t h e c o u n t a b l e i n t e r s e c To
F i r s t we show t h a t
is a
if
Z
n
F
f o r every
ao.
Then t h e r e e x i s t s
By t h e n o r m a l i t y of
sets.
Z0,
# @
F E
and
Z
X,
then
F c Z*
and
belongs t o that
Z
*
n
Z
3.
n
Z = @.
a0
Thus, X
i s fixed.
F c Z
Since
Therefore, Z
Now, s i n c e filter
Z = @.
Z0
z0
(6.8). Z
n
Suppose F = @.
F a r e completely s e p a r a t e d
Hence t h e r e e x i s t s a z e r o - s e t
*
Z E
Z E Z ( X ) and
such t h a t
FE$
on
Z-ultrafilter
To t h i s end, i t s u f f i c e s t o prove t h a t i f
Z #
=
Z ( X ) denote t h e c o l l e c t i o n of
Let
X.
that
3
be Hewitt-Nachbin complete and l e t
X
zero-sets i n
i s a Hewitt-Nachbin s p a c e .
X
*
E
*
Z it
E
Z ( X ) such t h a t
, i t follows t h a t
Z
*
Z0. This c o n t r a d i c t s t h e f a c t
is a
Z - u l t r a f i l t e r on
i s a Hewitt-Nachbin
space t h e
X.
Z-ultra-
Moreover, by t h e complete r e g u l a r i t y of
X, f o r each A E r , t h e r e e x i s t s a family s e t s i n X such t h a t
(Z
a : a
E
I\] of zero-
HEWITT-NACHBIN SPACES AND CONVERGENCE
90
n iza
F) =
Note t h a t f o r every
a
and hence
X
w i t h t h e countable
'a
' 0
is fixed.
Z0 b e a
Let
(2)
3
meets e v e r y m e m b e r of I,. W e then have
Za
5
so t h a t
: a c I],).
Z - u l t r a f i l t e r on
intersection property.
3
ultrafilter
3-
may be embedded i n a
So
Then
I t w i l l be shown t h a t
by Z o r n ' s Lemma.
5
r e t a i n s the countable i n t e r s e c t i o n property. L e t (Fi
Since
5
:
irN
1
b e any c o u n t a b l e s u b c o l l e c t i o n o f
w i t h o u t l o s s of g e n e r a l i t y t h a t i F i sequence.
n
Now, suppose t h a t
t h e r e e x i s t open neighborhoods X,
n
and s a t i s f y i n g
iElN
and
Fi
that
Zi
r
5
Zi
f o r each
is closed.
Z-ultrafilter.
: itN
3
i s a decreasing
iclN
1
=
:
containing
: i c N ) =
(Gi
6.
/Fi Gi
a.
Then by 8 . 1 4 f o r each
Fi
By t h e n o r m a l i t y of
a r e completely s e p a r a t e d s e t s .
X\Gi
e x i s t s a zero-set Zi
5.
i s c l o s e d under f i n i t e i n t e r s e c t i o n s , w e may assume
Hence,
E
Z ( X ) with
because
iclN Zi
Therefore,
Fi
C
Zi
5
C
Hence there I t follows
Gi.
is a
? - f i l t e r and
z0 s i n c e z0 i s i i - l N ) # 6 because a.
belongs t o
n
(Zi
:
a has
t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y and, t h e r e f o r e , by
i s f i x e d . I t f o l l o w s t h a t z0 i s f i x e d and i s a Hewitt-Nachbin s p a c e . T h i s c o n c l u d e s the proof o f t h a
assumption
5
X
theorem. Before w e p r e s e n t s e v e r a l i m p o r t a n t examples a s s o c i a t e d w i t h Hewitt-Nachbin c o m p l e t e n e s s and some of o u r p r e c e d i n g results,
i t w i l l be u s e f u l t o i n t r o d u c e t h e n o t i o n o f a "non-
measurable c a r d i n a l . '' 8.16
A c a r d i n a l number
DEFINITION.
a b l e i n case ther e e x i s t s a set countably a d d i t i v e , on t h e power s e t each nal. -
pcX.
X
m
i s s a i d t o be measur-
of c a r d i n a l i t y
( 0 , l ) - v a l u e d set f u n c t i o n
m
#
0
and a defined
k ( X ) = 1 and ~ ( p= ) o for i s s a i d t o b e a nonmeasurable c a r d i -
P(X) such t h a t
Otherwise
c~
m
PROPERTIES OF HEWITT-NACHBIN SPACES
91
For a d e t a i l e d t r e a t m e n t of nonmeasurable c a r d i n a l s t h e r e a d e r i s r e f e r r e d t o Chapter 1 2 of t h e Gillman and J e r i s o n
I n t h e n e x t c h a p t e r i t w i l l b e d e m o n s t r a t e d t h a t nonmeasurable c a r d i n a l s p l a y an i m p o r t a n t p a r t i n t h e i n v e s t i g a text.
t i o n of Hewitt-Nachbin s p a c e s from t h e p o i n t of view of u n i form s t r u c t u r e s .
I n t h e i n t e r e s t of c o m p l e t e n e s s w e w i l l
s t a t e t h e r e s u l t s c o n c e r n i n g nonmeasurable c a r d i n a l s t h a t a r e needed i n o u r development, b u t w e omit most of t h e s t r a i q h t forward p r o o f s s i n c e they a p p e a r i n Gillman and J e r i s o n .
3
Now, l e t
be a Bourbaki u l t r a f i l t e r on a non-empty
x3 : 63 ( X ) 3 (0,1]by x3 ( A ) = 1 5 , and 0 o t h e r w i s e . Then x5 i s a nonz e r o , f i n i t e l y a d d i t i v e , { O , 11-valued s e t - f u n c t i o n . T h i s i s set if
and d e f i n e t h e mapping
X
belongs t o
A
e a s y t o show from t h e f a c t t h a t A
n
B = $3.
O n t h e o t h e r hand,
3 F
f i l t e r on
= ‘A c X
satisfying
X
11, t h e n
=
: b(A)
x,
if
x5(B)
is a (0,lj-valued
p
b ( X ) , and i f w e
f i n i t e l y a d d i t i v e s e t - f u n c t i o n d e f i n e d on define
+
U B) = x5(A)
x3(A
if
3
i s a Bourbaki u l t r a -
L4
The proof i s s t r a i q h t f o r -
= w. LL
ward i f one o b s e r v e s t h a t b(A
n
B)
.
p
5
Thus, t h e correspondence
+
on
----
t h e s e t of a l l Bourbaki u l t r a f i l t e r s
x3 X
+
b(A U
B\B) +
i s one-one from
onto the s e t of
a l l non-zero, f i n i t e l y a d d i t i v e , ( O , l ] - v a l u e d s e t f u n c t i o n s d e f i n e d on X . i s a non-zero,
If then
countably a d d i t i v e set f u n c t i o n ,
w i l l b e c a l l e d s i m p l y a measure.
The f o l l o w i n g r e s u l t
r e l a t e s Bourbaki u l t r a f i l t e r s and measures f o r d i s c r e t e s p a c e s and i s proved i n Gillman and J e r i s o n (page 1 6 2 ) . 8.17
set
-
Suppose t h a t
LEMMA. X.
Then
and o n l y i f
x3
5
&a
Bourbaki u l t r a f i l t e r on a
has the countable i n t e r s e c t i o n property
5 =
is a measure.
The s e t f u n c t i o n
x3
if
i s s a i d t o be f i x e d i n c a s e t h e
c o r r e s p o n d i n g Bourbaki u l t r a f i l t e r Now, f o r a d i s c r e t e s p a c e
5
is fixed.
p c 2 5
which i s e q u i v a l e n t t o t h e c o n d i t i o n t h a t
i f and o n l y i f [ P I ;3 x,(p)
= 1.
More-
o v e r , by d e f i n i t i o n , t h e c a r d i n a l i t y o f a s p a c e i s nonmeasura b l e i f and o n l y if e v e r y measure
x5
is fixed.
Since i n a
92
SPACES AND CONVERGENCE
HEWITT-NACHBIN
d i s c r e t e space t h e c o n c e p t s of Bourbaki u l t r a f i l t e r and a z e r o - s e t u l t r a f i l t e r a r e t h e same,
t h e f o l l o w i n g r e s u l t s have
been e s t a b l i s h e d . 8.18
and o n l y 8.19
d i s c r e t e space i s a Hewitt-Nachbin s p a c e i f
THEOREM.
i f its cardinal
is
nonmeasurable.
The c a r d i n a l
COROLLARY.
number
No
IN
of the s e t
nonmeasurable. The c l a s s o f nonmeasurable c a r d i n a l s i s a v e r y e x t e n I n f a c t , i t i s c l o s e d under a l l of t h e s t a n d a r d
s i v e one.
operations of c a r d i n a l a r i t h m e t i c :
addition, multiplication,
t h e f o r m a t i o n o f suprema, e x p o n e n t i a t i o n ,
and t h e p a s s a g e
from a given c a r d i n a l t o i t s immediate s u c c e s s o r o r to any s m a l l e r c a r d i n a l (Gillman and J e r i s o n , 1 2 . 5 ) .
A celebrated
unsolved problem i s whether o r n o t e v e r y c a r d i n a l i s nonmea s u r a b l e
.
W e conclude t h i s s e c t i o n by p r e s e n t i n g s e v e r a l impor-
t a n t examples r e l a t e d t o our p r e v i o u s r e s u l t s . 8.20
&
EXAMPLE.
q u o t i e n t o f a Hewitt-Nachbin space need n o t
be 2 Hewitt-Nachbin
space.
T h i s example i s Problem 81 i n t h e Gillman and J e r i s o n text. [O,n)
X =
Let
[0,n] x
D
where
D
i s the o r d i n a l space
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
topology.
Now
D
X
have t h e p r o d u c t
i s a Hewitt-Nachbin space b y 8.18.
i s Hewitt-Nachbin complete by 7.15(1) and 8 . 1 0 ( 7 ) . t h e subspace
Y = ( ( a, p ) : a
Thus
X
Moreover,
p ) i s Hewitt-Nachbin complete
s i n c e i t i s a c l o s e d subspace o f a Hewitt-Nachbin space. Moreo v e r , t h e space [O,n) i s t h e q u o t i e n t of t h e Hewitt-Nachbin space (U,P)
Y
into
under t h e p r o j e c t i o n mapping t h a t maps t h e p o i n t
a.
However,
it was e s t a b l i s h e d i n 7 . 1 5 ( 5 ) t h a t
[0, a) f a i l s t o b e Hewitt-Nachbin complete.
8.21
EXAMPLE.
The union of a Hewitt-Nachbin s p a c e w i t h a
Hewitt-Nachbin space may f a i l t o b e a Hewitt-Nachbin s p a c e . /
T h i s example i s found i n t h e 1958D p a p e r of Mrowka.
SPACES
PROPERTIES OF HEWITT-NACHBIN
lN
Let
93
d e n o t e t h e s e t of p o s i t i v e i n t e g e r s and l e t
lN.
n o t e t h e c o l l e c t i o n of a l l i n f i n i t e s u b s e t s of
r
there exists a subcollection
c 6
B
de-
Then
s a t i s f y i n g t h e following
conditions:
r
(i)
is i n f i n i t e :
( i i ) For e v e r y
and
N1
the intersection ( i i i ) For each
that
N
n
Nl
1
6
E
r
in
N2
n
N1
such t h a t
N1
#
N2,
is finite;
N2
there exists a set
N2
E
r
such
is infinite.
N2
r r
Condition (iii) s t a t e s t h a t c o n d i t i o n s ( i f and ( i i ) and
i s maximal w i t h r e s p e c t t o is e s t a b l i s h e d using t h e
Hausdorff Maximal P r i n c i p l e (Gillman and J e r i s o n , 0 . 7 ) . Let D = {x y ~ dge n o t e a new s e t c o n s i s t i n g o f d i s t i n c t p o i n t s ,
-
Y .
and d e f i n e t h e space t h e p o i n t s of x
Y
E
3N
9 = IN U D
a r e i s o l a t e d , and a neighborhood of a p o i n t
i s any s e t c o n t a i n i n g
D
p o i n t s of t h e s e t D
The s p a c e
(3)
't
9 9
and a l l b u t f i n i t e l y many
Y
Tvchonoff s p a c e and b o t h
are d i s c r e t e
The s p a c e
(2)
x
YET.
(1) The s p a c e
and
w i t h t h e f o l l o w i n g topology:
subspaces
of
IN
?t.
i s n o t c o u n t a b l y compact.
is pseudocompact,
b u t is n o t a
Hewitt-Nachbin s p a c e . The s p a c e \I. is n o t normal. The s t a t e m e n t (1) i s a n immediate consequence o f t h e f a c t t h a t (4)
e v e r y neighborhood f o r every p o i n t of
O(x) i s b o t h open and c l o s e d ( i . e . , c l o p e n )
~ € 9 .Moreover, D IN
i s open i n
is closed i n
9 s i n c e every
9. T h e r e f o r e , a s D i s a c l o s e d , 9, D c a n n o t be c o u n t a b l y
i n f i n i t e , and d i s c r e t e subspace of compact.
This proves ( 2 ) .
Our n e x t aim i s t o p r o v e t h a t f
E
C(U). Note f i r s t t h a t
f
\I/
i s pseudocompact.
Let
i s bounded on e v e r y s u b s e t o f
r.
For, i f Y E T , l e t f ( x 1 = r E IR. If G = ( r - c,ry + E ) Y Y i s a n open s e t of containing r t h e n f - l \ G ) i s an open Y' set containing x and t h e r e f o r e m e e t s y a t a l l b u t f i n i t e Y l y many p o i n t s of y. I t now f o l l o w s t h a t f i s bounded on
y.
Next w e show t h a t
f
i s bounded on
N.
I f not, then
94
HEWITT-NACHBIN
SPACES AND CONVERGENCE
F
IN
W e then have t h a t
f
t h e r e e x i s t s an unbounded i n c r e a s i n g sequence [ f ( x n ) : x n and
1
ncIN
But ( x
of r e a l numbers.
r
hence by t h e maximality of
n
such t h a t fxn : n6Td i
i s unbounded on
f
there exists a set
g: N~
and
r
E
which c o n t r a d i c t s t h e above o b s e r v a t i o n .
N2
F i n a l l y , note t h a t IN,
n
ncm] is i n
is infinite.
N2
IN
hood o f every p o i n t i n bounded o n
:
E
9
i s dense i n D
IN.
meets
C*(Q) and
9
s i n c e e v e r y neighborTherefore,
since
i s pseudocompact.
f
is \k
Thus
i s n o t a Hewitt-Nachbin space by 7 . 1 4 , which proves ( 3 ) .
The
s t a t e m e n t ( 4 ) i s a l s o immediate s i n c e e v e r y normal pseudocomp a c t Hausdorff space i s c o u n t a b l y compact
both
Note, however, t h a t by 8.18
3D.2).
Hewitt-Nachbin subspaces
of
of nonmeasurable c a r d i n a l . LindeLof, and o-compact.
Q
(Gillman and J e r i s o n , N
and
D
s i n c e each i s 5 d i s c r e t e space
I n fact,
IN
is paracompact,
T h e r e f o r e , w e have t h e f o l l o w i n g
result: (a)
The union of a
(3-compact space w i t h a Hewitt-Nach-
-
b i n space n e e d n o t be a Hewitt-Nachbin s p a c e .
0-compactness i m p l i e s L i n d e l o f which i n t u r n i m p l i e s
Since
paracompactness, w e can r e p l a c e "o-compact" i n ( a ) by " L i n d e l o f " o r by ttparacompactlt and have a t r u e s t a t e m e n t . Moreover, i t h a s a l s o been shown t h a t : (b)
The union of a Hewitt-Nachbin s p a c e w i t h a H e w i t t Nachbin s p a c e need n o t be a Hewitt-Nachbin s p a c e .
Finally, s e t Xn
Xo = D
Xn = ( n ) (nelN) s o t h a t each
and
i s a c l o s e d Hewitt-Nachbin subspace of
00
U Xn n=o
q.
=
Then
which e s t a b l i s h e s t h a t t h e assumption of n o r m a l i t y i n
8 . 1 3 ( 2 ) c a n n o t be dropped.
8.22
EXAMPLE.
A normal Hausdorff space t h a t f a i l s t o be
9-
Hewitt-Nachbin complete and such t h a t i t s Hewitt-Nachbin p l e t i o n i s n o t normal. This example a p p e a r s i n t h e 1948 paper by A . S t o n e . I = [0,1] d e n o t e t h e u n i t i n t e r v a l , and l e t
where 7N
ma
d e n o t e s a copy of
IN
f o r each
i s Hewitt-Nachbin complete by 7 . 1 1 ,
so that
T
T = n[lNa ac1.
:
Let
acI)
The space
since i t i s countable,
i s a Hewitt-Nachbin space by 8 . 1 0 ( 7 ) .
Let
X
PROPERTIES OF HEWITT-NACHBIN
SPACES
95
d e n o t e t h e s e t of a l l p o i n t s (xaIaiI i n T f o r which x = 1 a e x c e p t f o r c o u n t a b l y many c o o r d i n a t e s . I n (1948, Theorem 3 ) A.
Stone proves t h a t
i s n o t normal i n t h e f o l l o w i n g way:
T
For each p o s i t i v e i n t e g e r points (x )
a
in
a
o t h e r than
that
xy
loint. sets
=
n.
k,
d e n o t e t h e s e t of a l l
such t h a t f o r e v e r y p o s i t i v e i n t e g e r
T
~
n-m
Ak
let
k,
‘(1
t h e r e i s a t most one index
Now, t h e s e t s
such
a r e c l o s e d and p a i r w i s e d i s -
Ak
S t o n e then proves by an i n d u c t i v e argument t h a t t h e and
A’
A2
c a n n o t b e s e p a r a t e d by d i s j o i n t open s e t s .
I n C o r o l l a r y 2 of h i s 1959 p a p e r , H a Corson p r o v e s t h e non-trivial r e s u l t that 8.23
vX = T .
i s normal and t h a t
X
T h i s ex-
A non-normal Hewitt-Nachbin s p a c e .
EXAMPLE.
ample w i l l also d e m o n s t r a t e t h a t c l o s e d Hewitt-Nachbin s p a c e s of a Hewitt-Nachbin s p a c e need n o t b e
r
Let
denote the s u b s e t I ( x , y ) : y
2
&-
C-embedded.
IR x IR
0 ) of
provided w i t h t h e f o l l o w i n g e n l a r g e m e n t of t h e p r o d u c t topology:
for
r
0
>;
the sets
a r e a l s o neighborhoods of t h e p o i n t ( x , ~ )(see Gillman and Jerison,
3K).
The s p a c e
h a s a f i n e r topology than t h e
u s u a l one on t h e c l o s e d upper h a l f C a r t e s i a n p l a n e and h e n c e
m u s t b e a Hausdorff s p a c e .
With t h i s topology
i s called
t h e Niemytzki p l a n e o r sometimes t h e Moore p l a n e . the r e a l l i n e andi tis -
D = ( ( x , O ) : X E D )i s a d i s c r e t e s u b s p a c e
r.
---
2 zero-set i n
nim,
For each define the --
space
topology from
r.
A n = [; (
let
m
X = ( U An) U D nEm
, ) ;1
: (m
X
is
The s n a c e
X
i s n o t normal.
X
i s n o t paracompact.
(3)
The s p a c e
+
1)
E
of r
IN ) and
endowed w i t h t h e r e l a t i v e
(1) The s p a c e (2)
Note t h a t
s e p a r a b l e Tychonoff s p a c e .
i s Hewitt-Nachbin c o m p l e t e . To prove (1) w e f i r s t e s t a b l i s h t h a t I? i s a Tychonoff s p a c e . (4)
The s p a c e
Consider t h e c a s e ing
p.
X
p = (x,O)
Then t h e r e e x i s t s
E
E
and
D
>
0
U
an open s e t c o n t a i n -
such t h a t
p
E
VE ( p )
c U.
96
HEWITT-NACHBIN
Define a real-valued f(p)
Let
= 0,
ments from linear. X
of
p
let
SPACES AND CONVERGENCE
function f(x) = 1
U
An
x.
a d m i t s a t most
Vc(p) define f
E
C(r).
f
t o be
Moreover t h e s p a c e
i s a countable dense s u b s e t
2
NO
= c
From (1) i t f o l l o w s t h a t
IR ( s i n c e c o n t i n u o u s
f u n c t i o n s t h a t a g r e e o n t h e d e n s e subspace
m u s t a g r e e on
X).
However, D
of c a r d i n a l i t y
p l e t e (8.18).
Thus
U
An
ncm i s a closed d i s c r e t e subspace
and a s such i s Hewitt-Nachbin com-
c D
X
continuous r e a l - v a l u e d f u n c t i o n s ,
denotes the c a r d i n a l i t y of
c
real-valued
X
i n t h e f o l l o w i n g way:
x ,d V c ( p ) , and on a l l seg-
nEm
Next w e e s t a b l i s h ( 2 ) .
of
r
on
t o t h e boundary of
Then one can show t h a t
i s s e p a r a b l e because
where
f if
admits e x a c t l y
2'
d i s t i n c t continu-
ous r e a l - v a l u e d f u n c t i o n s and i s t h e r e f o r e n o t C-embedded i n I t f o l l o w s t h a t X f a i l s t o be normal which p r o v e s ( 2 ) .
X.
The s t a t e m e n t ( 3 ) i s now immediate because
is a regular
X
Hausdorff s p a c e and e v e r y paracompact r e g u l a r Hausdorff s p a c e
i s normal. The f a c t t h a t
X
i s a Hewitt-Nachbin s p a c e f o l l o w s from
t h e o b s e r v a t i o n t h a t t h e i d e n t i t y mapping from
IR x IR
into
i s c o n t i n u o u s coupled w i t h t h e r e s u l t 8.18 i n t h e
Gillman and J e r i s o n t e x t .
( W e wish t o postpone t h e p r o o f o f
t h i s l a t t e r r e s u l t u n t i l 16.16 of C h a p t e r 4 i n o r d e r t h a t t h e r e s u l t s c o n c e r n i n g Hewitt-Nachbin s p a c e s and c o n t i n u o u s mapp i n g s appear t o g e t h e r i n a s i n g l e c h a p t e r . ) I t f o l l o w s i m m e d i a t e l y from t h i s example t h a t c l o s e d Hewitt-Nachbin subs p a c e s of a Hewitt-Nachbin s p a c e need n o t b e
C-embedded s i n c e
t h a t property c h a r a c t e r i z e s normality. I n t h e n e x t s e c t i o n w e w i l l t u r n o u r a t t e n t i o n to f o c u s
on t h e i m p o r t a n t q u e s t i o n of embedding a Tychonoff s p a c e densel y i n an a p p r o p r i a t e Hewitt-Nachbin s p a c e . Section 9:
Hewitt-Nachbin Completions
I n h i s 1964 p a p e r 0 . F r i n k i n t r o d u c e d t h e n o t i o n o f a
normal b a s e ( 6 . 2 0 )
8
i n o r d e r t o c o n s t r u c t h i s Hausdorff
c o m p a c t i f i c a t i o n u(8) c o n s i s t i n g o f a l l t h e 9 - u l t r a f i l t e r s on t h e s p a c e X i n t h e f o l l o w i n g way: The c o l l e c t i o n w ( 2 )
COMPLETIONS
HEWITT-NACHBIN
97
i s made i n t o a t o p o l o g i c a l space by taking a s a base f o r t h e
w ( 8 ) a l l s e t s of t h e form
closed s e t s i n
w(8)
Zw = [ $ E
:
Z E ~ ) . To s e e t h a t t h e s e s e t s do indeed form a b a s e , observe w w u) t h a t z1 w~ z 2 0 = (zl u z 2 ) . A l s o note t h a t zl n zZu) =
(zl n z 2 )
. 8
Since
i s a d i s j u n c t i v e c o l l e c t i o n of c l o s e d s u b s e t s
3 = ( Z E ~: pcZ] i s t h e unique P 8 - u l t r a f i l t e r converging t o the p o i n t P E X . I t i s easy t o
of
by 6 . 6 the
X,
8-filter
v e r i f y t h a t t h e mapping
cp
from
w ( 8 ) d e f i n e d by Furthermore, cp 5 into
X
cp(p) = 3 i s an i n j e c t i v e mapping. P homeomorphism from X onto q ( X ) . To see t h i s observe t h a t
cp(z) = cp(x) n z w . I t w i l l be shown t h a t
c p ( X ) i s dense i n
w(@
l i s h i n g t h a t every non-empty b a s i c open s e t i n
cp(x).
But a b a s i c open set of Uw =
m ( 8 ) i s of t h e form
(8 E ~ ( 8 :) t h e r e e x i s t s and (X\u)
Analogously one h a s t h a t
U
s a t i s f y i n g (X\U)
any
The space
a2 Z1
€ o r any
PEZ
are distinct E
g1
and
E
A c
u
8).
n
Uw
f o r every open s e t
i s non-empty, then s e l e c t Uu), and 3 E c p ( U ) . P Hausdorff. For suppose t h a t $l and E
Then t h e r e e x i s t s e t s Z1 n Z 2 = @ a s a consequence of i s a normal c o l l e c t i o n , t h e r e e x i s t sets
8-ultrafilters.
Z2 E
Since
6.8(2).
is
A E ~such t h a t
v(U) = v ( X )
E 8. I f Uw Zf5 where 3
w(8)
by e s t a b -
w ( 8 ) meets
8
Z2
with
( X \ C 2 I W = @.
~(8)
Finally,
of c l o s e d sets i n I t suffices for Q = [ Z c g : Zw
property. If
ZcQ,
a
n
For l e t
aw
be a c o l l e c t i o n
w ( 8 ) with t h e f i n i t e i n t e r s e c t i o n p r o p e r t y .
aW
t o c o n s i s t of b a s i c c l o s e d s e t s . Let Q W ) . Then has the f i n i t e i n t e r s e c t i o n
Therefore, by Z o r n ' s Lemma t h e r e e x i s t s a
3
filter
E
compact.
such t h a t
then
d".
8-ultra-
# c 3 ( r e c a l l o u r remarks following 6 . 1 ) .
Z E ~so t h a t
3
E Zu).
I t follows t h a t
Therefore, i t h a s been e s t a b l i s h e d t h a t
w ( 8 ) i s indeed
98
S P A C E S AND CONVERGENCE
HEWITT-NACHBIN
a compact Hausdorff s p a c e t h a t c o n t a i n s a d e n s e homeomorphic copy of t h e s p a c e
X.
i s the collection
Z ( X ) of a l l z e r o - s e t s on
8 ~ ( 8 i)s
Moreover, F r i n k e s t a b l i s h e d t h a t i f
then X ( t h i s i s exacti s c o n s t r u c t e d i n t h e Gillman and J e r i s o n X,
V
p r e c i s e l y t h e Stone-Cech c o m p a c t i f i c a t i o n of l y t h e way text).
px
Moreover, i f
3
i s t h e s u b c o l l e c t i o n of
Z ( X ) con-
s i s t i n g of t h e z e r o - s e t s of t h o s e f u n c t i o n s t h a t a r e c o n s t a n t on t h e complement of some compact s u b s e t o f X , then ~ ( 8 i)s t h e A l e x a n d r o f f o n e - p o i n t c o m p a c t i f i c a t i o n of t h e l o c a l l y compact Hausdorff s p a c e
X.
W e n e x t want t o c o n s i d e r t h e c o r r e s p o n d i n g i d e a f o r
Hewitt-Nachbin c o m p l e t e n e s s .
Throughout t h i s s e c t i o n , by
completion of t h e Tychonoff s p a c e
X
w e w i l l mean a H e w i t t -
Nachbin s p a c e t h a t c o n t a i n s a d e n s e homeomorphic copy o f
The Hewitt-Nachbin p l e t i o n of
X.
completion
uX
2 X.
i s one example o f a com-
S i n c e e v e r y compact Hausdorff s p a c e i s a
Hewitt-Nachbin s p a c e , t h e Stone-&ch
compactification
pX
X. ( W e w i l l i n v e s t i g a t e a n o t h e r and i t s r e l a t i o n s h i p t o Hewitt-Nach-
g i v e s a n o t h e r completion of n o t i o n of " c o m p l e t e n e s s , It b i n completeness,
i n the n e x t c h a p t e r where w e c o n s i d e r t h e
uniform s p a c e c o n c e p t . )
I n c o n s t r u c t i n g w ( 8 ) f o r some normal b a s e 8 on t h e X, F r i n k n o t o n l y gave a c o m p a c t i f i c a t i o n of t h e s p a c e b u t a l s o a completion i n t h e Hewitt-Nachbin sense ( s i n c e e v e r y compact s p a c e i s a Hewitt-Nachbin s p a c e ) . The q u e s t i o n a r i s e s a s t o whether e v e r y completion Y of a s p a c e X can be o b t a i n e d by u t i l i z i n g and a d j u s t i n g t h e n o t i o n of a normal b a s e and then c o n s t r u c t i n g from t h i s a d j u s t m e n t a n e w s p a c e p ( 8 ) t h a t i s homeomorphic t o Y . Since the H e w i t t Nachbin completion UX i s i n g e n e r a l n o t e q u a l t o t h e StoneV Cech c o m p a c t i f i c a t i o n pX, w e c a n n o t hope t o u s e m ( 8 ) f o r one 8 (even a s a modified normal b a s e ) f o r a g e n e r a l complet i o n method. Thus, w e t u r n our a t t e n t i o n t o non-compact comple tions. I t w i l l be shown t h a t c e r t a i n s u b c o l l e c t i o n s o f t h e c o l l e c t i o n Z ( X ) of a l l z e r o - s e t s on a Tychonoff s p a c e X Tychonoff s p a c e
which a r e a l s o normal b a s e s w i l l g e n e r a t e a c o m p l e t i o n o f t h e s p a c e which i n g e n e r a l i s n o t compact (see Theorem 9 . 3 ) .
HEWITT-NACHBIN
99
COMPLETIONS
Normal b a s e s t h e m s e l v e s w i l l y i e l d compact c o m p l e t i o n s . W e now i n t r o d u c e a g e n e r a l i z a t i o n of t h e normal b a s e
c o n c e p t i n o r d e r t o c o n s t r u c t t h e Wallman-Frink c o m p l e t i o n o f
X.
a space
With r e f e r e n c e t o d e f i n i t i o n s 6 . 3 , 6 . 1 5 , and 6 . 2 0
t h e f o l l o w i n g d e f i n i t i o n i s made. 9.1
Let
DEFINITION.
8 8
A collection
base i n
case
b e an a r b i t r a r y t o p o l o g i c a l s p a c e .
X
c P ( X ) i s s a i d t o b e a s t r o n q d e l t a normal i s a d e l t a r i n g o f s e t s t h a t i s a normal b a s e
and complement g e n e r a t e d
.
I t i s immediate t h a t t h e c o l l e c t i o n
sets i n a Tychonoff space Moreover, i f normal,
X
Z(X) o f a l l zero-
i s a s t r o n g d e l t a normal b a s e .
X
i s a normal Hausdorff s p a c e t h a t i s p e r f e c t l y
then t h e c o l l e c t i o n o f a l l c l o s e d s u b s e t s of
a s t r o n g d e l t a normal b a s e .
X
is
I t w i l l b e shown i n 9 . 3 t h a t
e v e r y s t r o n g d e l t a normal b a s e i s a s u b c o l l e c t i o n o f t h e collection
Z(X) o f a l l z e r o - s e t s on
X.
W e remind t h e r e a d e r of t h e o b s e r v a t i o n t h a t i f
normal c o l l e c t i o n t h a t i s a ( d e l t a ) r i n g of s e t s ,
8
is a
then e v e r y
& u l t r a f i l t e r with t h e countable i n t e r s e c t i o n property i s c l o s e d under c o u n t a b l e i n t e r s e c t i o p s by 6 . 1 4 . W e may now d e f i n e t h e subspace
P
8)
=
3; E
w(8)
:
3;
PEX,
8-ultrafilter,
f i l t e r converging t o
the c o l l e c t i o n and moreover p
by 6 . 7 .
from
X
into
p ( 8 ) d e f i n e d by
from
x
onto
cp(X) a s b e f o r e .
where
Z
is r e a l ) j
~(8).
p ( 8 ) w i t h t h e r e l a t i v e topology o b t a i n e d from
F o r each real
~ ( 8 ) .D e f i n e
h a s t h e c o u n t a b l e i n t e r s e c t i o n prop-
e r t y ( i . e . , 3: and endow
p ( 8 ) of
and
X\U
are i n
3 = ( Z E ~: PEZ] i s a P i s t h e unique 8 - u l t r a -
5P T h e r e f o r e t h e mapping
cp
cp(p) = 3 i s a homeomorphism P L e t us set
3.
U t i l i z i n g the above d e f i n i t i o n s one may r e a d i l y show t h e f o l l o w i n g theorem (see Alo and S h a p i r o , 1969B, Theorem 1 ) .
100
9.2
HEWITT- NACHBIN SPACES AND CONVERGENCE
THEOREM ( A l o and Shapiro)
with 2
.
and l e t
q
X 5 Tychonoff space ( r e s p e c t i v e l y normal b a s e ) ,
& e &
2
stronq d e l t a normal base
of x into p ( 8 ) (re-
be t h e n a t u r a l embedding
~ ( 8 ) )I .f
spectively,
U,
v, & {un
:
ntm j
=
complements
of members of 3, and i f iZn n t m ) are members of -then the followinq p r o p e r t i e s hold: 2,
:
8,
(1) ~fu c V , then U P c V P ( r e s p e c t i v e l y , uu) c v'). (x\z)P = p ( 8 ) \zP ( r e s p e c t i v e l y , (x\z) u, = w ( 5 )\z') (2) (4)
n
~ l ~ ( ~r) Z ) nq ) = ( ( fi Z,)P
m
n znp:
=
n=l
0
=
u
[
n=l
n zn
(5)
or
C ~ ~ ( ~ ) V ( Z e~q u) i v a l e n t l y ,
n=l
n=l
cD
u unP
x
(respective1L
n=l
if
i f and only
p(8) .
covers
00
Un)P =
n=l
n=l ( 6 ) I Z n : n t l N j covers
.
n znp
=
6.
n= 1
i f and only i f [ Z n p : n E m )
I n the d i s c u s s i o n of w ( 8 ) i n Section 6 w e remarked t h a t t h e normal b a s e s used i n t h e c o n s t r u c t i o n s of w e l l known c o m p a c t i f i c a t i o n s were always s u b c o l l e c t i o n s of t h e c o l l e c t i o n Z ( X ) of a l l z e r o - s e t s . I t w i l l now be shown t h a t : I f 8 & 2 s t r o n q d e l t a normal base i n a Tychonoff space X, then 8 i s a s u b c o l l e c t i o n of Z ( X ) 9.3
REMARK.
-
.
For l e t
268.
Then s i n c e
8
is complement generated,
t h e r a e x i s t s a countable c o l l e c t i o n ( C n : n c m ] of complements
8
.
Z = fl ( Cn : n c m ) Then t h e r e i s a sequence ( Z n : n t m ) in 8 such t h a t Z n c Cn c Zn-l for a l l n such t h a t n [cn : n e m ) = f~ { Z , : n c l N ] . Thus, z"' = n iznUI : n e m ) = t l [ C n w : n € m ) by (1) and ( 4 ) of 9 . 2 . Consequently, f o r each n c m t h e r e e x i s t s a function u) f n F C ( ( u ( 8 ) ) ( s i n c e w ( 8 ) i s normal) such t h a t w ( 8 ) \ C n c u) Z ( f n ) and Z ( f n ) fl w ( 8 ) \ C n = 6 by 3 . 1 1 ( 1 ) . Hence, '2 c Z ( f n ) c Cn' f o r every n t m so tha t of members of
such t h a t
zw c
n nclN
z(fn) c
n nem
C,
W
=
zw .
101
HEWITT-NACHBIN COMPLETIONS
Therefore, Z
111
i s a countable i n t e r s e c t i o n of z e r o - s e t s i n
u(8) and hence i s i t s e l f a z e r o - s e t i n
~ ( 8 ) .Let
where
f E C(w(8)).
Z(f0cp)
i s a zero-set i n
where
cp
X
w(B),
Then
Z =
i s the embedding o f
into
Zu = Z(f) X,
establishing
8
c Z(X). I n the next r e s u l t i t w i l l be e s t a b l i s h e d t h a t the subspace p ( 8 ) of w ( 8 ) i s a Hewitt-Nachbin space. The r e s u l t i s found i n t h e 1969B paper of Alo and S h a p i r o . that
THE COMPLETION THEOREM ( A l o and Shapiro)
9.4
.
s t r o n g d e l t a normal base i n 2 Tychonoff space
8 is 2
If X,
then
is
X
homeomorphic t o a dense subspace of a Hewitt-Nachbin space
P(8)* Since q ( X ) i s dense i n w ( 8 ) i t i s a l s o dense i n I t w i l l be shown t h a t p ( 8 ) i s Hewitt-Nachbin complete
Proof.
p(8).
by proving t h a t i t i s
5
w(8) (8.7).
in
G -closed
6
w ( 8 ) \ p ( 8 ) , then we want t o f i n d a
E
5
that contains
and such t h a t
n
G
G -set
6
p(8)
=
Now, i f
@.
w(8)
in
G
R e c a l l from
our opening d i s c u s s i o n concerning t h e Frink c o m p a c t i f i c a t i o n t h a t t h e c o l l e c t i o n (Uw : (X\U) E 8 ; i s a base f o r t h e open
~(8).
sets i n
3
3 i s a 8 - u l t r a f i l t e r on X t h a t f a i l s t o have the countable i n t e r s e c t i o n p r o p e r t y . Hence, t h e r e e x i s t s a sequence ( Z n : nE7N) of members of 5 s a t i s If
n
fying
U I ( ~ ) \ P ( ~ ) then ,
E
[Zn
n c I N ) = @.
:
m e n t generated, f o r each (Cn,
fl
Hence, f o r each
implies t h a t t h e set
C:,i
8-ultrafilter G -set
G =
8
5
F i n a l l y , w e claim t h a t belongs t o
n
Zn
C
such t h a t Zn = which
Cn,i
belongs t o t h e b a s i c open i,n
E
n cn, UI
Therefore,
IN.
in
w(8)
ncm icm
6
Q
icIN,
f o r every p a i r o f i n d i c e s
belongs t o the
i s comple-
t h e r e e x i s t s a sequence
nEIN
: i c l N ) of complements of members of
( c ~ :, i c~I N ) .
8
Furthermore, s i n c e
w Cn,
G f7
p(8)
=
.
@. For i f
f o r every p a i r of i n d i c e s
then
QEG
i,n
IN.
E
Hence, f o r each p a i r of such i n d i c e s t h e r e e x i s t s a member
bn , i
belonging t o Therefore, n n
8
$?
such t h a t . c n
i c m ncm n , l follows t h a t
G
bn , i n c
icm ncm
E
Q
and
?! n , i
~ = , n ~zn
=
c Cn,i.
6.
ncm
f a i l s t o have the countable i n t e r s e c t i o n
3
It
l o2
HEWITT-NACHBIN SPACES AND CONVERGENCE
property.
Hence, G
p(8)
does n o t belong t o
completing t h e
proof of t h e theorem. The p r e v i o u s theorem y i e l d s an a d d i t i o n a l i n t e r n a l c h a r a c t e r i z a t i o n of a Tychonoff s p a c e : namely, 2
is c o m p l e t e l y r e q u l a r i f and o n l y i f i t h a s a s t r o n g d e l t a normal
base.
For i f
T1-space
i s a Tychonoff s p a c e , then t h e c o l l e c t i o n
X
Z ( X ) i s a s t r o n g d e l t a normal b a s e .
Conversely, i f a
T1-
s p a c e h a s a s t r o n g d e l t a normal b a s e , then by F r i n k ’ s compact i f i c a t i o n i t i s homeomorphic t o a d e n s e subspace of a compact Hausdorff s p a c e . An i n t e r p r e t a t i o n of t h e above theorem i s now a t hand.
If
8
Z ( X ) of a l l z e r o - s e t s on
is the collection
then
X,
p a r t s ( 3 ) and ( 5 ) of Theorem 9 . 2 g i v e u s c o n d i t i o n ( 3 ) of Theorem 8 . 4 .
is
Consequently, X
C-embedded i n
p(Z(X)).
vX
i s t h e unique Hewitt-Nachbin s p a c e i n which
d e n s e and
C-embedded, w e have proved t h e n e x t r e s u l t .
Since
9.5
and i f
8
i s the collection
p(8) i s
then
If
(Alo and S h a p i r o ) .
COROLLARY
Z(X)
is
X
i s a Tychonoff s p a c e
X
of a l l z e r o - s e t s on
X,
vX.
t h e Hewitt-Nachbin completion
The n e x t r e s u l t a l s o a p p e a r s i n t h e 1969B paper of A l o and S h a p i r o . 9.6
COROLLARY (Alo and S h a p i r o )
space.
. Let
be a Tychonoff
X
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 t r u e :
(1)
If 8 is a s t r o n g d e l t a normal b a s e p ( 8 ) is p r e c i s e l y the G 6 - c l o s u r e of w(8)
Wallman-Frink c o m p a c t i f i c a t i o n q(X)
is
-
G -closure
the
in (2)
6 vx.
then
q ( X ) i n the
.
Moreover,
X
in
pX
and
X
is
is
UX
G -dense
6-
Every non-empty z e r o s e t i n t h e Hewitt-Nachbin completion
Proof.
X,
~ ( 8 ) In . particular
G6-dense i n
of
in
(1) I f
3
vx
meets
X.
i s any element o f
w(8)
which f a i l s t o
have t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y , t h e n t h e p r o o f of 9.4 exhibits a Therefore,
G
G -set
6
G
t h a t contains
m i s s e s t h e subset
q ( X ) of
5
and misses
~ ( 8 ) .I t
p(8).
follows
HEW I TT- NACHBIN COMPLETIONS
t h a t the
G 6 - c l o s u r e of
~(8).
w ( 8 ) i s contained i n
in
p(X)
103
To show t h e o t h e r d i r e c t i o n i t s u f f i c e s t o c o n s i d e r o n l y sets
which a r e t h e i n t e r s e c t i o n of b a s i c open s e t s
G
where t h e complement of
zn
in
8
G h UJ ,
2 . I f G i s such a p ( 8 ) , then f o r e a c h n c m Z n c Un and Zn i 3. S i n c e
belongs t o
Un
5
s e t t h a t c o n t a i n s a member t h e r e is a
Un
of
such t h a t
3 has t h e countable i n t e r s e c t i o n property, t h e r e e x i s t s a point
p
F
n
p ( 8 ) i s contained i n the
i n which c a s e
q(X)
G - c l o s u r e of
6
~ ( 2 ) . T h i s p r o v e s t h e f i r s t s t a t e m e n t of ( 1 ) .
in
T o prove t h e second s t a t e m e n t of
is a
G -set
6
set i n G
q(p) c G P q(X)
1t f o l l o w s t h a t
: nclN].
:Zn
n
p ( 8 ) then
in
~(8). By
G =
(1) o b s e r v e t h a t i f
p(8) n
H , where
the f i r s t statement, H
(i
cp(X)
is a
H
# @
G
t-
so t h a t
q ( X ) # @. T h e r e f o r e , q ( X ) i s G - d e n s e i n ~(8). 6 The f i n a l s t a t e m e n t of (1) i s immediate from 9 . 5 and
what h a s j u s t been proved. (2)
Note t h a t e v e r y z e r o - s e t i n
Since
X
is
immediate.
vX
is a
G -set i n
,X.
6 by p a r t (1) t h e r e s u l t i s
G -dense i n UX 6 This concludes t h e p r o o f .
G - c l o s u r e of a s e t i s
G - c l o s e d , and s i n c e 6 6 e v e r y G - c l o s e d s u b s e t of a Hewitt-Nachbin space i s H e w i t t 6 Nachbin complete by 8 . 7 , Theorem 9 . 4 can be deduced from 9 . 6 .
Since the
However t h e approach taken above i s j u s t i f i e d by e x p o s i n g t h e c o n s t r u c t i o n of
~ ( 3 ) W. e
remark t h a t Gillman and J e r i s o n
p r o v i d e an a l t e r n a t i v e proof t o p a r t ( 2 ) of 9 . 6 (see Gillman and J e r i s o n , 8 . 8 ( b ) ) . The f o l l o w i n g example i s found i n t h e 1969B p a p e r o f Alo and S h a p i r o .
I t w i l l demonstrate t h a t d i s t i n c t s t r o n g d e l t a
--normal bases on of t h a t s p a c e . -Let
X
a space
may p r o d u c e d i f f e r e n t c o m p l e t i o n s
X
be a d i s c r e t e t o p o l o g i c a l s p a c e of c a r d i n a l i t y
c ( t h e c a r d i n a l i t y of
IR) .
I t was shown i n 8.18 t h a t such a
space i s always Hewitt-Nachbin complete. c o l l e c t i o n of a l l s u b s e t s
A
cX
complement XW i s c o u n t a b l e . i s a s t r o n g d e l t a normal b a s e .
Let
B1
f o r which e i t h e r
denote t h e A
or its
~t i s e a s y t o v e r i f y t h a t 81 (Observe t h a t 3, d o e s n o t
r e p r e s e n t the c o l l e c t i o n of a l l z e r o - s e t s i n
X.)
L e t the
HEWITT-NACHBIN SPACES AND CONVERGENCE
104
p(B1) be given a s i n the proof of 9 . 4 , i n i s homeomorphic t o c p ( X ) . I t w i l l be shown, cp(X) # ~ ( 8 ~ To ) . t h i s end, l e t 3 d e n o t e t h e
cp : X
mapping
which c a s e
--f
X
however, t h a t
X
B 1 - f i l t e r c o n s i s t i n g of a l l s u b s e t s of
i s countable. A c
5,
b l e , then
A
or
A
is a
3
Then
either
is a
al-ultrafilter
because f o r each
X’+
i s countable: i f
A
i s c o u n t a b l e , then
~and i f
E
e i t h e r event, 5
whose complement
i s countaX U E 3. I n
X\F\
by 6.8(3). Moreover,
31-ultrafilter
has the countable i n t e r s e c t i o n property.
For suppose { A n :
n c m ) belongs t o 3. Then, s i n c e t h e complement of n € m ] i s c o u n t a b l e i t cannot e q u a l t h e e n t i r e space which c a s e set to
ll (An
:
3
a.
n
(An
X,
in
:
neIN] # F i n a l l y , f o r each PEX t h e 3 so t h a t n 3 = Hence, 3 belongs
a.
X\[pj belongs t o
p(B1) \ c p ( X ) . Since
i s Hewitt-Nachbin complete i t i s t h e c a s e t h a t
X
9 is
X = p ( f j ) , where
t h e c o l l e c t i o n of a l l z e r o - s e t s of
p ( 3 ) i s t h e Hewitt-Nachbin completion
Hence
each a r e d i s t i n c t completions of
t h i s f a c t again s t r e s s e s t h a t
of a l l z e r o - s e t s i n
Z(X) . )
c o l l e c t i o n of
X
by 9 . 5 .
How-
p ( 8 ) i s not homeomorphic t o
e v e r , i t h a s been shown t h a t
~ ( 8 so~ t)h a t
VX
X.
a1
X.
( N o t e that
i s not the c o l l e c t i o n
Z(X)
and t h a t i t m u s t be a proper subOn t h e o t h e r hand,
s i n c e Lindelof
spaces a r e c h a r a c t e r i z e d by t h e p r o p e r t y t h a t every c o l l e c t i o n of c l o s e d s e t s with t h e countable i n t e r s e c t i o n p r o p e r t y i s
f i x e d , i t i s c l e a r t h a t a Lindelof space w i l l always be homeomorphic t o p ( 3 ) f o r e v e r y s t r o n g d e l t a normal base
8
X.
on
The n e x t r e s u l t i s u s e f u l . THEOREM ( A l o and S h a p i r o ) .
9.7
If
----
normal b a s e o n t h e Tychonoff space
Bp
8
i s a stronq d e l t a then t h e c o l l e c t i o n
X,
= ( Z p : Z E ~ )i s a s t r o n q d e l t a normal b a s e on
over, every
gP-ultrafilter
s
p(8).
More-
p ( 8 ) with t h e c o u n t a b l e i n t e r -
section property i s fixed. That
Bp
from 9 . 2 ( 4 ) .
If
Proof.
the point A
in
AP
n
5
5
E
i s a d e l t a r i n g of s e t s f o l l o w s immediately i s any b a s i c c l o s e d s e t of p ( 8 ) and
Zp
p ( 3 ) does n o t belong t o
such t h a t
z p = (A
n
ZIP =
A c X\Z.
Hence,
e.
8P
Thus
Zp
a
then t h e r e i s an
is i n
Ap
i s disjunctive.
and
105
HEWITT- NACHBIN COMPLETIONS
If Z1
n
and
F1
and
ZlP
Z1 c X \ F 1
that (X\F,)’
=
i s normal.
of
8
I f (Cn
Z2 c X\F2.
and
and
I t follows t h a t
ZlP
Z 2 p C (X\F2lp = P ( ~ ) \ F ~ T~h e. r e f o r e ,
n c m ) i s a sequence of complements o f members
:
z = n
such t h a t
n
n , and such t h a t
9,
: ncm7) E
:Cn
8
quence ( Z n : n c m ) of members o f for a l l
8 p , then
there a r e sets
whose complements a r e d i s j o i n t and such
p ( 8 ) \FlP
$
8
By t h e n o r m a l i t y of
8
in
F2
a r e two d i s j o i n t s e t s i n
Z2p
i s empty.
Z2
[Cn
:
1
=
then t h e r e i s a se-
such t h a t
nim
1
Zn c Cn c Zn-l
= r~ ( Z n
: ncN ) .
Thus,
n (z,P
zp =
by (1) and ( 4 ) of 9 . 2 . If
ncN
i n t e r s e c t i o n p r o p e r t y , then
p(8)
8p
Hence
n
jcnP : ncm!
i s complement g e n e r a t e d .
p ( 8 ) with the countable
BP-ultrafilter on
is a
A*
:
A
*
i s a p r i m e z e r o - s e t f i l t e r on
w i t h t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y by 9 . 3 and t h e
zp
fact that
zu) n ~ ( 8 ) .H e n c e ,
=
A
*
is fixed s i n c e
p(8)
i s a Hewitt-Nachbin s p a c e . Many t i m e s and i f Z (fx)
x
Fix
in
Z(f)
8
E
8
# Z(f).
such t h a t
i s any s t r o n g d e l t a normal b a s e
x # Z(f) there is a
then f o r each
Z(X)\8
E
,9
If
X.
such t h a t
= p
i s t h e o n l y s t r o n g d e l t a normal b a s e on
Z(X)
a Tychonoff s p a c e
X\Z(fx)
n
z(f) =
and
Then t h e r e a r e z e r o - s e t s
p c Z ( g ) c X\z(h) c x \ Z ( f )
xcx\z(fx). Z ( g ) and
.
Thus,
# z(f)j u x \ z ( g ) i s an open c o v e r o f
X.
i s L i n d e l o f , then a c o u n t a b l e subcover w i l l c o v e r
X,
(x\z(fx) : x Z(f) =
n
(Z(fx )
: iEm)
n
Z(h),
Z (h)
~f
x
t h a t is
Thus w e have shown t h e
i f o l l o w i n g r e s u l t which may b e found i n t h e 1 9 7 1 p a p e r by A . S t e i n e r and E .
Steiner.
THEOmM ( S t e i n e r and S t e i n e r ) .
9.8
If
X
i s a Tychonoff
s p a c e t h a t i s L i n d e l o f , t h e n t h e o n l y s t r o n g d e l t a normal b a s e on -
X
i s the collection
Z ( X ) of a l l z e r o - s e t s .
N o w t h e o n e - p o i n t c o m p a c t i f i c a t i o n (and t h e r e f o r e com-
p l e t i o n ) IN
*
of t h e p o s i t i v e i n t e g e r s
IN
cannot be obtained
SPACES AND CONVERGENCE
106
HEWITT-NACHBIN
a s a space
p ( 8 ) f o r any s u i t a b l e s t r o n g d e l t a normal b a s e
8
( t h i s i s found i n t h e 1971 p a p e r by A . S t e i n e r and E . S t e i n e r ) .
IN
From t h e above r e s u l t t h e o n l y s t r o n g d e l t a normal b a s e on is
However, w e have a l r e a d y i n d i c a t e d a way o f o b t a i n -
Z(lN).
i n g any o n e - p o i n t c o m p a c t i f i c a t i o n (and t h e r e f o r e c o m p l e t i o n )
w(3) for a
of a l o c a l l y compact Hausdorff s p a c e a s a s p a c e p a r t i c u l a r normal b a s e was used t o o b t a i n
iJm
N
*
.
8.
N e v e r t h e l e s s a Wallman- t y p e method Of course w e note t h a t
~ ( Z ( I N ) )=
= N .
Another i n t e r e s t i n g example of a completion of a Tycho-
p(8) is t o
n o f f s p a c e t h a t c a n n o t be o b t a i n e d a s a s p a c e c o n s i d e r the space on
IR.
Now
Q
of r a t i o n a l s i n t h e r e l a t i v e topology
i s L i n d e l o f and hence
Q
s t r o n g d e l t a normal b a s e on Nachbin space s o t h a t completion of
IR
that
Q.
Z(Q) is t h e only Q
Moreover,
Q = uQ = p ( Z )
.
is a Hewitt-
The r e a l l i n e
However, by o u r p r e v i o u s remarks, w e see
Q.
i s not o b t a i n a b l e a s a space
s t r o n g d e l t a normal b a s e on
Q.
Clearly
p ( 8 ) where
IR
8
is a
cannot b e obtain-
"(3) b e c a u s e i t f a i l s t o b e compact.
ed a s a s p a c e
is a
IR
Conse-
q u e n t l y , an a p p r o p r i a t e s t r e n g t h e n i n g of t h e c o n c e p t of normal b a s e s o a s t o have a Wallman-type method o f o b t a i n i n g a l l c o m p l e t i o n s o f a Tychonoff s p a c e must be weaker t h a n the conc e p t of a s t r o n g d e l t a normal b a s e . W e remark t h a t i n h i s 1969 p a p e r J . Van d e r S l o t h a s
a l s o provided a g e n e r a l completion c o n s t r u c t i o n which i s based on t h e work o f J . D e Groot and J . A a r t s ( 1 9 6 9 ) . We conclude t h i s s e c t i o n w i t h t h e f o l l o w i n g e x t e n s i o n theorem a s s o c i a t e d w i t h t h e completion 9.9
Let
THEOREM.
--d e l t a normal b a s e s
X
F,&
and and
c o n t i n u o u s mapping from whenever
of
f
Proof.
4,
Z E
from Let
pQ%) p
X
Y
~ ( 8 ) .
mchonoff spaces with s t r o n q
q ,r e s p e c t i v e l y .
into
Y
such t h a t
If
f-'(Z)
is a E
%
then t h e r e e x i s t s a c o n t i n u o u s e x t e n s i o n
into
denote a n a r b i t r a r y p o i n t i n p(%)
(2 E
f
p(&).
denote the following s u b c o l l e c t i o n :
al=
f
4:
P E cl p($-l(Z)
1.
and l e t B1
*
Q1
Then
is a
&-filter
on
because, by 9 . 2 ( 4 )
Y
hp
h P - f i l t e r on
h),
and l e t alp We claim t h a t a l p is a
Let denote t h e c o l l e c t i o n i Z p denote t h e c o l l e c t i o n ( Z p : z E ‘Y1). prime
107
COMPLETIONS
HEWITT-NACHBIN
: Z E
p ( & ) w i t h t h e countable i n t e r s e c t i o n
property . For suppose t h a t ( Z n p : n 6 . N ) i s a countable subcollect i o n of
alp
(f-l(zn)
:
n
(ci
P
with empty i n t e r s e c t i o n .
Then the c o l l e c t i o n
n c m ) has empty i n t e r s e c t i o n which i m p l i e s t h a t This i s a
f - l ( Z n ) : n c m ) i s empty by 9 . 2 ( 5 ) .
(8.x)
c o n t r a d i c t i o n s i n c e the p o i n t
p
belongs t o t h e i n t e r s e c t i o n
f-’(Zn) : n c m ) by the d e f i n i t i o n of al. n [ci P(8X) alp has t h e countable i n t e r s e c t i o n p r o p e r t y . I t i s t h a t alp i s a q P - f i l t e r .
alp
To e s t a b l i s h t h a t ZlP
U Z 2 p c (Z1
p
cl
E
P(+)
cl
so t h a t
p or
z1
z2p
E
to f*
n alp.
p.
E
E
n
a1
@Jl
E
so t h a t
Thus, p
belongs t o
(Z,)
p
or
t
c l p ( Gf)- l ( Z , ) .
by d e f i n i t i o n , so t h a t
alp
Therefore,
zlp
Hence,
alp
E
or
i s prime.
By 6 . 1 6 and 9 . 7 t h e r e e x i s t s a unique p o i n t belonging We d e f i n e f * ( p ) E n a l p , and we w i l l show t h a t i s a continuous extension of t h e f u n c t i o n f . The mapping
of
f-
c lp ( 9 X ) f - 1 ( ~ 1 ) Z2
U Z2)
(Z1
U Z 2 ) by d e f i n i t i o n .
P (8,)
immediate
i s prime, suppose t h a t
Hence,
U Z2)p.
fP1(Zl
Therefore,
f*
from
f , f o r if the p o i n t
{cip ( 4 ) Z
: Z E
.S,
and
p
p(&) i n t o belongs t o p
E
f-’(Z)).
p(&) X,
then
i s a n extension
f(p) is i n
Since t h e l a t t e r
i n t e r s e c t ion i s p r e c i s e l y and t h i s implies t h a t
.
f (p) = f * ( p ) * To e s t a b l i s h t h a t f i s continuous, l e t
p
E
p(%)
be
108
a r b i t r a r y and l e t
p(&)
containing
exists a set 3 Zp =
ZlP
sets
SPACES AND CONVERGENCE
HEWITT-NACHBIN
ZlP
6.
j+p
E
b e a b a s i c open s e t i n
hP i s
such t h a t
disjunctive there
ft(p)
hp
belonging t o
C2p
and
ZlP
E
j+p
Then by t h e n o r m a l i t y o f
and
ClP
p ( & ) \Zp Since
Up =
f*(p).
there exist
such t h a t
Zp
c
(p(h)
P ( & ) \clp, Z l P c p ( 4 ) \C2’ and ( p ( j + ) \ClP) \C,h = 6. 1 f- ( C , ) . W e c l a i m t h a t PEV and Define V = p ( & ) \ c l P (iQ f*(V) c Up. For i f pkv then p E c l f-l(C,) so t h a t C2 P(%) b e l o n g s t o a1 = ( Z E : p c cl f - l ( Z ) ) and C 2 p E alp. P(&) Now, f * ( p ) E n alp which i m p l i e s t h a t f * ( p ) E c 2 p c o n t r a -
4
d i c t i n g the f a c t t h a t suppose t h a t
xcV
c P(&)\C,~.
f * ( p ) E Z1p
i n which c a s e
x
# c l p ( Gf)- l ( C , ) .
C2p
f a i l s to belong to t h e c o l l e c t i o n
and
x F cl
f-l(Z)].
P (&) i s a prime q P ; f i l t e r implies t h a t
maps
V
into
S e c t i o n 10 :
f
(x)
E
axp =
Therefore, Clp
p(&)
on
and
so t h a t
ClP
Finally,
QXp
E
Clp
f*(x)
Hence
[Zp : Z E
axp
because
U C2p = p ( & ) .
#
Zp.
Hence
This f*
T h i s c o n c l u d e s t h e p r o o f of t h e theorem.
Up.
z-Embeddinq and
u-Embeddinq
*
I n S e c t i o n 4 t h e n o t i o n s of C- and C -embedding were i n t r o d u c e d and it was observed t h a t t h e s e p a r a t i o n axiom of n o r m a l i t y is c h a r a c t e r i z e d i n t e r m s o f t h o s e c o n c e p t s . Furt h e r on ( S e c t i o n 8 ) i t was e s t a b l i s h e d t h a t UX i s the l a r g e s t subspace o f BX i n which X i s C-embedded. Several o t h e r t y p e s of embeddings p l a y an i m p o r t a n t p a r t i n c o n n e c t i o n w i t h t h e s t u d y of Hewitt-Nachbin s p a c e s t h a t a r e weaker s t i l l * than C -embedding. I t i s the i n t e n t of t h i s section t o i n v e s t i g a t e t h e s e embeddings.
The f i r s t p a r t o f o u r development
c l o s e l y f o l l o w s t h a t found i n t h e 1 9 7 4 book by R. Alo and H . L. S h a p i r o wherein t h e r e l a t i o n s h i p b e t w e e n
z-embedding and
normality is studied extensively. 1 0 . 1 DEFINITION.
Let
t r a r y t o p o l o g i c a l space
x
i f every z e r o - s e t
some z e r o - s e t
Z1
in
b e a non-empty s u b s e t of an a r b i -
S
Z
S i s z-embedded & I i s o f t h e form S n Z f f o r X ( t h a t is, i f every z e r o - s e t i n S is
X.
The s u b s e t
in
S
2-EMBEDDING AND
the i n t e r s e c t i o n of
with a z e r o - s e t i n
S
a r e two s u b s e t s of
then
X,
Z1
i f there e x i s t zero-sets A c
zl,
B c
X
z-embedded i n
and
z1 n z2
z2, and
Notice t h a t i f
and
A
is
109
U-EMBEDDING
;s
If
B
in
X
0.
=
C -embedded i n
X
then
b e c a u s e e v e r y z e r o - s e t of
S
i s t h e zero-
S
is
S
However
z-embedded s u b s e t s t h a t a r e n o t
*
and
such t h a t
s e t of a bounded 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 . examples abound of
A
S-separated
X
of
Z2
.
X)
are
B
*
C -embed-
ded:
any non
ded.
The l a t t e r o b s e r v a t i o n f o l l o w s from t h e f a c t t h a t i n a
C -embedded
X
p e r f e c t l y normal s p a c e see t h i s l e t
S
of
Z
S.
Then
G -set i n
6
i s a zero-set
z- embedded i n x
C -embedding.
every s u b s e t is
be a s u b s e t of
X
F
of
such t h a t
X
z-embed-
z-embedded.
and l e t
is a closed subset of
a closed subset a
subset of the r e a l l i n e i s
To
be a z e r o - s e t
Z
and h e n c e t h e r e i s
S
n
Z = S
But
F.
F
is
and e v e r y c l o s e d G 6- s e t i n a normal s p a c e (see Gillman and J e r i s o n , 3 D . 3 ) . Thus S i s
X,
X. Consequently z- embedding i s weaker than I n t h e f i n a l c h a p t e r w e w i l l see t h a t z-embed-
d i n g i s h e l p f u l i n t h e p r e s e r v a t i o n o f Hewitt-Nachbin comp l e t e n e s s under c l o s e d c o n t i n u o u s mappings. The f o l l o w i n g res u l t c h a r a c t e r i z i n g t h e c o n c e p t o f z-embedding i n a manner a n a l o g o u s t o Theorem 4 . 8 is due t o R . B l a i r ( 1 9 6 4 ) . 10.2
If
THEOREM ( B l a i r ) .
t o p o l o g i c a l space
X,
i s a non-empty s u b s e t o f a
S
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 -
lent: (1) The s u b s e t
is
z-embedded
B
are
completely s e p a r a t e d
(2)
If
(3)
and g ( x ) # 0 if xcB. If A and B are c o m p l e t e l y t h e y a r e S - s e p a r a t e d in X .
A
and
S
there e x i s t s
g
E
X.
--
C ( X ) such t h a t
in
g(x) = 0
S
then
S
then
if
xeA
Proof.
separated
in
W e w i l l show t h a t (1) i m p l i e s ( 2 ) i m p l i e s ( 3 ) i m p l i e s
( 1 ) . Assuming (1) suppose t h a t
separated i n the
A
and
z-embedded s u b s e t
S
B
of
a r e completely X.
I t follows
i n Z(S) such t h a t A C Z and Z n B = By assumption t h e r e i s a z e r o - s e t Z ' = Z ( g ) i n Z(x) such t h a t Z = s l l Z ' Note t h a t g from 3 . 1 1 t h a t t h e r e i s a z e r o - s e t
a.
Z
.
1l o
HEWITT-NACHBIN SPACES AND CONVERGENCE
belongs t o
C(X),
g(x) = 0
This e s t a b l i s h e s Next assume separated i n
S.
z e r o - s e t s of
S
assume t h a t
X E A , and
( 2 ) h o l d s and t h a t
g(x)
#
if
0
Thus
and
A
and
A
B
XEB.
a r e completely
a r e contained i n d i s j o i n t
B
by 3 . 1 1 s o w i t h o u t loss of g e n e r a l i t y and
A
if
(2).
may
WE
a r e themselves d i s j o i n t z e r o - s e t s .
B
assumption t h e r e e x i s t s a zero- s e t
in
Z1
Z(X)
By
such t h a t
a,
and Z1 n B = Then ( S n Z1) and B a r e d i s j o i n t 1 in z e r o - s e t s s o t h a t a g a i n by ( 2 ) t h e r e i s a z e r o - s e t Z 2
A c Z
Z(x)
such t h a t
B c
z2
z 2 n (s n zl)
and
a.
=
This e s t a b -
lishes (3). F i n a l l y , suppose ( 3 ) h o l d s and l e t A = Z ( f ) b e l o n g t o n , d e f i n e t h e s e t Bn -
Z(S), For each p o s i t i v e i n t e g e r
2 ;),1
{xcS : f ( x )
Then A and Bn a r e c o m p l e t e l y s e p a r a t e d s o t h a t by ( 3 ) t h e r e e x i s t s a Zn i n Z ( X ) such
n
f o r each
A c Zn
that
and
of a l l such contains then f(x)
x
Zn.
a.
Let
Z1 Z1
n
x
belongs t o
Zn
B =
A.
On t h e o t h e r hand,
Bn
for a l l positive integers
p
<
n
f o r every
so t h a t
n
Then t h e z e r o - s e t
S
is
if
and hence
z-embedded i n
be t h e i n t e r s e c t i o n
z-embedding.
n.
Z(S) c l e a r l y Z 1 fl S
It follows t h a t
f ( x ) = 0.
Thus
Z1
n
S = A
and (1) h o l d s .
X
The n e x t two r e s u l t s c h a r a c t e r i z e
i n terms of
in
S
C-
and
*
C -embedding
Note t h e s i m i l a r i t y o f 1 0 . 3 w i t h
4.8(1). THEOREM ( B l a i r ) .
10.3
t o p o l o g i c a l space
Let
b e a non-empty s u b s e t of a
S
Then t h e f o l l o w i n g s t a t e m e n t s
X.
equivalent: S
(1) The s u b s e t (2)
is is
The s u b s e t S A c s and each
--
S-separated
in
suppose are
If
z-embedded
2 E Z(x)
X
p l e t e l y separated Proof.
*
C -embedded
then
in
and
S-separated i n
belonging t o
Z E Z(X)
X.
X
and for each
if A and s n z are and S n Z a r e com-
A
X.
(1) h o l d s then c l e a r l y
A c S
& I
in
are
S
is
z-embedded.
a r e such t h a t
A
and
Next S
n
Z
X. Then t h e r e e x i s t z e r o - s e t s Z1, Z 2 Z(x) such t h a t A c s f l zl, s n z c s n z2, and
Z-
(s n zl) fl ( s n
EMBEDDING AND
z2) = @ ,
Since
S.
*
S.
Z1 and
Z2 i n
Z2 = 0 .
X
a r e completely
Then by 10.2(3) t h e y a r e
S-separated i n
S
B u t then
Z1 fl S
A
and
A
S,
C
B
and
B
is
C
S
z1
s n
Z1,
B c Z2, and
are
S-separated i n
X
-embedded i n
fl
so
X
Finally,
X.
a r e completely s e p a r a t e d i n &
T h e r e f o r e by 4 . 8 ( 1 )
X
that is there are zero-sets
X:
such t h a t
X
and
A
t h a t by ( 2 ) they a r e c o m p l e t e l y s e p a r a t e d i n
Z1
are
in
B
z-embedded i n
S
A C
z
Tl
Thus (2) h o l d s .
X.
because
since
s
and
C -embedded
a r e completely
and
A
Assuming ( 2 ) suppose t h a t is
is
S
n Z
i t f o l l o w s from 4 . 8 ( 1 ) t h a t
separated i n
111
EMBEDDING
I n o t h e r words, A
completely s e p a r a t e d i n separated i n
U-
X.
and (1) h o l d s .
Note t h e s i m i l a r i t y o f t h e n e x t r e s u l t w i t h 4 . 8 ( 2 ) . 10.4
Let
THEOREM ( B l a i r ) .
t o p o l o q i c a l space
b e a non-empty s u b s e t of 2
S
are
Then t h e f o l l o w i n q s t a t e m e n t s
X.
equivalent: (1)
The s u b s e t
S
(2)
The s u b s e t
S
is
in in
C-embedded z-embedded
X. X
and
completely
s e p a r a t e d from e v e r y z e r o - s e t d i s j o i n t from i t . proof.
T h a t (1) i m p l i e s
observation t h a t
(2) i s immediate
C-embedding i m p l i e s
Next assume t h a t ( 2 ) h o l d s . i t i s s u f f i c i e n t t o prove t h a t S
b y 4 . 8 ( 2 ) and t h e
z-embedding.
Then a c c o r d i n g t o 4 . 8 ( 2 ) is
*
in
C -embedded
w i l l e s t a b l i s h t h a t t h e c o n d i t i o n i n lo.3 (2) h o l d s .
and
z
S- s e p a r a t e d i n
X.
suppose are
in Then
c S
A
Z ( x ) such t h a t
Z1
n
E Z(X)
a r e such t h a t
A c zl,
s n z c z2,
Z2 i s a z e r o - s e t i n
X
x
z1 fl z
x
s c z and s n
z* n (zln z 2 ) * z c z2 fl z so t h a t
*
C -embedded
10.5
in
X
X.
C-embedded
=
0.
A
X.
z2 = 0 .
in
But then and
s
Tl
S.
Z(X) A c
z
are
I t f o l l o w s from 1 0 . 3 t h a t
. If
S
i s non-empty, X, then
G6-dense i n t h e t o p o l o g i c a l s p a c e
in
*
Z
Z2
S
is
which e s t a b l i s h e s ( 1 ) .
COROLLARY ( B l a i r - H a g e r )
ded, and
Z
and
t h a t i s d i s j o i n t from
and
completely s e p a r a t e d i n
n
S
Z1
s n z1 fl
and
T h e r e f o r e by assumption t h e r e e x i s t s a z e r o - s e t such t h a t
Thus
and
A
Then t h e r e a r e zero- se ts
We
X.
z-embedS
is
112
HEWITT-NACHBIN
Proof.
SPACES AND CONVERGENCE
Since every z e r o - s e t i n
i s d i s j o i n t from
is a
X
G -set,
no z e r o - s e t
b and t h e c o n d i t i o n i n 10.4(2) i s s a t i s f i e d
S
vacuously.
I n t h e i r 1974 book, Alo and S h a p i r o show t h a t a topol o q i c a l s p a c e i s normal i f and o n l y i f e v e r y c l o s e d s u b s e t i s z-embedded.
Coupling t h a t r e s u l t w i t h t h o s e g i v e n i n Gillman
and J e r i s o n ,
3D.1,
we see t h a t f o r normal s p a c e s t h e c l o s e d
s u b s e t s s a t i s f y a l l t h r e e p r o p e r t i e s of
C-,
C
*
-,
and
z-embed-
For any t o p o l o g i c a l s p a c e t h e c o n d i t i o n s a r e e q u i v a l e n t
ding.
f o r s u b s e t s t h a t a r e zero- s e t s . COROLLARY ( B l a i r ) .
10.6
the topological
space
X,
If
is a non-empty z e r o - s e t of
Z
then t h e f o l l o w i n g s t a t e m e n t s
are
equivalent:
(1) T h e set
Z
The s e t
Z
The s e t
Z
(2) (3)
Proof.
is
C-embedded
*
z-embedded
& I
X.
X.
t h a t i s d i s j o i n t from
X
C-embedded i n
X.
( 3 ) i m p l i e s (1). Thus l e t
a r e completely s e p a r a t e d i n
Z’
,&
C -embedded
is
W e need o n l y prove t h a t
be a z e r o - s e t i n and
is
Two i m p o r t a n t c l a s s e s of
THEOREM.
(1) ( B l a i r ) .
---
then i t i s
(2)
=
(1) S i n c e
Proof. f o r some
f
in
S.
in
Z ( f ) and
E
C(X)
If
z-embedded
X.
If
Tychonoff s p a c e
X,
X.
.
s
-
i s a c o z e r o - s u b s e t of
S
& I
i s a Lindelof
S
Define a f u n c t i o n
h
Z ( 9 ),
on
h ( x ) = ( f A g) (x) i f
g
negative real-valued functions.)
E
h(x) = 0
by
X
x Since
The c o n t i n u i t y of
h
s
X,
sub-
z-embed= x\Z(f)
C(S) , be a z e r o - s e t
is i n
S.
and
f
g
if
a t points of
S
h
x
is
(Without a r e non-
Z(g) = Z(h)
proof w i l l be completed once i t i s shown t h a t X.
&
S
i s a c o z e r o - s e t w e may s e t
Now l e t
l o s s of g e n e r a l i t y w e may assume t h a t
on
Z
z-embedded s u b s e t s a r e pro-
(Henriksen and J o h n s o n ) .
-set of ded i n --
Z
X.
v i d e d by t h e n e x t result. 10.7
Clearly
Z.
s o t h a t by 1 0 . 4
X
2’
n
S, t h e
i s continuous
is clear s i n c e
2- EMBEDDING AND
113
U-EMBEDDING
i t i s t h e infimum of two c o n t i n u o u s f u n c t i o n s on
Now l e t
S.
Z(f) and E > 0 . Then o b s e r v e t h a t t h e s e t N = (XEX : h ( x ) < t i i s simply t h e union [xtX : f ( x ) < E j U (XES : g(x) < E ] . The f i r s t s e t i n t h i s union i s open i n X and khe second s e t i s open i n S , hence i s open i n X. Thus N is a p
E
p
neighborhood o f hood of (2)
which
h
maps i n t o t h e g i v e n
z
upp pose
i s a z e r o - s e t of
F -set i n
s e t i t i s an
Since
S.
5
! (s\z) n z *
=
:
n 5
show t h a t
6.
=
z
Z(x) and
E
x ,d c l x Z .
x
Suppose
whose i n t e r s e c t i o n w i t h Thus
*
z
is
S
S\Z.
E
We w i l l
S\Z.
Any open s e t i n
X
w i l l be d i s j o i n t from
S\Z
-
a
c z*].
Z.
Consequently by t h e complete r e g u l a r i t y of
t h e r e i s a c o n t i n u o u s f u n c t i o n f i n C ( X ) such t h a t and
F
Let
i s a c o l l e c t i o n of c l o s e d s u b s e t s o f
3
i s a cozero-
S\Z
S ( i t i s e a s y t o show t h a t e v e r y
a
s u b s e t of a Lindelof space is L i n d e l o f ) .
Thus
E-neighbor-
0.
f(y) = 0
y
for a l l
belong t o
z(f)
e v e r , S\Z
i s an
n (s\z),
E
F -set i n
S
x
Thus t h e p o i n t
clxZ.
31, so
an e l e m e n t o f
X
f(x) = 1
n 3
does n o t
= gi.
and h e n c e L i n d e l o f .
HOW-
It fol-
a : ncN ) o f z e r o - s e t s lows t h a t t h e r e i s a c o u n t a b l e f a m i l y [ Z n i n X such t h a t Z n fl (S\Z) i s i n 3 f o r a l l n , and 00
(I)
n [zn n (s\z) J
gj =
n zn n ( s \ z ) .
=
n=l
n= 1 Let
Z
*
= fl ( Z n
Z c Zn
ncm.
Z
*
n
S = Z
Z*
i s a z e r o - s e t on
X
and
Therefore,
Z c Z*
Hence
Then
: n+z7N].
for a l l
and
so t h a t
z * fl S
is
(s\z)
= gi,
z-embedded
X.
in
z- embedding a r e worth mentioning, and a p p e a r i n Alo and S h a p i r o ’ s book. F o r example, S e v e r a l o t h e r r e s u l t s concerning
every
normal t o p o l o q i c a l -
F -subset of
a--
-i s z-embedded ded i n
X.
I n fact, X
in
X.
space
X
i s normal i f and o n l y i f e v e r y
z-embedF -set
a-
Next w e o b t a i n a c h a r a c t e r i z a t i o n of
z-embedding i n terms o f z e r o - s e t f i l t e r s .
114
SPACES AND CONVERGENCE
HEWITT-NACHBIN
10.8 D E F I N I T I O N . I f 3 i s a Z - f i l t e r on X and non-empty s u b s e t of X , then by t h e t r a c e of 3
meant the collection
S S = ‘ Z fi S : Z c 3 1 .
forms a b a s e f o r a z e r o - s e t f i l t e r on
X
z n s #
3.
$3
z
€ o r every
belonging t o
S, b u t if
Ss
Note t h a t
Ss z-embedded i n
is
S
S,
i f and o n l y i f
I n g e n e r a l i t is not t r u e t h a t the t r a c e z e r o - s e t f i l t e r on
is a is
S
on
w i l l be a the
X
s i t u a t i o n i s improved a s t h e f o l l o w i n g theorem d e m o n s t r a t e s .
Let
THEOREM ( B l a i r ) .
10.9
Tychonoff s p a c e
b e a non-empty
S
subset of the
Then t h e f o l l o w i n q s t a t e m e n t s
X.
are
equiva len t :
(1) The s u b s e t (2) (3)
is
S
Z-ultrafilter
[ i8 (Q)], =
G,
If
3
z n
S
is 5 # $3
filter --
i s the i n c l u s i o n S c X. 2-ultrafilter X such t h a t
on
Then
S.
Q
Z-ultra-
belonging t o
i-’(zT
)
G I so
Z(X)
.
is a zero-set u l t r a f i l t e r
c l e a r l y [ i8 (Q)], =
such t h a t
S fl Z
# @
Q.
Q
ultrafilter
Zs
c Q. Hence Z 3 = i # (G) because
8 [ i (Q)],
s
on
= Q
E
with
i H (G)
3
c
so t h a t
6.
Z(X)
= (Z E
is a
is a
Z E ~ . Then
If
Z E ~ t, h e n
: i-’(Z)
2-ultrafilter. gS
Z-ultrafilter
zs
is a
so t h a t t h e r e e x i s t s a
S
as
:
Thus ( 2 ) h o l d s .
3
f o r every
b a s e f o r a z e r o - s e t f i l t e r on
for
S = i-’(Z’)
i# ( Q ) = ( Z ’ E Z ( X )
But
Next assume ( 2 ) h o l d s and t h a t X
n
Z = ZT
i f and o n l y i f
ZEQ
Z’
=
is a
S.
some
Zs
Zs
Z E ~ ,then
(1). Assuming (11, suppose t h a t
on
X. S, the trace
W e w i l l show t h a t (1) i m p l i e s ( 2 ) i m p l i e s ( 3 ) i m p l i e s
Proof. on
on
Q
i
where
f o r every
on
&I
z-embedded
For e v e r y
is a
E
Z-
( Z fl ,S)
Q).
E
Thus
F i n a l l y , by ( 2 ) ,
s. This
Z - u l t r a f i l t e r on
establishes ( 3 ) . Assuming t h a t ( 3 ) h o l d s w e w i l l show t h a t c o n d i t i o n ( 2 ) of 1 0 . 2 is s a t i s f i e d . s u b s e t s of
S
A
and
B
A
# $3
t h a t a r e completely separated i n
and c o n s i d e r t h e f i x e d (see 6 . 6 ) .
Thus suppose t h a t 2-ultrafilter
Then by (3), S s
is a
3
and S.
= ( Z E Z(X)
Z - u l t r a f i l t e r on
are completely s e p a r a t e d i n
S
B
Let
are PEA
: PEZ)
S.
Since
t h e r e e x i s t zero-
Z-
sets
and
Z1 Z2 =
Z1
of t h e Z
n
0.
in
Z2
Then
Z1
f o r some
g(x) = 0
if
zs
F
Z = z ( g ) where
and
XFA
B c Z2,
and
meets e v e r y m e m b e r
Z1
By d e f i n i t i o n of t h e t r a c e ,
g(x)
#
( 2 ) i m p l i e s (l), t h a t
10.2,
A c Z 1,
because
3,.
Z-ultrafilter
S
such t h a t
S
115
U- EMBEDDING
EMBEDDING AND
g if
0
S
belongs t o
is
=
Z1
Thus
C(X).
I t f o l l o w s from
XEB.
z-embedded i n
This
X.
completes t h e proof of t h e theorem.
10.10
If
on
is a Z-ultrafilter X with t h e zc o u n t a b l e i n t e r s e c t i o n p r o p e r t y and i f S i s a non-empty COROLLARY.
3
embedded s u b s e t of
X
then t h e t r a c e zs ---
is a
ble intersection Proof.
Z
n
S
Z-ultrafilter
# @
on
f o r every
Z E ~ ,
w i t h t h e counta-
S
property.
3
Since
such t h a t
i s c l o s e d under c o u n t a b l e i n t e r s e c t i o n s by
6 . 1 4 , t h e proof i s immediate from (1) i m p l i e s ( 3 ) of t h e theorem. W e n e x t r e l a t e t h e concept of
z-embedding t o t h e counta-
b l e union o f Hewitt-Nachbin s p a c e s . 10.11 THEOREM ( B l a i r ) .
If
Tvchonoff space such t h a t
X
X = U ( X n : n c N ] where each
--
that is
Proof.
z-embedded Let
3
X,
be a
@
zn
i s a Hewitt-Nachbin
subspace
i s a Hewitt-Nachbin s p a c e .
Z - u l t r a f i l t e r on
intersection property.
is a zero-set
Xn then X
X
with the countable
n
I f f o r each p o s i t i v e i n t e g e r
a
in
with
zn n xn
=
@,
then
z
=
nE m
c o n t r a r y t o the? countable i n t e r s e c t i o n p r o p e r t y of
3.
n
3.
Therefore, f o r some
lo. 10
the trace
n,
Z
is a
Xn
#
@
f o r every
2 - u l t r a f i l t e r on
countable i n t e r s e c t i o n property.
Therefore
Z
Xn
@ #
in
X
By
with t h e
n ZX n
and
there
n zn=
C
n
3;
i s a Hewitt-Nachbin s p a c e . Note t h a t s i n c e every c l o s e d subspace of a normal space
is
z-embedded t h e r e i n w e o b t a i n Mrdwka’s r e s u l t 8.13(2) a s a
c o r o l l o r y t o 10.11.
However our approach i n o b t a i n i n g 8.13(2)
i s j u s t i f i e d by t h e c o n s t r u c t i v e proof t h a t was u t i l i z e d t h e r e . W e now focus our a t t e n t i o n on s t i l l a n o t h e r embedding
concept t h a t t u r n s o u t t o be weaker even than
z-embedding.
116
SPACES AND CONVERGENCE
HEWITT-NACHBIN
I n o r d e r t o s i m p l i f y t h e n o t a t i o n throughout t h e remainder o f
r
t h i s section, we w i l l let
2s
tension
+
SX
d e n o t e t h e Hewitt-Nachbin ex-
of the inclusion
subset
S
Of a Tychonoff
space X i s s a i d t o b e 2-embedded jJ a homeomorphism from US o n t o r ( u . 5 ) .
X
if
10.12
A non-empty
S c X.
DEFINITION.
7
:
2s
-$
is
uX
Li-embedding i s i n v e s t i g a t e d e x t e n s i v e l y
The c o n c e p t of
i n t h e 1 9 7 4 p a p e r by R . B l a i r .
I t is certainly a natural
n o t i o n t h a t d e s e r v e s a t t e n t i o n i n t h e s t u d y o f t h e Hewitt-NachThe main r e s u l t 1 0 . 1 7 w i l l p r o v i d e t h e formu-
b i n completion. lation that
is
S
notion f o r is
*
P
in
;-embedded
( u p t o a homeomorphism).
y i e l d s n o t h i n g new:
in
C -embedded
i f and o n l y i f
X
QS c 'JX
Observe t h a t t h e c o r r e s p o n d i n g
PS c pX
i f and o n l y i f
(see Gillman and J e r i s o n , 6 . 9 ( a ) ) .
X
t h e n e x t s e c t i o n w e w i l l see t h a t
i n t h e s t u d y of t h e e q u a l i t y
S
In
j~-embedding i s s i g n i f i c a n t
u ( X x Y ) = UX x v Y .
The f o l l o w i n g n o t i o n i s b a s i c t o o u r development. 10.13
n o f f space
S
be a non-empty s u b s e t o f a Tycho-
By t h e d i l a t i o n
X.
of a l l p o i n t s i n on
Let
DEFINITION.
X
of
It is clear that i f W e w i l l see l a t e r t h a t i f
diluxS.
X
t h a t a r e l i m i t s of r e a l
We d e n o t e t h e d i l a t i o n by
S.
jJ
S
One might c o n j e c t u r e t h a t
Z-ultrafilters
dilXS.
S c X c Y,
vS c uX,
i s m e a n t t h e set
then
d i l S = X fl d i l y S . X
then n e c e s s a r i l y dil
UX
S
US =
m u s t always be a
Hewitt-Nachbin s p a c e , b u t B l a i r p r o v i d e s an example t o t h e c o n t r a r y i n h i s 1972 p a p e r (see Example 2.6 i n t h a t p a p e r ) . Before proving t h e main r e s u l t g i v i n g s e v e r a l e q u i v a l e n t v-embedding a few o b s e r v a t i o n s a r e i n o r d e r
f o r m u l a t i o n s of
which should c l a r i f y t h e g e n e r a l s i t u a t i o n : For
s c
X
i t i s always t h e c a s e t h a t
S c d i l ux s c G 6 - c l ux s c c l u x S .
W e need o n l y e s t a b l i s h t h e second i n c l u s i o n : I f p E d i l u X S then t h e r e i s a r e a l Z - u l t r a f i l t e r 3 on S t h a t c o n v e r g e s
z-EMBEDDING AND
in
p
to
and
space by 8 . 7 ,
Z - f i l t e r on A = G 6 - c l CXS A i s a Hewitt-Nachbin
S denote t h e
Let
LX.
t h a t i s g e n e r a t e d by
5.
The subspace
i s a prime
Q
Z - f i l t e r on
Q
countable i n t e r s e c t i o n p r o p e r t y because
(in fact, G 6 . 1 7 and 6 . 1 9 because i t i s a prime i s the i n c l u s i o n
in
q
is a
S c A
under c o u n t a b l e i n t e r s e c t i o n s ) . some p o i n t
with the
A
= id
(5), where
2 - f i l t e r t h a t is closed
q
converges t o
p = q.
I t was e s t a b l i s h e d i n 8.11 t h a t t h e e q u a l i t y
it occurs i f
occurs q u i t e r a r e l y : and o n l y i f
S
is
i
Z - u l t r a f i l t e r by
Therefore,
Necessarily
A.
117
u-EMBEDDING
is
S
'JS = c l , , S
dX
C-embedded p r o v i d e d t h a t e i t h e r
i s normal (Gillman and J e r i s o n , 8 . l O ( b ) ) .
X;
C-embedded i n or
X
YX
,AS =
The e q u a l i t y
G6-clUXS o c c u r s much more f r e q u e n t l y . 10.14
If
THEOREM ( B l a i r ) .
X,
Tychonoff space z-embedded Proof.
&
If
then
Gb-cl!
JX
C-embedded i n
also
T
Nachbin s p a c e by 8 . 7 , US = T
then
is
S
US = G - c l , , S
onlyif
by 1 0 . 5 .
T = G -clXxS, then
6 Moreover, T
and t h e r e f o r e C-embedded
US = T .
S
is
S
is
is a H e w i t t Conversely, i f
z-embedded) i n
(and hence
The n e x t r e s u l t e s t a b l i s h e s t h a t than
i f and --
dX
6
S.
z-embedded i n
is
S
i s a non-empty s u b s e t of t h e
S
u-embedding
T.
i s weaker
z-embedding.
10.15
COROLLARY ( B l a i r - H a g e r )
Tychonoff s p a c e
X,
then
. If
S
S
z-embedded
+embedded
X
and
i n the US =
G -cluxS.
6'
Proof. ding, S
in
T.
l o . 16
Let
is
6 z-embedded i n
By t h e t r a n s i t i v i t y o f WX, and hence
By t h e p r e c e d i n g theorem COROLLARY ( B l a i r - H a g e r )
noff space Proof.
T = G - c l uxS.
x is
u-embedded
.
in
S
is
z-embedz-embedded
US = T c uX. Every c o z e r o - s e t i n a TychoX.
T h i s i s immediate from 1 0 . 7 ( 1 ) and 10.15. The f o l l o w i n g r e s u l t g i v e s s e v e r a l c h a r a c t e r i z a t i o n s
of
u-embedding and a p p e a r s i n t h e 1974 p a p e r o f B l a i r .
118
SPACES AND CONVERGENCE
HEWITT-NACHBIN
. The
THEOREM ( B l a i r )
10.17
Tychonoff s p a c e
X.
(1) The space
be a non-empty
S
are e q u i v a l e n t :
followins statements u-embedded
S
s u b s e t of a
X.
on
qenerate d & -
(2)
D i s t i n c t real
(3)
The s p a c e
(4)
There e x i s t s a Hewitt-Nachbin subspace
tinct
Z-ultrafilters
Z-filters
on
9
S
S
X.
z-embedded
diluxS.
of
in
UX
which S i s d e n s e and C-embedded. Moreover, i f any one o f t h e above c o n d i t i o n s _is s a t i s f i e d ,
then
d i l u X S i s t h e unique Hewitt-Nachbin
which
i s d e n s e and
S
Proof.
subspace
We w i l l establish that
implies (4) implies (1). L e t
u : US
+
f i r s t that
T = diluXS, l e t
7
verges t o
3 on
T(uS) c T
The i n c l u s i o n
T(uS) = T.
t h a t converges t o
S
q c US.
Hence
Now assume t h a t (1) h o l d s s o t h a t H e n c e w e i d e n t i f y T w i t h US. I f 3, I d i s t i n c t points
p1
S, then
p2
and
T.
and Note
i s immediate
p; b u t then
in
Z1 T
Z-
3 con-
~ ( 3 c)o n v e r g e s t o ~(u.5). a i s a homeomorphism. and 3, a r e d i s t i n c t L
p = ~ ( q ) It follows t h a t
2 - u l t r a f i l t e r s on
+ uX
;f: =
.
T ( q ) , and t h u s
real
US
P E T , then t h e r e i s a r e a l
If
T .
f o r some
q
:
S c X,
T ( u S ) be t h e s u r j e c t i v e map induced by
from t h e c o n t i n u i t y of ultrafilter
uX &
(1) i m p l i e s ( 2 ) i m p l i e s ( 3 )
be t h e Hewitt-Nachbin e x t e n s i o n of t h e i n c l u s i o n
let
of
C-embedded.
and
T c
Z2
converge t o
by 8 . 5 ( 5 ) .
The p o i n t s
Z1 and z 2 i n ux, and t h u s Z1 n X and Z 2 D X a r e d i s j o i n t members of t h e Z - f i l t e r s on X g e n e r a t e d by z1 and a 2 . Next suppose ( 2 ) h o l d s . I t w i l l be shown t h a t S i s C-embedded i n T by e s t a b l i s h i n g t h a t e v e r y p o i n t o f T i s t h e l i m i t of a unique r e a l 2 - u l t r a f i l t e r on s (8.4, ( 5 ) i m p l i e s ( 2 ) ) . L e t P E T and assume t h a t Z1 and Z2 a r e If 8 ,l and r e a l 2 - u l t r a f i l t e r s on S t h a t converge t o p . p1
z2I
and
have d i s j o i n t z e r o - s e t neighborhoods
p2
are the
2 - f i l t e r s on
3,l
r e s p e c t i v e l y , then by 6 . 1 7 and 6 . 1 9 ( 5 inclusion
S c X)
l 1
=
and
i# (
X
g e n e r a t e d by
Zl and
Z2,
3.,’ a r e r e a l Z- u l t r a f i 1t e r s ~ ~ j1 = , 1,2, ~ where i i s t h e
and t h e r e f o r e converge i n uX. I t follows b o t h converge to p so t h a t Sll = z 2 l ; hence
t h a t Z l l and z2l Sl = Z 2 by assumption.
Thus ( 3 ) h o l d s .
Z-EMBEDDING
Assume t h a t G -dense i n
is
is
S
T c G - c l , J x ~ ,t h e s e t
Since
C-embedded i n
s o i t s u f f i c e s t o show t h a t
T,
i s a homeomorphism. L e t - S , and l e t Z1 and Z 2
p1
and
p2
S
*
n
u(pi) c clT(S
n
for
Zi)
C -,embedded i n
T + T
From t h e d e n s i t y
n
S , and
S
i = 1,2.
Zi), S fl
But
i = 1,2.
are d i s j o i n t zero-sets i n
Z2
1 ; s
d e n o t e d i s j o i n t z e r o - s e t neighbor-
p1 and p 2 , r e s p e c t i v e l y , i n vS. i n US i t f o l l o w s t h a t pi c c l d S ( S
Hence
:
be d i s t i n c t p o i n t s o f
hoods o f S
a
I t w i l l be shown t h a t
i s a Hewitt-Nachbin s p a c e .
of
S
I t f o l l o w s from t h e assumption and 1 0 . 5
T.
6
that
(3) holds.
119
U-EMBEDDING
AND
Z1 and
i s d e n s e and
I t f o l l o w s from Gillman and J e r i s o n ( 6 . 4 )
T.
that
c i T ( s n zl) n Thus, a f p , ) # ~ ( p , ) , so Now l e t
h
n
z2)
=
6.
is a b i j e c t i o n .
a
denote t h e i n c l u s i o n
f E C ( L S ) . Since
any
ciT(s
C-embedded i n
is
S
S C liS,
and c o n s i d e r
T t h e composite
g E C ( T ) , and ( g o a ) ( x ) = f ( x ) f o r X C S . H e n c e g o a = f and t h e r e f o r e u ( Z ( f ) ) = Z ( g ) . Now s i n c e a i s b i j e c t i v e and t h e z e r o - s e t s of LIS form a f
0
h
h a s an e x t e n s i o n
every
b a s e f o r t h e c l o s e d s e t s of
vS, w e c o n c l u d e t h a t
c l o s e d , and hence a homeomorphism.
embedded, onto cp = 0.
TI
TI
of
UX
i n which
Then t h e r e i s a H e w i t t S
i s d e n s e and
Thus t h e r e e x i s t s a homeomorphism t h a t leaves
Then
TI
S
= diluXS
from
cp
p o i n t w i s e f i x e d by 8 . 5 . and
is
S
is
Thus ( 3 ) i m p l i e s ( 4 ) .
F i n a l l y , assume t h a t ( 4 ) h o l d s . Nachbin subspace
u
u-embedded
CUS
Clearly
in
X.
Furthermore, t h e f i n a l a s s e r t i o n of t h e theorem i s now c l e a r ,
so t h e proof i s c o m p l e t e . Now i f
S
is
u-embedded i n
X,
then b e c a u s e of t h e
f i n a l a s s e r t i o n of t h e p r e c e d i n g theorem, w e m a y i d e n t i f y with
US
d i l u X S (whenever t h e r e i s no p o s s i b i l i t y o f c o n f u s i o n )
and t h u s w r i t e simply
US
g a t e s many a d d i t i o n a l
u-embedding p r o p e r t i e s :
C
uX.
B l a i r ’ s 1974 paper i n v e s t i f o r instance,
u-embedding p r o p e r t i e s t h a t a r e p e c u l i a r t o cozero- s e t s , and
I n t h e n e x t s e c t i o n we w i l l
t h e u n i o n s of
u-embedded s e t s .
c o n s i d e r some
u-embedding problems i n p r o d u c t s p a c e s .
W e end
120
SPACES AND CONVERGENCE
HEWITT-NACHBIN
t h i s s e c t i o n w i t h t h e f o l l o w i n g u s e f u l t r a n s i t i v i t y theorem
is is
Let
THEOREM ( B l a i r ) .
10.18
-assume t h a t
S
~ e m b e d d e d&
T:
and i f
u-embedded
X,
then
Proof.
Assume f i r s t that
US = d i l uxS.
Let
: uT
T
S
is is
9
S
cp : US
S'
= US.
S'
Now t h e mapping
that
Then
Both cp'
7 ' 0
and
cp'
be t h e
cX
cp'
i n d u c e s a map
cp
S c T,
and
: US
3
2-ultrafilter
3 = ~ ( 3 )c o n v e r g e s t o
~ ( p E) d i l b X S = US.
hence US.
p.
so t h a t
X
uT T
T
= d i l U T S . W e want t o show t h a t
p c s ' , then t h e r e e x i s t s a r e a l converges t o
S
in T and in X .
--f
Hewitt-Nachbin e x t e n s i o n s of t h e i n c l u s i o n s r e s p e c t i v e l y , and l e t
then
X,
u-embedded
u-embedded
and
VX
& I
u-embedded
u-embedded i n
is
S +
be a Tychonoff s p a c e and
X
If
S c T c X.
9'0
Thus
r'
If
T I
: S'
--f
pointwise f i x e d , s o
S
i s a homeomorphism: i . e . , S
.
that
S
~ ( p ) and ,
i n d u c e s a map
T
leave
S'
+
on
is
u-embedded i n
T.
The second a s s e r t i o n of t h e theorem is o b v i o u s . Hewitt-Nachbin Completions of p r o d u c t s
S e c t i o n 11:
I n t h i s s e c t i o n w e a r e c h i e f l y i n t e r e s t e d i n examining the equation
u ( X x Y ) = UX x uY,
The q u e s t i o n o f when t h a t
equality holds has a t t r a c t e d considerable attention:
various
r e s u l t s have been o b t a i n e d by W . W. Comfort (1968B), M . Hugek (197lA and 1972A), A . Hager (1969A, 1969B, and 1972A), W . M c A r t h u r (1970 and 1 9 7 3 ) , and R.
Blair
(1971 and 1 9 7 4 ) .
This
q u e s t i o n i s m o t i v a t e d by t h e G l i c k s b e r g - F r o l l / k Theorem: If X and Y i n f i n i t e Tychonoff s p a c e s , p ( X x Y ) = pX x BY
are
-i f and only if
X
x
Y
is pseudocompact
A c o r r e s p o n d i n g c o n d i t i o n on
X
x Y
(Glicksberg, 1959).
i n order t h a t
u ( X X Y) =
uX x UY
h a s n o t been found, and t h e r e a p p e a r s t o b e no s i m p l e
answer.
A s was p o i n t e d o u t i n t h e p r e c e d i n g s e c t i o n ,
notion of
the
u-embedding h a s a d i r e c t b e a r i n g on t h e problem,
and i t t u r n s o u t t h a t a c o n s i d e r a t i o n of t h e p o s s i b l e e x i s t ence of measurable c a r d i n a l s must b e taken i n t o a c c o u n t .
w i l l a l s o a p p e a l t o t h e c o n c e p t of "P-embedding" and s t u d i e d by H . L.
We
a s introduced
S h a p i r o i n h i s 1966 paper.
The f o l l o w i n g r e s u l t coupled w i t h t h e G l i c k s b e r g - F r o l l k Theorem p r o v i d e s a s u f f i c i e n t c o n d i t i o n t h a t
u (X x Y ) = wX x wY.
COMPLETIONS OF PRODUCTS
11.1
11
THEOREM (Gillman and J e r i s o n ) .
pseudocompact i f and o n l y i f Proof.
Assume t h a t
121
Tychonoff s p a c e
i s pseudocompact s o t h a t
X
C(X) = C
f c C ( X ) , then t h e r e e x i s t s a unique S t o n e e x t e n s i o n
If from
into
PX
embedded i n
fp,X = f .
i n which
>LX = p X
a r b i t r a r y function i n
C(X)
.
unique c o n t i n u o u s f u n c t i o n = f.
f
If
space, then
f'"
from
Proof.
X x Y
x Y)
v(X
fp C-
b e an
Then by 8 . 5 ( 2 ) t h e r e e x i s t s a
IR
into
,X
p X i s a compact Hausdorff s p a c e . so t h a t X i s pseudocompact.
COROLLARY.
(x).
LX = p X .
and l e t
bounded b e c a u s e
11.2
*
i s t h e l a r g e s t sub-
satisfying
f': E C ( p X ) which i m p l i e s t h a t
Therefore,
C(X) = C*(X)
is
X
C-embedded s o t h a t
is
X
C o n v e r s e l y , suppose t h a t
f'/X
Hence
However, by 8 . 2 (l), ;X
PX.
pX
space o f
satisfying
IR
is
X
= pX.
;X
is
f'
Therefore,
pseudocompact Tvchonoff
= d x aY.
From t h e theorem
;(X
x Y ) = p ( X x Y ) and by t h e
4
G l i c k s b e r g - F r o l i k Theorem, P ( X x Y ) = pX x BY.
Since the
c o n t i n u o u s image o f a pseudocompact s p a c e i s pseudocompact, uX = P X
and
c o m p l e t i n g t h e argument.
irY = BY,
The n e x t r e s u l t a p p e a r s a s Theorem 2 . 8 i n t h e 1966 p a p e r Comfort and S . N e g r e p o n t i s .
by W . W .
Let
THEOREM ( C o m f o r t - N e g r e p o n t i s ) .
11.3
s p a c e and l e t
C
*
continuous functions space
C
*
noff space Proof.
on
with the
Y
s u p norm.
i s a Hewitt-Nachbin s p a c e ,
(Y)
X
b e a Tychonoff
Y
(Y) d e n o t e t h e s p a c e o f bounded r e a l - v a l u e d
the
equality
I f t h e Banach
then € o r e v e r y Tycho-
u ( X x BY) = uX x pY
Without loss of g e n e r a l i t y we may assume t h a t
s i n c e w e a r e o n l y concerned w i t h f u n c t i o n s i n r e l a t i o n involving
BY.
C
shown t h a t f
E
X
x Y
C ( X X Y ) be an a r b i t r a r y f u n c t i o n .
t i o n (?x) ( y ) = f ( x , y )
e x i s t s a neighborhood that
-
define the function
1
.
fx
Moreover, U(x)
(?x) ( y ) - (?x) ( y l ) 1
=
Y
from
Y =
py
( Y ) and a Y
is
I t w i l l be
LIX x Y .
C-embedded i n
is
*
H e n c e , C*(Y) = C ( Y ) s i n c e
compact Hausdorff and t h e r e f o r e pseudocompact.
XEX
holds.
Hence,
let
Then f o r e a c h p o i n t
IR
into
f o r each
E
>
by t h e equathere
0
x ~ ( y o) f t h e p o i n t ( x , y ) such /f(x,y)
-
f (x,yI)
1 <
E
whenever
HEWITT-NACHBIN SPACES AND CONVERGENCE
122
( x , y ' ) c U(x) x V ( y ) because of t h e c o n t i n u i t y of
f.
f o r e , f o r each
c (Y),
-
Hence
f
x;-X
X
d e f i n e s a mapping from
a t e s w i t h each
xtX
b o t h be g i v e n .
u
X V
Y
By t h e c o n t i n u i t y of
y1 ri
(u
y2
,..., V :
Yk i
A
1
Yi
lf(x,y.) 3
-
f
kj of t h e p o i n t
(XI
,y)
' -i,
'
\
Therefore,
y.
X
belongs t o
IR g
x;X
x
and
i
(Y). ' 0
E
and
y
whenever
(XI
of
5
llyx
U: whence
- rxl
f
x.
YEY
respec-
,y') E
Then,
(XI
f (x' , y )
1 <
11 <
,y) c
c
-g l X = -f .
by t h e r e l a t i o n
u
x
i s continuous.
whenever (x', y )
x'
-
g : ux
g
c
+
i
(Y)
x Y
: UX
+
I t w i l l be shown t h a t
g ( p , y ) = (gp) ( y ) . f.
t
T h e r e f o r e , by 8 . 5 ( 2 )
Hence, d e f i n e t h e mapping
i s a c o n t i n u o u s e x t e n s i o n of
v
whenever t h e p o i n t
c
t h e r e e x i s t s a unique c o n t i n u o u s e x t e n s i o n satisfying
c
in
f , f o r each p o i n t
whenever
-
which i m p l i e s t h a t If ( x , y )
u
fx
Y i s compact, t h e r e e x i s t s a f i n i t e s u b c o v e r of Y ; hence d e f i n e t h e neighborhood u =
Since
Y'
,V
(Y) t h a t a s s o c i -
-
i s continuous: f o r l e t
f
t h e r e e x i s t neighborhoods Uy and V Y t i v e l y such t h a t I f ( x , y ) - f ( x ' , y t ) 1 < V
k
c
into
There-
b
belongs t o
t h e continuous f u n c t i o n
-
Now, t h e mapping
-f x
the function
To t h i s end, l e t
F
>
0
b e g i v e n , and l e t ( p , y ) b e a f i x e d , b u t a r b i t r a r y , p o i n t i n UX x Y .
Because of t h e c o n t i n u i t y of
borhood
U
whenever
p'
every p o i n t hood
v
of
of t h e p o i n t
Hence,
U.
F
y'
E
y
such t h a t
Y
in
p
1
LIX
g
such t h a t /Igp
(gp)( y ' ) -
whenever
p'
t h e r e e x i s t s a neigh-
E U.
-
4p' \ / < $
c
(Tp' ) ( y ' ) < for Now, choose a neighbor~
Then the following r e l a t i o n s hold :
Therefore, g
i s continuous.
glX x Y = f : hence
X
x Y
is
Moreover, i t i s c l e a r t h a t C-embedded i n
uX x Y.
123
COMPLETIONS OF PRODUCTS
UX x Y
Finally, since
d e n s e l y , i t i s the c a s e t h a t
X X Y
8.5.
.,(X
X Y)
= JX
x Y
by
T h i s c o n c l u d e s t h e proof o f t h e theorem. if
NOW,
my
i s a Hewitt-Nachbin s p a c e c o n t a i n i n g
Y
i s of nonmeasurable c a r d i n a l , t h e n t h e s e t
of a l l r e a l - v a l u e d f u n c t i o n s from
*
into
Y
IR
i s non-
T h e r e f o r e . C ( Y ) i s a m e t r i c space w i t h c a r d i -
measurable.
my,
n a l i t y no l a r g e r t h a n t h a t o f
and hence i s a l s o of non-
I n t h e next c h a p t e r i t w i l l be e s t a b -
measurable c a r d i n a l .
l i s h e d t h a t such m e t r i c s p a c e s a r e always Hewitt-Nachbin spaces.
T h e r e f o r e , an a p p l i c a t i o n of t h e p r e v i o u s theorem
y i e Id s t h e r e l a t i o n s , L(X
x Y) = ,(X x BY) = LX x BY =
assuming t h a t
,x
x Y
I n o t h e r words w e have e s t a b l i s h e d t h e
Y = BY.
following c o r o l l a r y . 11.4
If
COROLLARY.
measurable c a r d i n a l , Tvchonoff s p a c e X .
Y
is a compact Hausdorff s p a c e o f non-
then
LJ(X x Y) = VX
X
Y
for every
I t t u r n s o u t t h a t t h e assumption o f t h e nonmeasurable
Y
c a r d i n a l i t y of ped.
i n t h e p r e c e d i n g c o r o l l a r y c a n n o t b e drop-
W e w i l l a p p e a l t o t h e c o n c e p t of "P-embedding" a s i n t r o -
duced i n S h a p i r o ' s 1966 p a p e r i n c o n s t r u c t i n g an example e s t a b l i s h i n g t h e n e c e s s i t y of t h e nonmeasurable c a r d i n a l i t y condition i n 11.4.
X
A p s e u d o m e t r i c on a s e t X
x X
JR
into
need n o t imply
d(x,y) = 0 If
(X,T)
is a f u n c t i o n
d
from
t h a t d i f f e r s from a m e t r i c o n l y i n t h a t
x = y.
i s a t o p o l o g i c a l s p a c e , then a p s e u d o m e t r i c
d
on X i s s a i d t o b e c o n t i n u o u s i n c a s e i t i s c o n t i n u o u s a s a f u n c t i o n from X x X i n t o IR. E q u i v a l e n t l y , d i s c o n t i n u ous i f and o n l y i f t h e topology fies
rd c If
g e n e r a t e d by
d
satis-
T.
dl
and
d2
a r e p s e u d o m e t r i c s on t h e s e t
i t i s easy t o v e r i f y t h a t X.
T~
dl
V
d2
X,
then
is a l s o a p s e u d o m e t r i c on
124
11.5
SPACES AND CONVERGENCE
HEWITT-NACHBIN
A non-empty s u b s e t
DEFINITION.
l o g i c a l space
i s s a i d t o be
X
every continuous pseudometric on con tinuous pseudome t r i c on
X
of an a r b i t r a r y topo-
S
X
P-embedded
can be extended t o a
S
.
Using t h e above terminology,
R . Arens
(1952) h a s shown
t h a t every c l o s e d subspace of a m e t r i c space i s therein.
l a t e d t o c o l l e c t i o n w i s e normality a s
P-embedded
P-embedding i s re-
S h a p i r o (1966) h a s shown t h a t
C-embedding i s r e l a t e d
More p r e c i s e l y , 2 t o p o l o g i c a l space
t o normality.
i n case
X
is
c o l l e c t i o n w i s e normal i f and only i f every c l o s e d s u b s e t of
is
X.
P-embedded
X
W e w i l l now s t a t e some i m p o r t a n t r e -
l a t i o n s h i p s concerning
C-embedding and
P-embedding a l l of
which a r e proved i n S h a p i r o ’ s 1966 p a p e r .
W e omit t h e p r o o f s
h e r e because t h e problems which would a r i s e , i f pursued, t a k e
u s f a r a f i e l d from our b a s i c aim. 11.6
(1) I f
REMARKS.
S
a r b i t r a r y t o p o l o g i c a l space however,
If
n a l and i f (3) S
then
X,
S
is
C-embedded i n
X;
i s dense i n
S
If
is
i s a Tychonoff space of nonmeasurable c a r d i -
X
i f and only i f
3.2,
P-embedded s u b s e t of an
the converse f a i l s t o hold i n t h e g e n e r a l c a s e .
(2)
then
is a
is
S S
X,
then
C-embedded i n
is
S
P-embedded i n
X
X.
i s a compact s u b s e t of a Tychonoff space
P-embedded i n
3 . 3 , and 3 . 7 ,
X,
(See Shapiro, 1966, Theorems
X.
respectively,
f o r the d e t a i l s . )
The n e x t two r e s u l t s a r e due t o S h a p i r o (1966) and L . Imler (1969) r e s p e c t i v e l y .
The p r o o f s r e q u i r e s e v e r a l i d e a s
concerning t h e r e l a t i o n s h i p s between
P-embedding and l o c a l l y
f i n i t e c o z e r o - s e t c o v e r s on a t o p o l o g i c a l s p a c e .
Hence we
omit t h e p r o o f s h e r e . 11.7
If
THEOREM ( S h a p i r o ) .
-then t h e
followinq
(1) The space
X
The space
X
(2)
completion
X
i s a d i s c r e t e Tychonoff space,
are e q u i v a l e n t :
i s of nonmeasurable c a r d i n a l . P-embedded i n its Hewitt-Nachbin
ux.
125
COMPLETIONS OF PRODUCTS
11.8
(Imler)
THEOREM
following s t a t e m e n t s
. If
are
(1) The space (2)
The space
(3)
The
NOW,
cardinal.
i s a Tychonoff s p a c e , then t h e equivalent: X
&
P-embedded
X X x pX
uX.
&
C-embedded
u ( X x p X ) = uX x px
equation
VX X p X .
holds.
suppose t h a t
D
i s a d i s c r e t e space of measurable
Then by 1 1 . 7
D
cannot be
follows from 1 1 . 8 t h a t t h e r e l a t i o n f a i l s t o hold.
P-embedded i n u(D
uD.
It
= uD X pD
x pD)
T h e r e f o r e , t h e c o n d i t i o n of nonmeasurable
cardinality i n 11.4 is essential.
( A n a l t e r n a t i v e proof
for
t h i s example i s given by Comfort i n 1968B, 4 . 8 ) . I f the product t h e d e n s i t y of implies t h a t
i s c-embedded i n VX x vY, then i n t h e Hewitt-Nachbin space uX x VY
X X Y
X X Y
u ( X X Y) = LIX X x Y
remark i n 1 1 . 6 ( 2 ) , i f C-embedded i n
X
uY, by 8 . 5 .
i s of nonmeasurable c a r d i n a l and
then i t i s
WX X vY,
Moreover, by t h e
P-embedded t h e r e i n .
How-
e v e r , t h e following r e s u l t w i l l e s t a b l i s h t h a t a c r i t e r i o n a s C-embedding i s n o t r e q u i r e d .
strong a s 11.9
THEOREM (Comfort-Negrepontis).
ded i n --
S x uY,
X
then
%
Moreover, i f t h e c a r d i n a l
is -*-C
X x Y
vx x
-embedded
in
Y
of VX
.&
If
C-embedded
%
*
C -embedUX X uY.
x Y is nonmeasurable and i f x >Y, then it i s P-embedded in
X
2Y.
By 4 . 8 ( 2 ) i t s u f f i c e s t o show t h a t
Proof. Z
n
(X
x Y)
=
a.
Now, X
and
X % Y
2 E Z(vX X
p l e t e l y s e p a r a t e d from every z e r o - s e t which
X X Y
Y
are
i s com-
uY) f o r
G -dense
6 I t follows t h a t
in
VX and uY, r e s p e c t i v e l y , by 9.6(1). X x Y i s G -dense i n t h e product space UX X UY because 6 fl (Ui x v 1 . ) = fl Ui x n vi. T h e r e f o r e , no G 6- s e t and, i e IN icN i cm i n p a r t i c u l a r , no z e r o - s e t i n vX x uY can be d i s j o i n t from X x Y. The second s t a t e m e n t i s an immediate consequence of the r e s u l t s t a t e d i n 1 1 . 6 ( 2 ) . The n e x t r e s u l t a p p e a r s i n t h e 1966 paper b y Comfort and Negrepontis.
HEWITT-NACHBIN SPACES AND CONVERGENCE
126
11.10
Let
COROLLARY ( C o m f o r t - N e g r e p o n t i s ) .
be
Tychonoff s p a c e s , and l e t lYl +
C -embedded
in
X x BY,
Proof.
f
C*(x x
If
F
then
follows t h a t
f
x
d(X
Y), then
Hence, s i n c e
assumption.
extends t o
Y) =
f
,X
dX
If
x PY
%
be
Y
X X Y
x ,Y.
extends t o
x BY) = JX
,(X
and
X
nonmeasurable.
by
X X PY
BY by 1 1 . 4 ,
by 8 . 5 ( 2 ) .
It
Thus
f
VX x JY s i n c e JY C BY. Therefore, X X Y 1s i n UX x 3Y and t h e c o n c l u s i o n now f o l l o w s by
extends t o i
C -embedded
11.9.
I n t h e 1966 p a p e r by Comfort and N e g r e p o n t i s i t i s shown t h a t i f t h e p r o j e c t i o n mapping F~ from X x Y o n t o X i s c l o s e d , then x x Y i s C -embedded i n X x BY. Moreover i t i s w e l l known t h a t i f t h e s p a c e Y i s compact, then t h e proj e c t i o n mapping i s c l o s e d (see Dugundji, Chapter X I , X 7
Theorem 2 . 5 ,
page 2 2 7 ) .
f ol lowing r e s u l t
Coupled w i t h 11.10 t h i s p r o v e s t h e
.
11.11 COROLLARY ( C o m f o r t - N e g r e p o n t i s ) . Tychonoff s p a c e s . -T
X x Y
x - from
I f either onto
X
&&
9 compact
Y
X
and
Y
o r the projection L I ( X x Y ) = ,JX
i s c l o s e d , then
X
LJY.
The n e x t s e v e r a l r e s u l t s a p p e a r i n B l a i r ’ s 1974 paper and w i l l be b a s i c t o r e l a t i n g
u-embedding t o t h e e q u a t i o n
u ( X x Y) = UX x 2 Y .
11.12
If
b i n space, Proof. UX
X x Y
then
T C vX x
C
T = uX
and
Y
cY, and i f
T
X
d e n o t e Tychonoff a Hewitt-Nach-
x uY.
Suppose t h e r e e x i s t s a p o i n t ( p , q ) b e l o n g i n g t o
x YY\T.
say, p
Let
LEMMA ( B l a i r - H a g e r ) .
spaces.
t
Without l o s s of g e n e r a l i t y w e may assume t h a t , c l T ( X x 141) i s a p r o p e r Hewitt-Nachbin
Thus
uX\X.
vX x [ q j t h a t c o n t a i n s
subspace of
X
x (q).
But t h i s i s
impossible. 11.13
LEMMA ( B l a i r ) .
-Assume that in
Y,
ded i n --
A
and t h a t X X Y
9
Let
X
v-embedded
and in
Y X,
u ( X x Y ) = VX x vY.
i f and o n l y &
v(A
&
Tychonoff s p a c e s .
that
B
Then
A
v-embedded
x B
x B ) = VA x vB.
is
uembed-
127
COMPLETIONS OF PRODUCTS
Proof.
A x B
If
X x Y , then
-\-embedded i n
is
X B) C ,(X
A X B C ;(A
Y)
X
a s well a s A X B C UA X UB
uX
C
uY = u ( X X Y ) .
X
x B) :A x "B i s i t s e l f an i n c l u s i o n map.
T h e r e f o r e , t h e Hewitt-Nachbin e x t e n s i o n of t h e i n c l u s i o n
A
x ,B
X B
,A
C
v ( A X B)
: u(A
T
Therefore, A X B
u ( A x B) = JA
s o t h a t by 1 1 . 1 2 ,
;A
C
;B
X
x GB.
The c o n v e r s e r e s u l t i s
trivial. 11.14
--s e t s i n the v ( X X Y)
Proof.
=
V X X uY,
x
A
and 1 0 . 1 4
,>(A X B )
Since
x Y)
then
x B)
,J(A
and
A
and
X
Tychonoff s p a c e s
Since L(X
. If
(Blair-Haqer)
COROLLARY
is the
= SX
x ;.Y
a r e cozero-
Y , r e s p e c t i v e l y , and i f
= uA
x ;B. X x Y , by 1 0 . 7 (1)
is a cozero-set i n
B
B
G - c l o s u r e of
A X B
6
in
.;(X
by assumption, and s i n c e t h e
c l o s u r e o f a p r o d u c t i s t h e p r o d u c t of t h e
G -closures,
b
X Y)
G
6 it
.
-
x B ) i s t h e p r o d u c t o f t h e G - c l o s u r e of b A i n LIX w i t h t h e G g - c l o s u r e o f B i n v Y . Moreover, by 1 0 . 7 (1) A and B a r e z-embedded i n X and Y , r e s p e c t i v e l y . Appealing a g a i n t o 1 0 . 1 4 w e o b t a i n follows t h a t
,(A
c o m p l e t i n g t h e argument. The n e x t theorem shows t h a t
u-embedding p r o v i d e s a
n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r t h e e q u a l i t y
ux x
u(X
x Y )=
SY.
1 1 . 1 5 THEOREM ( B l a i r ) . Then X X Y is v-embedded U ( X x Y ) = ux x UY.
-
Proof.
If
X
x Y
is
X
and vX
x
u-embedded i n
Y
&
Tychonoff s p a c e s .
~JY i f and o n l y i f UX
x uY, t h e n by 1 0 . 1 7
HEWITT-NACHBIN SPACES AND CONVERGENCE
128
x Y c u ( X x Y ) c vx x UY. I t f o l l o w s from 1 1 . 1 2 t h a t ';(X X Y) = sX X d Y . The c o n v e r s e i s t r i v i a l .
X
11.16
COROLLARY (Hager)
Then
';(X
pX
proof.
We have t h a t
X
Y c LIX
x Y
is
,-embedded
;-embedded
in
UX x uY.
theorem.
i f and o n l y
be
Y
if
Tychonoff s p a c e s . 3-embed-
X X Y
PY.
x x X
Jgx and
x Y) = v X x uY
ded &
If
.
x ;Y c p x x BY. pX x pY
in
t h e n by 10.18 i t i s
The r e s u l t now f o l l o w s from t h e
The c o n v e r s e f o l l o w s from t h e second s t a t e m e n t o f
lo.18 and t h e theorem. The n e x t r e s u l t g e n e r a l i z e s Theorem 5 . 3 i n t h e 1966 p a p e r o f Comfort and N e g r e p o n t i s .
and Y are Tychonoff -and i f IY/is nonmeasurable, then u ( X x Y ) = UX x uY o n l y i f X x Y & ;-embedded in X X PY. THEOREM ( B l a i r ) .
11.17
Proof.
Suppose t h a t
1 1 . 4 w e have
X
x Y
is
X
the l a t t e r r e s u l t ) .
and
Y
so t h a t
Conversely,
.
if
spaces i f and
X x pY.
in u(X
p l a y t h e r o l e of
u ( X x Y ) c dX x pY = u(X x P Y )
in
u-embedded
x BY
x BY) = ',X
;(X
by 1 1 . 1 3 (where
X
If
By
x Y ) = uX x and
A
UY
in
B
i ; ( X x Y ) = WX x uY, t h e n
Hence
X x Y
is
u-embedded
x BY.
X
I n t h e i r 1966 p a p e r , Comfort and N e g r e p o n t i s assume t h e * s t r o n g e r c o n d i t i o n of C -embedding in 1 1 . 1 7 . Comfort (1968B) e s t a b l i s h e s t h e n e x t two r e s u l t s i n which he a t t e m p t s t o c h a r a c t e r i z e t h o s e p a i r s of s p a c e s ( X , Y ) f o r which
u(X
x Y)
x uY.
= uX
I t w i l l b e shown f o r example
t h a t , b a r r i n g t h e e x i s t e n c e of measurable c a r d i n a l s , t h e r e l a t i o n h o l d s whenever
Y
is a
uX
k-space and
is locally
compact. 11.18
THEOREM ( C o m f o r t ) .
-
If
Y
is 2
d o r f f s p a c e o f nonmeasurable c a r d i n a l , embedded
uX x Y
l o c a l l y compact Haus-
then
X X Y
f o r e v e r y Tychonoff s p a c e
X.
is
C
*
-
COMPLETIONS O F PRODUCTS
Proof.
For each f u n c t i o n
f
c*(X
E
129
x Y ) and each p o i n t
t h e r e e x i s t s a unique c o n t i n u o u s r e a l - v a l u e d f u n c t i o n
SX x { y )
IR
such t h a t t h e r e s t r i c t i o n
f ( X X [ y ] by 8.5(2).
cisely the restriction g : LX x Y
tion and
ytY.
IR
-f
by g
striction with
f
f / X
on
(p,z) is
g,(p,z)
x K.
ux
by t h e l o c a l
K ) = JX
X
;X
p c
YEY, then
x K
by 1 1 . 4 .
Now, t h e reg
agrees
Observe t h a t t h e o n l y p o s s i b l e v a l u e t h e f'
:
bX
X
K
--f
can have a t e a c h p o i n t
IR
because of t h e u n i q u e n e s s p r o p e r t y of t h e m u s t coincide with the extension
Therefore, g
f u n c t i o n on
y
i s c o n t i n u o u s , and moreover
K
X x K.
extension function extension.
C-embedded i n
is
X
b(X
i n which c a s e
Y,
Therefore, X x K
For i f
of
K
:
D e f i n e t h e func-
i s continuous.
t h e r e e x i s t s a compact neighborhood
9Y
x ' y ) i s pre-
g ( p , y ) = g y ( p , y ) f o r each
W e claim t h a t
compactness of
IX
g Y
ycy
LIX x K.
I t follows t h a t
i s c o n t i n u o u s on
g
,X X K . H e n c e , g i s c o n t i n u o u s a t t h e a r b i t r a r y p o i n t ( p , y ) i n uX x Y . F i n a l l y , i t i s immediate from t h e d e f i n i t i o n t h a t the r e s t r i c t i o n g / X x Y coincides with the o r i g i n a l function f.
11.19
If
COROLLARY.
i s a l o c a l l y compact Hewitt-Nachbin
Y
s p a c e of nonmeasurable c a r d i n a l , then The s p a c e
theorem.
is
X
x
x
is
Y
uX x UY
c-embedded i n
X
x Y
i n uX x Y by t h e by 4.4. H e n c e , X x Y
by 1 1 . 9 and t h e c o n c l u s i o n f o l l o w s in
UX x vY
and 8 . 5 .
The f o l l o w i n g r e s u l t s i n v o l v e t h e c o n c e p t o f a I t is said that
X
for
C -embedded
Moreover, UX x Y = V X x uY
from the d e n s i t y o f
x Y ) = VX x vY
X.
e v e r y Tychonoff s p a c e Proof.
u(X
is a
X
k-space i f anc? o n l y i f
k-space. has the
weak topology d e t e r m i n e d by i t s c l a s s o f compact s u b s e t s : e x p l i c i t l y , a s e t F is c l o s e d i n X i n c a s e F I7 K i s closed i n
K
f o r e v e r y compact s u b s e t
K
X.
in
I t i s w e l l known (Dugundji, C h a p t e r X I ,
249) t h a t t h e t o p o l o g i c a l p r o d u c t of k-space.
However,
the p r o d u c t
compact Hausdorff s p a c e i s a Theorem 4 . 4 , page 263).
pf 2 k-space
9 . 5 , Ex. 1, page
k - s p a c e s need n o t be a k-space w i t h a l o c a l l y (Dugundji, C h a p t e r XII.4,
Moreover, whenever
X
is a
k-space
130
a mapping
f
from
the r e s t r i c t i o n K
SPACES AND CONVERGENCE
HEWITT-NACHBIN
in
into
X
i s c o n t i n u o u s i f and o n l y i f
Y
i s c c n t i n u o u s f o r e v e r y compact s u b s e t
f(K
X (Dugundji, Chapter V I ,
Theorem 8 . 3 , page 1 3 2 ) .
The f o l l o w i n g r e s u l t i s due t o Comfort (1968B, Theorem 2.3).
11.20
If
THEOREM ( C o m f o r t ) .
--
2 Tychonoff
Y
k - s p a c e each
of whose compact subsets i s of nonmeasurable c a r d i n a l , and i f
:,X
is
l o c a l l y compact,
then
is
X x Y
*
C -embedded
&
x Y. Proof.
A s i n t h e proof of 11.18 e a c h f u n c t i o n
defines a function g(p,y) = gy(p,y).
from
g
IR
into
I t w i l l be shown t h a t
in
K
LJX
g
E
C (X
by t h e i d e n t i t y
i s continuous.
x Y
TO t h i s end,
let
ping.
T ~ ( K ) i s compact f o r each compact s u b s e t
Then
'rY : ,;X
X Y
x Y)
glK i s c o n t i n u o u s f o r e v e r y s i n c s vX x Y i s a k - s p a c e .
Hence i t s u f f i c e s t o show t h a t compact s u b s e t
x Y
-X
*
f
+
d e n o t e t h e p r o j e c t i o n map-
Y
in
K
';X x Y
i n which c a s e t h e r e l a t i o n 9 ( X x ryK) = SX x T K Y T h e r e f o r e , g i s c o n t i n u o u s o n vX x .;ryK by t h e same argument used i n t h e p r o o f o f 11.18 w i t h K r e p l a c e d by T ~ K . H e n c e , s i n c e K c VX x T ~ K , t h e f u n c t i o n g i s
h o l d s by 1 1 . 4 .
c o n t i n u o u s on
11.21
K
COROLLARY
completing t h e p r o o f .
is l o c a l l y c a r d i n a l then u ( X if
I
'JX
Proof.
If
(Comfort).
i s a Tychonoff
Y
x
Y ) = ax
x uy.
By t h e theorem, X x Y
is
is of
X x Y
compact, and i f
k-space,
nonmeasurable
*
C -embedded i n
UX
x Y.
i s l o c a l l y compact of nonmeasurable c a r d i n a l , i t * i s t h e c a s e t h a t UX x Y i s C -embedded i n uX x uY by
Since
11.18.
uX
It follows t h a t
X
*
by t h e t r a n s i t i v i t y o f immediate 11.22
x Y
is
COROLLARY.
If
Y
Tychonoff
and pseudocompact, and i f b l e c a r d i n a l , then u ( X x Y ) = ux x The r e l a t i o n
UX = pX
UX
x uY
The r e s u l t is now
C -embedding.
.
Tychonoff Proof.
*
C -embedded i n
k-space,
X x Y
if
X
is
i s of nonmeasura-
vy.
h o l d s by 11.1 i n which case
i s l o c a l l y compact s i n c e e v e r y compact s p a c e i s l o c a l l y com-
UX
COMPLETIONS AND PRODUCTS
The r e s u l t i s now immediate by t h e p r e v i o u s c o r o l l a r y .
pact.
If
11.23
COROLLARY.
spaces
of nonmeasurable
then
X x Y
X
are pseudocompact
Y
c a r d i n a l and i f
2
X
x uY
Tychonoff
k-space,
pseudocompact. LI(X x Y) =
By t h e p r e c e d i n g c o r o l l a r y , t h e r e l a t i o n
Proof. LIX
1 31
Moreover, VX x iiY = p X x BY
holds.
follows t h a t
x Y
X
by 11.1.
*
i s d e n s e and
p X x pY.
in
C -embedded
It
p ( X x Y ) i s t h e unique compact Hausdorff s p a c e i n which * i s d e n s e and C -embedded, t h e l a t t e r st.atement i m p l i e s p ( X x Y ) = p X x pY. T h e r e f o r e , p ( X x Y ) = v ( X x Y) so X x Y i s pseudocompact by 11.1.
Since X x Y
that that
A s Comfort p o i n t s o u t i n h i s 1968B p a p e r ,
the c o n d i t i o n
UX b e l o c a l l y compact i n 1 1 . 2 0 d o e s seem a b i t a r t i f i -
that
X
c i a l : i t would be d e s i r a b l e t o have a c o n d i t i o n on
itself.
Comfort d o e s e x p l o r e t h i s problem and e s t a b l i s h e s t h e r e s u l t I t i s due
The n e x t theorem i s b a s i c t o what f o l l o w s .
11.26.
t o A . Hager and D . Johnson ( 1 9 6 8 ) . THEOREM (Hager-Johnson).
11.24
t h e Tychonoff Then clxU Proof.
space
be an open s u b s e t o f
U
suppose t h a t
X,
f t c(clxU
lf(~,+~) I E
/f(x)
on
If(xn)i f o r which
C(X)
X.
-
f(xn)I
2
.
Beginning w i t h any p o i n t
compact.
n
=
g =
The f u n c t i o n
g
The c o n t i n u o u s e x t e n s i o n o f clxU.
2
con-
f o r which
n c m , an e l e m e n t
gn (x;$ = 0
and
x1 F U ,
xn E U
There i s , f o r each
1.
gn (x,)
1/4
i t s e l f , unbounded on of
&
he c o n t r a r y , t h a t t h e r e i s an unbounded
s t r u c t i n d u c t i v e l y a sequence of p o i n t s gn
clbxU
pseudocompact.
Suppose, on
function
Let
qn
n= 1 t o uX
whenever
i s continuous is, l i k e
g
T h i s c o n t r a d i c t s t h e compactness
ClUXU.
The f o l l o w i n g i s Problem 8 E . 1 i n Gillman and J e r i s o n . 1 1 . 2 5 THEOREM. X,
For any s u b s e t
if t h e r e s t r i c t i o n
clxS Proof.
&
f IS
is
S
of a Hewitt-Nachbin s p a c e
bounded f o r a l l
f
E
C(X),
then
compact.
Suppose t h a t
p
E
clpxS\clxS.
Then by 8 . 8 ( 3 ) t h e r e
HEWITT-NACHBIN SPACES AND CONVERGENCE
132
f(x) > 1 g = -; f whence g c c ( X ) . For each n c m , l e t un = (q E px : f (9)< . Then f o r each nc IN t h e r e e x i s t s a p o i n t xn b e l o n g i n g nt o un f' S b e c a u s e p E c l p x S . Therefore, g ( x n ) > n . 1t
e x i s t s a function
f(p) = 0
f E C ( p X ) such t h a t
xcX.
for a l l
0
Define t h e f u n c t i o n
g
on
and
by
X
- 3
follows t h a t
i s unbounded on t h e s u b s e t
g
i m p o s s i b l e so t h a t
c l PX S = c lX S .
This i s
S.
i s compact.
Thus, c l x S
The f o l l o w i n g r e s u l t i s due t o Comfort (1968B, Theorem 4.6). 11.26
I n order t h a t
THEOREM ( C o m f o r t ) .
pact, i t i s n e c e s s a r y
and
clUf.,
E
Proof. of
p
E
and
and
A
are c o m p l e t e l y
X\B
of
B
Given a compact neighborhood
SSX, l e t
b e a c o n t i n u o u s mapping of
f and
c (1).
f (uX\K)
p
in in
K
ux
UX
E
f o r which
X
separated
Necessity. f (p) = 0
with
and
A
b
s u f f i c i e n t t h a t f o r each
-t h e r e e x i s t pseudocompact p
b e l o c a l l y com-
2X
X. UX
o n t o [0,1]
Let
and
A c f - l ( [0,1/3])
Observe t h a t
n
Since
X.
A
K
n
X
n
X c K
X\B
c f - l ( [2/3,1])
i s compact i t f o l l o w s t h a t t h e c l o s e d s e t
i s a compact s u b s e t of
X.
Therefore, A
a r e completely separated i n
r'l X
and
and
f - I ( [2/3,1])
by 3 . 1 1 ( 3 ) , so t h e same
X
holds t r u e of A and X \ B . Furthermore, p E clu* because X i s dense i n uX and f - l ( [0,1/3) ) i s an open s e t i n uX that contains
p.
Finally, since
closed s u b s e t s of
c l U 2 and
hence compact, t h e s e t s
K,
are
cluXB
and
A
are
B
pseudocompact by 1 1 . 2 4 . To f i n d a compact neighborhood o f t h e p o i n t
Sufficiency. p c uX,
let
t i v e function
(1). L e t and s e t of
p
and
A
g
f
E
pX.
be a s h y p o t h e s i z e d and f i n d a nonnega-
C* ( X ) f o r which
f ( A ) c ( 0 ) and
d e n o t e t h e c o n t i n u o u s e x t e n s i o n of
K = g-'(
in
B
[ O , 1/21 )
.
Then
K
I t w i l l b e shown t h a t
compact by 1 1 . 2 5 .
Thus, t o show t h a t
f (X\B) f
to
C
PX,
i s a compact neighborhood K c uX.
K
Now, c l u X B
is
c uX, i t need o n l y b e
13 3
COMPLETIONS O F PRODUCTS
K c clpXB.
shown t h a t q
cl
E
PX
(X\B)
q
But i f
i n which c a s e
PX
E
g ( q ) = 1.
q k , clpXB
and
then
I t follows t h a t
q#K
completing t h e argument. The f o l l o w i n g i s t h e f i n a l r e s u l t o f t h i s s e c t i o n and i s due t o Comfort (1968B, Theorem 2 . 7 )
11.27
and If
Let
THEOREM ( C o m f o r t ) .
x Y
X
uX
are
! i;rY
&
Y
then
k-spaces,
Y ) there e x i s t s a function on
X
x Y.
X
2X x Y
Y ( X X Y) = VX x uY.
g
C
E
*
f o r each p o i n t
Now,
on [ p ) x Y .
h : VX X LW
Since
p
IR
x \JY i s a
X :
f
it
E
C
(X x
( u X x Y ) which a g r e e s w i t h t
let
be a P which a g r e e s w i t h
LX,
c o n t i n u o u s r e a l - v a l u e d f u n c t i o n on ( p ] x sY g
Tychonoff s p a c e s .
i s of nonmeasurable c a r d i n a l , and i f b o t h
A s i n t h e proof o f 1 1 . 2 0 f o r e a c h f u n c t i o n
Proof. f
.
k-space,
h
the function
d e f i n e d by
h ( p , q ) = h p ( p , q ) belongs t o C ( v X x uY) u s i n g t h e same argument a s t h a t i n t h e p r o o f of
*
-$
*
is
Therefore, X X Y
11.20.
C -embedded
in
x VY
X :
com-
p l e t i n g t h e argument by 1 1 . 9 . The f o l l o w i n g example i s p r e s e n t e d i n C o m f o r t ’ s 1968B paper. 11.28
k-space
EXAMPLE.
f o r which
X
uX
f a i l s t o be a
k- space.
Let let
Y
w2
d e n o t e t h e s m a l l e s t o r d i n a l of c a r d i n a l i t y
d e n o t e t h e compact p r o d u c t s p a c e [ 0 , w 2 ]
H2,
x [0,w2] and
define
x Y
The c l o s u r e i n
=
[(a,P)
E
Y
: a
<
P).
of t h e l o c a l l y compact Hausdorff s p a c e
i s a c o m p a c t i f i c a t i o n of
X.
X
H e n c e t h e r e i s a c o n t i n u o u s func-
I t i s t h e n shown by Comfort, t i o n f mapping PX o n t o cl?. i n a somewhat l e n g t h y argument, t h a t t h e s u b s e t A = [p
E
uX : f ( p ) = (a,a)
f o r some
a
<
w2)
i s n o t c l o s e d , a l t h o u g h i t meets each compact s u b s e t of
i n a closed set.
Thus, VX
f a i l s t o be a
k-space.
vX
HEWITT-NACHBIN SPACES AND CONVERGENCE
134
I n h i s 197lA and 1972A p a p e r s , M . Hugek a l s o c o n s i d e r s t h e problem u ( X x Y) = VX x UY under t h e assumption t h a t measurable c a r d i n a l s e x i s t . I f ml stands f o r the f i r s t measurable c a r d i n a l , then a c c o r d i n g t o I s b e l l (1964) a s p a c e X i s s a i d t o b e pseudo-m -compact i f and o n l y i f e v e r y l o c a l 1 l y f i n i t e d i s j o i n t f a m i l y o f open s e t s i n X i s of nonmeasurab l e c a r d i n a l . Huzek t h e n d e m o n s t r a t e s t h a t if X i s a l o c a l -
&
compact Hewitt-Nachbin s p a c e , t h e n
and o n l y
u ( X X Y) = VX x ;.Y
or
if
Y is pseudo-ml-compact. Husek a l s o s t a t e s an a d d i t i o n a l r e s u l t t h a t somewhat g e n e r a l i f either
v
1x1 <
ml
i z e s t h e r e s u l t s of 1 1 . 2 0 and 11.27 by u t i l i z i n g t h e i d e a of pseudo-m -compactness. The r e s u l t i s a s f o l l o w s : X 1 -a Tychonoff k-space and e i t h e r LY & l o c a l l y compact
or
ax x >Y
2
k-space.
e v e r y compact s u b s e t o f
-i f either
is
X
I f either
A . Hager
equation
S(X
formities.
If
& pseudo-ml-compact or
i s o f nonmeasurable c a r d i n a l , and o r e v e r y compact subset of
X
pseudo-ml-compact
i s of nonmeasurable c a r d i n a l ,
3)Y
Y
&
then
u ( X x Y) = 2X x LY.
(1969A, 1969B, and 1972A) i n v e s t i g a t e s t h e
x Y) = ux x
YY
from t h e p o i n t o f view o f uni-
aC d e n o t e s t h e weak u n i f o r m i t y g e n e r a t e d by
t h e 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 h e Tychonoff s p a c e
X
( i n t h e sense of Tukey, 1 9 4 0 ) , and i f aCX d e n o t e s t h e a s s o c i a t e d uniform s p a c e , then t h e q u e s t i o n o f when v ( X x Y) = uX
x
UY
aCX x aCY
h o l d s i s e q u i v a l e n t t o t h e problem of when a C ( X x Y ) = holds. ( W e w i l l t r e a t uniform s t r u c t u r e s and t h e i r
r e l a t i o n s h i p t o Hewitt-Nachbin s p a c e s i n t h e n e x t c h a p t e r . ) Hager i n v e s t i g a t e s t h e l a t t e r problem i n S e c t i o n 6 of h i s 1972A p a p e r , and h i s r e s u l t s e x t e n d t h o s e o b t a i n e d e a r l i e r i n t h e 1960 p a p e r of N . Onuchic W.
G . McArthur s t u d i e s t h e e q u a t i o n
i n h i s 1970 and 1973 p a p e r s .
v ( X 3 Y) =
vx x
UY
I n t h e e a r l i e r paper h e presents
a t e c h n i q u e f o r d e a l i n g w i t h s p e c i f i c i n s t a n c e s by i n t r o d u c i n g a " r e c t a n g l e condition"
which t u r n s o u t t o be a n e c e s s a r y and
s u f f i c i e n t c o n d i t i o n on t h e p a i r ( X , Y ) i n o r d e r t h a t t h e equa-
I n h i s second p a p e r h e t r e a t s t h e e q u a l i t y v ( X x Y) = UX x uY i n t h e c o n t e x t of l i n e a r l y o r d e r e d topol o g i c a l s p a c e s . H e p r o v e s , f o r example, t h a t if X i s a n y
tion holds.
13 5
COMPLETIONS OF PRODUCTS
s e p a r a b l e Hewitt-Nachbin s p a c e and i f
Y
i s any l i n e a r l y
o r d e r e d Tychonoff s p a c e o f nonmeasurable c a r d i n a l , t h e n
,(X x Y) = vX LY
if
X
X
and
Another r e s u l t i s t h a t
JY.
Y
a r e well-ordered
x
L ( X x Y ) = ,X
Tychonoff s p a c e s .
One f i n a l remark i s i n o r d e r i n c o n n e c t i o n w i t h
~n
_X.
t h e i r 1 9 7 0 paper R . Alo and A . de Korvin prove t h e f o l l o w i n g
result:
Let
G
be a s e p a r a t i n g a l g e b r a of complex-valued
f u n c t i o n s on a non-empty s e t self-adjoint).
Then
G
s
(G
need n o t n e c e s s a r i l y be
can b e viewed a s an a l g e b r a of con-
t i n u o u s f u n c t i o n s o n t h e Hewitt-Nachbin completion of S
i s endowed w i t h an a p p r o p r i a t e t o p o l o g y .
S
when
This r e s u l t i s
r e l a t e d t o t h e c l a s s i c a l Gelfand R e p r e s e n t a t i o n Theorem which + a s s e r t s t h a t e v e r y Abelian C - a l g e b r a c o n t a i n i n g t h e i d e n t i t y i s isometric-isomorphic p a c t Hausdorff s p a c e .
t o a l l c o n t i n u o u s f u n c t i o n s on a com-
Chapter 3 HEWITT-NACHBIN SPACES, U N I F O R M I T I E S , AND RELATED TOPOLOGICAL SPACES
The n o n - t o p o l o g i c a l n o t i o n o f c o m p l e t e n e s s , i n t h e sense of convergence of Cauchy sequences o r n e t s , the m e t r i c space s e t t i n g .
is appropriate i n
Uniform s p a c e s a r e t h e n a t u r a l gen-
e r a l i z a t i o n s of metric s p a c e s and a r e t h e c a r r i e r s f o r t h e n o t i o n s of uniform convergence, uniform c o n t i n u i t y , completen e s s , and t h e l i k e .
W e have s e e n how e v e r y Tychonoff s p a c e
h a s a Hausdorff c o m p a c t i f i c a t i o n and a Hewitt-Nachbin completion.
Analogously e v e r y such s p a c e h a s a Hausdorff uniform
completion ( i n t h e s e n s e o f C a u c h y ) .
Not s o a n a l o g o u s i s t h e
u n i q u e n e s s o f t h i s uniform c o m p l e t i o n , b u t i t d o e s resemble V
t h e uniqueness o f t h e Stone-Cech c o m p a c t i f i c a t i o n and t h e Hewitt-Nachbin c o m p l e t i o n . There a r e two p r i n c i p a l o b j e c t i v e s i n t h i s c h a p t e r ,
The
f i r s t i s t o i n t r o d u c e t h e n o t i o n of a " u n i f o r m space" and t o s t u d y i t s i n t e r a c t i o n w i t h t h e c o n c e p t of a Hewitt-Nachbin space.
O n e of t h e main r e s u l t s t h a t w i l l b e e s t a b l i s h e d i s
t h e Nachbin-Shirota Theorem which a s s e r t s t h a t t h e H e w i t t Nachbin s p a c e s o f nonmeasurable c a r d i n a l a r e p r e c i s e l y t h o s e W e w i l l then
s p a c e s t h a t admit a complete uniform s t r u c t u r e .
u t i l i z e t h a t r e s u l t t o e s t a b l i s h t h a t e v e r y paracompact Hausd o r f f space of nonmeasurable c a r d i n a l i s a H e w i tt-Nachbin space. Second, w e w i l l b r i n g t o g e t h e r t h e many r e s u l t s rel a t i n g t h e c l a s s o f Hewitt-Nachbin s p a c e s t o o t h e r c l a s s e s o f t o p o l o g i c a l s p a c e s , such a s t h e a l m o s t r e a l c o m p a c t s p a c e s , t h e cb- and weak
cb-spaces,
the
q - s p a c e s and t h e
M-spaces.
In
s t u d y i n g F r o l f k l s n o t i o n of an " a l m o s t r e a l c o m p a c t s p a c e " f o r example, i t w i l l be shown t h a t e v e r y Hewitt-Nachbin s p a c e i s a l m o s t realcompact
(see 1 4 . 1 1 ) .
The a l m o s t r e a l c o m p a c t s p a c e s
p l a y an i m p o r t a n t r o l e i n t h e s t u d y o f t h e i n v a r i a n c e and i n v e r s s i n v a r i a n c e of H e w i tt-Nachbin c o m p l e t e n e s s under c o n t i n u o u s mappings which i s t o b e i n v e s t i g a t e d i n t h e n e x t c h a p t e r . The
cb- and weak
c b - s p a c e s a l s o p l a y an i m p o r t a n t r o l e i n
t h a t s t u d y and t h e y w i l l a l s o b e i n t r o d u c e d h e r e .
~t w i l l b e
137
UNIFORM SPACES
shown t h a t e v e r y a r e weak
c b - s p a c e i s a weak
c b - s p a c e and t h a t t h e r e
c b - s p a c e s which f a i l t o b e Hewitt-Nachbin s p a c e s .
Many a d d i t i o n a l r e s u l t s and examples w i l l b e p r o v i d e d .
For
p u r p o s e s o f q u i c k and e a s y r e f e r e n c e , w e w i l l p r o v i d e a c h a r t summarizing t h e v a r i o u s r e l a t i o n s h i p s t h a t w i l l b e e s t a b l i s h e d i n t h i s chapter. A Review o f Uniform Spaces
Section 1 2 :
The n o t i o n of a uniform s p a c e was f i r s t i n t r o d u c e d by Andrg W e i l i n 1937 a s t h e n a t u r a l m a t h e m a t i c a l s t r u c t u r e i n which t o c o n s i d e r such p r o p e r t i e s a s completeness and uniform convergence.
W e i l l s d e f i n i t i o n f o r a uniform s p a c e looked a t
a p a r t i c u l a r f i l t e r on X x X f o r which he had a c e r t a i n b a s e of sets generated b y a family of pseudometrics. However, t h e r e i s some i n c o n v e n i e n c e t o W e i l ’ s axioms.
Currently there
a r e t h r e e w i d e l y a c c e p t e d a p p r o a c h e s t o t h e uniform s p a c e c o n cept:
The Tukey-Smirnof u n i f o r m i t y which d e f i n e s a uniform
s t r u c t u r e i n t e r m s of c o v e r s ;
t h e uniform s t r u c t u r e a s d e f i n e d
i n terms o f e n t o u r a g e s ; and t h e uniform s t r u c t u r e a s d e f i n e d i n terms of p s e u d o m e t r i c s .
The approach of G i l l m a n and J e r i -
son ( C h a p t e r 1 5 ) , and t h e one t h a t w e s h a l l a d o p t , u t i l i z e s It is
pseudometrics and i s t h e most c o n v e n i e n t f o r o u r work.
n o t o u r i n t e n t t o p r e s e n t t h e t h e o r y of uniform s p a c e s , b u t t o e s t a b l i s h how i t r e l a t e s t o Hewitt-Nachbin s p a c e s .
Therefore,
w e s h a l l f e e l f r e e t o draw upon many o f t h e fundamental res u l t s c o n c e r n i n g uniform s p a c e s a s t h e y a r e p r e s e n t e d i n t h e Gillman and J e r i s o n t e x t , J . K e l l e y ’ s 1955 t e x t , and S . W i l l a r d ‘ s 1970 t e x t . The f o l l o w i n g d e f i n i t i o n s a r e b a s i c t o o u r i n v e s t i g a t i o n .
12.1
DEFINITION.
Let
s t r u c t u r e , o r uniformity,
9
of p s e u d o m e t r i c s on
(1) I f (2)
If
dl
and
e
on
X
d2
x
By a u n i f o r m
i s meant a non-empty f a m i l y
with the properties:
are i n
P, t h e n
dl V d2
i s i n 9; E
>
0
x,y
E
X,
i s a p s e u d o m e t r i c , and i f f o r e v e r y
there exists a d(x,y) then
be a non-empty s e t .
X
e
dE9
6 implies is i n 8 .
and a
6
>
e(x,y)
5
E
0
such t h a t for a l l
138
H E W I T T - N A C H B I N SPACES AND RELATED SPACES
The p a i r (X,$) d e n o t e s
X
c a l l e d a uniform s p a c e .
19, and i s is called
w i t h t h e uniformity
B
A uniform s t r u c t u r e
Hausdorff i f x # y , t h e r e e x i s t s a pseudometric
Whenever
(3)
P
If S i s any non-empty f a m i l y of p s e u d o m e t r i c s on t h e r e e x i s t s a s m a l l e s t uniform s t r u c t u r e 1G c o n t a i n i n g We c a l l
0 , and w e s a y t h a t i s c a l l e d a base f o r P
a subbase f o r
8
63
d
in
d(x,y) # 0.
satisfying
0
X,
8.
i s generated
i f f o r every e 6 > 0 such that d(x,y) b implies e ( x , y ) E f o r a l l x,y i n X. I f f i s a mapping from t h e uniform s p a c e ( X , B ) t o t h e uniform s p a c e ( Y , & ) then c l e a r l y , f o r any e i n & t h e funct i o n e o ( f x f ) i s a p s e u d o m e t r i c on X . I f f o r every e i n E , t h i s pseudometric b e l o n g s t o 0 , then f i s s a i d t o be uniformly c o n t i n u o u s . I f (Xa,Oa)acG i s a non-empty f a m i l y
by
8.
in
B
A subbase
and
E
b 0,
there exist
d
8
in
and
of uniform s p a c e s ,
t h e p r o d u c t uniform s t r u c t u r e
C a r t e s i a n product
X =
1: X
a
arG s t r u c t u r e i n which e v e r y p r o j e c t i o n
i s uniformly c o n t i n u o u s .
J Xa
ar G
0
on t h e i r
i s d e f i n e d t o be t h e s m a l l e s t -r
a
The n o t a t i o n
from
X
i n t o (Xa,Pa)
Il ( Xa , Oa ) means aiG
with t h e product u n i f o r m i t y .
A uniform s t r u c t u r e 8 on X i n d u c e s a topology on c a l l e d t h e uniform topology, d e f i n e d a s f o l l o w s : f o r each
point
a b a s i c neighborhood s y s t e m of
pcX
p
X,
i s g i v e n by
< E ] , (dcr9, c > 0 ) . P i s a u n i f o r m i t y on X, then r9 i s an a d m i s s i b l e u n i f o r m i t y on X i f t h e u n i f o r m topology c o i n c i d e s w i t h t h e g i v e n topology on X . A t o p o l o g i c a l s p a c e X a d m i t s a uniform s t r u c t u r e i f t h e r e i s an admiss i b l e u n i f o r m i t y on X . The u s u a l uniformity on IR i s gene r a t e d by d ( x , y ) = / X - y / f o r X , Y i n W t h e c o l l e c t i o n of a l l s e t s (yEX : d ( p , y )
If
X
i s a t o p o l o g i c a l s p a c e and i f
-
I n t h e d e f i n i t i o n of a uniform topology induced by a uniform s t r u c t u r e P, i t i s enough f o r d t o range o v e r a
base f o r
0.
C l o s u r e s i n t h e uniform topology a r e g i v e n by cl A =
n dcB
(xtX
:
d (x,A) = 0 ) .
UNIFORM SPACES
If
i s a s u b s e t of
A
139
t h e mapping
X,
6 : X
IR
+
defined by
6 ( x ) = d ( x , A ) i s c o n t i n u o u s r e l a t i v e t o t h e uniform topology on
T h e r e f o r e , cl A
X.
i s an i n t e r s e c t i o n of z e r o - s e t s on
X.
X i s a Hausdorff s p a c e , then X may admit o n l y Hausdorff uniform s t r u c t u r e s , and c o n v e r s e l y . The f o l l o w i n g f a c t s a r e u s e f u l and may b e found i n Chapt e r 1 5 o f t h e Gillman and J e r i s o n t e x t . If
12.2
Let (x,19)and ( y , e )
THEOREM.
The
uniform s p a c e s .
following statements a r e t r u e : function
(1)
from (x,&) i n t o ( Y , @ )
f
c o n t i n u o u s i f and o n l y i f f o r each there e x i s t
0,
c
d(x,y) in -
6
19
and
6
e(f(x),f(y))
‘j
and
@
such t h a t
0
-for a l l
E
(XaS&a)aFG i s a non-empty f a m i l y
s p a c e s and i f X
in
d
implies
in
e
x,y
X.
If
(’)
5
uniformly
n
=
acG
Xa,
19
then
of
uniform
i s the product uniformity
is
B
on
g e n e r a t e d by t h e f a m i l y
of
a l l pseudometrics
of t h e form ( x , y ) + d ( x a , y a ) , x = (x ) and d E Ba. a a&’ = (ya)acG’ composition o f two u n i f o r m l y c o n t i n u o u s func-
-
where (3) (4)
The -t i o n s i s uniformly c o n t i n u o u s . Let X & a Hausdorff t o p o l o g i c a l space
X
is
X
space.
The
a d m i t s a uniform s t r u c t u r e i f and o n l y
if
completely r e q u l a r .
The f o l l o w i n g d e f i n i t i o n s w i l l b e needed i n t h e n e x t s e c t i o n and remaining d i s c u s s i o n . A subset
12.3
DEFINITION.
(X,B)
i s s a i d t o be
where
of a Hausdorff uniform s p a c e d - c l o s e d f o r d i n B i n c a s e A = cdA , A
d e n o t e s t h e s e t (xcx
cdA
(Aa : acG] of s u b s e t s of ( X , O ) 6
gauqe
A subset XEA) is
(dE19, 6 A
is
>
0)
d - d i s c r e t e of gauge
d ( x , A ) = 01.
i s s a i d t o be
i n case
d-discrete
:
d(Aa,AP)
2
6
A family
d-discrete whenever
of a # P.
(dEr9) i n c a s e t h e c o l l e c t i o n ((x) :
6
f o r some
6
>
0.
Every p s e u d o m e t r i c s p a c e (X,d) h a s an a d m i s s i b l e u n i f o r m i t y which i s g e n e r a t e d by ( d } and c a l l e d a p s e u d o m e t r i c
140
SPACES AND RELATED SPACES
HEWITT-NACHBIN
5
A family
uniformity.
o f s u b s e t s of
t a i n a r b i t r a r i l y small sets i f f o r every
3
c o n t a i n s a s e t of
5
filter
5
on ( X , B )
19
in
d
d - d i a m e t e r less than
c
and
>
0,
A zero-set
E.
Z-filter i n case
i s s a i d t o b e a Cauchy
contains a r b i t r a r i l y small sets.
i s s a i d t o con-
(X,8)
A uniform s p a c e
s a i d t o be complete i n c a s e e v e r y c o l l e c t i o n
(X,&)
is
of c l o s e d
Ji
s e t s with t h e f i n i t e i n t e r s e c t i o n property t h a t contains a r b i -
n
t r a r i l y small sets s a t i s f i e s If
# #.
i s a t o p o l o g i c a l space, the f u n c t i o n s i n
X
can b e used t o d e f i n e v a r i o u s u n i f o r m i t i e s on f
E
C(X) let
Note t h a t
IR.
b e t h e p s e u d o m e t r i c on
= d
o ( f x f ) where
+f
It f o l l o w s t h a t
A family
19
$f
hf
on
of
X
X
For each
X.
d e f i n - ? d by
i s t h e u s u a l m e t r i c on
d
i s a c o n t i n u o u s p s e u d o m e t r i c on
on
functions ( f a : acG!
i n c a s e the family ( $ f
X
X.
qenerates g uniformity
8.
: a c G j generates
a A uniform s p a c e ( Y , & ) i s a uniform subspace of
Y
C(X)
i s contained i n
uniformity
@.
and i f ( d l Y x Y : d t B ] g e n e r a t e s t h e
X
Let
(X,8) i f
X
be a t o p o l o g i c a l space.
The u n i f o r m i -
t i e s g e n e r a t e d by a l l bounded 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
X,
by a l l 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 , and by a l l
c o n t i n u o u s p s e u d o m e t r i c s , a r e d e n o t e d by
@*( X ) ,
L0(x) r e s p e c t i v e l y .
i s c a l l e d t h e univer-
The s t r u c t u r e
Lo(X)
@ ( X ) , and
s a l uniformity. 12.4
REMARKS.
The f o l l o w i n g f a c t s a r e i m p o r t a n t and a p p e a r
i n t h e Gillman and J e r i s o n t e x t ( 1 5 . 1 5 ) . (1) Every
d - c l o s e d s u b s e t of a uniform s p a c e ( X , 8 )
is
a zero-set. (2)
The union of a
(3)
The i n t e r s e c t i o n of
sets i n (X,19)
d - d i s c r e t e f a m i l y o f c l o s e d sub-
i s closed. d-closed sets i s
A l s o , i f each s e t i n a closed i n (X,4), (4)
Every
d-closed.
d - d i s c r e t e family is
t h e n t h e union i s
d-closed.
d - d i s c r e t e s e t i n a uniform s p a c e ( X , S )
C-embedded i n
X.
Also,
d-
every p o i n t i n a
is d-dis-
141
UNIFORM SPACES
Crete s e t i s c l o s e d .
(5)
Every f i x e d
2-ultrafilter
and e v e r y c o n v e r g e n t
i s a Cauchy
Z - f i l t e r i s cauchy.
particular,
t h e neighborhood
b(p) = fZ
Z(X)
E
Z-filter
In
2- f i l t e r
i s a neighborhood o f
: Z
p)
Z - f i l t e r s i n c e i t converges t o
i s a cauchy
F i n a l l y , e v e r y Cauchy
p.
Z - f i l t e r converges t o each
Z-
of i t s c l u s t e r p o i n t s ; t h a t i s , e v e r y Cauchy f i l t e r is clusterable.
The f i r s t p a r t o f t h e n e x t r e s u l t i s found i n Gillman and J e r i s o n
The p a r t c o n c e r n i n g t h e u n i v e r s a l u n i -
(15.6).
formity i s easy t o v e r i f y . 12.5
and
THEOREM.
Lio(X)
are
If
2 Tychonoff s p a c e , t h e n
X
admissible uniformities
on
@*(X)
,
@(X),
X.
The n e x t r e s u l t r e l a t e s t h e c o n c e p t o f c o m p l e t e n e s s t o t h a t o f Cauchy f i l t e r s . 12.6
If
THEOREM.
statements
(X,fJ)
is 2
are e q u i v a l e n t :
uniform s p a c e , t h e n t h e followinq
(1)
The
(2)
Every Cauchy Bourbaki f i l t e r on
(3)
Every Cauchy
Z - f i l t e r on
(4)
Every Cauchy
Z-ultrafilter
uniform s p a c e (X,&) is c o m p l e t e . X
X
converges.
converqes.
on
X
converqes.
I t f o l l o w s immediately from t h e p r e v i o u s r e s u l t t h a t
e v e r y compact uniform s p a c e (X,&) i s c o m p l e t e . known t h a t t h e non-compact s p a c e
It is w e l l
I?? and i t s d i s c r e t e sub-
a r e b o t h complete r e l a t i v e t o t h e u s u a l m e t r i c .
space
The f o l l o w i n g r e s u l t s a r e found i n t h e Gillman and J e r i son t e x t
12.7
.
THEOREM.
(1) Every c l o s e d s u b s p a c e o f a complete
form s p a c e (2)
An
(X,&)
is
complete.
a r b i t r a r y .product
complete.
&-
of
complete uniform s p a c e s
is
HEWITT-NACHBIN SPACES AND RELATED SPACES
1 42
A
(3)
af
complete subspace closed.
2 Hausdorff uniform s p a c e i s
O n e of t h e fundamental r e s u l t s c o n c e r n i n g t h e t h e o r y of
uniform s p a c e s i s t h a t e v e r y Hausdorff uniform s p a c e ( X , B ) can b e embedded homeomorphically a s a d e n s e subspace o f a complete Hausdorff uniform space
may b e v regarded a s t h e q u o t i e n t of a subspace of t h e Stone-Cech com-
pX
pactification
struct X
and i s u n i q u e .
a r e extended t o t h e space
in
cX
d
to
CX
of
5
of a l l cauchy
CX
pX.
i s denoted by
If dC.
Z-ultrafil-
dcrY, t h e n t h e
Next, a l l p o i n t s
t h a t a r e c l u s t e r p o i n t s o f t h e same Cauchy
are identified; that is, 5 where
F i r s t t h e p s e u d o m e t r i c s on
which i s a subspace o f
X,
e x t e n s i o n of
Moreover, yX
Gillman and J e r i s o n con-
i n t h e f o l l o w i n g way.
'{X
t e r s on
yX.
5
and
by
. ' 3
G
belong t o The c l a s s e s
Q
i f and o n l y i f
cX.
3''
Z-filter
d C ( 5 , G ) = 0,
Denote t h e e q u i v a l e n c e c l a s s a r e t h e p o i n t s of
yX.
The
equation
defines
d.f
a s a pseudometric on
yX, and the c o l l e c t i o n
r d Y : d c 9 j g e n e r a t e s a Hausdorff uniform s t r u c t u r e on
yX.
For t h e d e t a i l s w e r e f e r t h e r e a d e r t o Theorem 1 5 . 9 of G i l l man and J e r i s o n . W e p o i n t o u t t h a t i t i s p o s s i b l e t o o b t a i n a completion
o f a non-Hausdorff uniform s p a c e ( X , & ) .
The c o n s t r u c t i o n f o r
such c o m p l e t i o n s i s g i v e n i n Theorem 2 7 and Theorem 28 of Chapter 6 of t h e K e l l e y t e x t . The n e x t theorem i s Theorem 1 5 . 1 1 of Gillman and J e r i s o n . 12.8
THEOREM.
If
i s d e n s e i n a uniform s p a c e ( T , & ) , then x i n t o a complete
X
e v e r y u n i f o r m l y c o n t i n u o u s f u n c t i o n from
uniform space h a s a u n i f o r m l y c o n t i n u o u s e x t e n s i o n
(T,fJ).
The f o l l o w i n g r e s u l t f o l l o w s immediately from t h e above theorem. 12.9
COROLLARY.
If
X
j s = uniform subspace
e v e r y uniformly c o n t i n u o u s f u n c t i o n
from
X
of
(T,&),
then
i n t o a complete
143
COMPLETENESS AND UNIFORM SPACES
uniform s p a c e h a s a u n i f o r m l y c o n t i n u o u s e x t e n s i o n t o t h e closure
of
(T,&).
X
The f o l l o w i n g r e s u l t i s problem 15.H o f Gillman and J e r i son. 12.10
THEOREM. X 2 Compact H a u s d o r f f s p a c e . (1) The o n l y a d m i s s i b l e uniform s t r u c t u r e X universal uniformity. (2)
Every c o n t i n u o u s mappinq from
X
i s the
i n t o a uniform
space i s uniformly continuous with r e s p e c t t o t h e unique a d m i s s i b l e u n i f o r m i t y Section 1 3 :
on
X.
Hewitt-Nachbin Completeness and Uniform Spaces
W e a r e now i n p o s i t i o n t o i n v e s t i g a t e t h e p r o p e r t y of
Hewitt-Nachbin completeness i n t h e c o n t e x t of u n i f o r m s t r u c V
t u r e s and t o s t u d y t h e r e l a t i o n s h i p s between t h e Stone-Cech compactification
pX,
t h e Hewitt-Nachbin c o m p l e t i o n
t h e uniform s t r u c t u r e completion
yX.
vX,
and
O n e of t h e p r i n c i p a l
r e s u l t s t o b e e s t a b l i s h e d i s t h e Nachbin- S h i r o t a Theorem a s s e r t i n g t h a t t h e Hewitt-Nachbin s p a c e s a r e p r e c i s e l y t h o s e Tychonoff s p a c e s t h a t admit a complete uniform s t r u c t u r e provided t h e c a r d i n a l i t y of t h e s p a c e i s nonmeasurable.
As a
c o r o l l a r y w e o b t a i n Katztovl s Theorem which s a y s t h a t e v e r y paracompact Hausdorff s p a c e of nonmeasurable c a r d i n a l i s Hewitt-Nachbin c o m p l e t e .
F i n a l l y t h e N a c h b i n - S h i r o t a Theorem
i s sharpened o b t a i n i n g a r e s u l t f o r Hewitt-Nachbin c o m p l e t e n e s s a n a l o g o u s t o t h e f a c t t h a t 2 uniform s p a c e i s compact i f and o n l y i f i t i s complete
and
t o t a l l y bounded.
I n o r d e r t o b e g i n o u r i n v e s t i g a t i o n some f a c t s concerning
C(X)/M
a s an o r d e r e d f i e l d a r e needed, where
a r b i t r a r y maximal i d e a l of
M
i s an
C(X).
The f o l l o w i n g d e f i n i t i o n s and r e s u l t s a r e b a s i c and may b e found i n most s t a n d a r d t e x t s on modern a l g e b r a . 13.1
DEFINITION.
A field
F
i s s a i d t o be t o t a l l y ordered
i n c a s e t h e r e e x i s t s a p a r t i t i o n of t h e non-zero e l e m e n t s of F
into disjoint classes
P
and
two c o n d i t i o n s a r e s a t i s f i e d :
h
such t h a t t h e f o l l o w i n g
HEWITT-NACHBIN SPACES AND RELATED SPACES
144
(1) I f
ach, then
If
(2)
I t i s said that
- a c P , and
a , b c 63, then a + b r 6 and a b c 6 . 6 ( r e s p e c t i v e l y , b) i s t h e c l a s s o f p o s i t i v e
( r e s p e c t i v e l y , n e q a t i v e ) e l e m e n t s of (a-b)
P, and
c
a
<
b
i f (a-b)
We write
F.
>
a
b
if
b.
E
I t i s customary t o r e f e r t o a t o t a l l y o r d e r e d f i e l d a s
simply an o r d e r e d f i e l d , and w e s h a l l a d o p t t h a t c o n v e n t i o n . I t i s e a s y t o show t h a t i f
and
b
<
b,
a
belong t o a = b,
a
>
i s an o r d e r e d f i e l d and i f
F
a
t h e n e x a c t l y one of t h e a l t e r n a t i v e s
F,
b
Moreover, i t can b e e s t a b l i s h e d
holds.
t h a t e v e r y o r d e r e d f i e l d c o n t a i n s an isomorphic copy o f t h e field 13.2
of r a t i o n a l numbers.
Q
An ordered f i e l d
DEFINITION.
i n f i n i t e l y l a r q e element ordered f i e l d
ment
acF If
F
a
if
a
2
n
i s s a i d t o b e archimedean i f f o r e v e r y e l e -
t h e r e e x i s t s an
n 2 a.
with
nclN
i s a maximal i d e a l i n
M
i s s a i d t o c o n t a i n an f o r every n c N . An
F
then
C(X),
C(X)/M
o r d e r a d i n such a way t h a t t h e c a n o n i c a l mapping of w i l l be o r d e r p r e s e r v i n g : namely, i f
C(X)/M
residue c l a s s of
f
in
tive i f there exists a f
E
C ( X ) modulo
g
in
M,
C ( X ) such t h a t
(mod M).
f
i s non-negative on some z e r o - s e t of
I t can b e shown t h a t
M(f)
C(X) onto
M(f) denotes t h e
then
g
if
can b e
2
M(f) i s p o s i g 0
>
0
and
i f and o n l y
M ( s e e Gillman and
Jerison, 5 . 4 ) . If 0
f
E
C(X),
according a s
then d e f i n e ( M ( f )1 t o be
M(f), -M(f), or
M(f) i s , r e s p e c t i v e l y , p o s i t i v e , n e g a t i v e , o r
zero. The f o l l o w i n g r e s u l t s a r e fundamental t o o u r f u t u r e work. 13.3
(2)
.
(1) The o r d e r e d f i e l d C(X)/M is archimedean i f and o n l y i f M is a r e a l maximal i d e a 1. For e v e r y f E C ( X ) the f o l l o w i n q s t a t e m e n t s are
THEOREM (Gillman and J e r i s o n )
equivalent: ( a ) lM(f) 1
(b)
The
infinitely larqe.
function
f
is unbounded
on e v e r y zero-
1 45
COMPLETENESS AND UNIFORM SPACES
s e t of --
(c)
M.
zn belonqs
t h e zero- set
nclN,
For each
= rx
to
: If(x)
1
L\: n j
Z[M] = f Z ( f )
IR ( s e e , f o r example, 0 . 2 1 i n G i l l -
f i e l d of the ordered f i e l d man and J e r i s o n ) .
If
i s a r e a l maximal i d e a l , then by
M
d e f i n i t i o n 7 . 4 the residue c l a s s f i e l d to
Now
2
M(f)
z e r o - s e t of
IR
5
n
i n t o i t s e l f i s the i d e n t i t y .
i f and o n l y i f
0
I t follows t h a t
M.
there e x i s t s a zero-set I f f x )1
On t h e o t h e r hand, i f M i s non-archimedean s i n c e t h e o n l y
C(X)/M
non-zero isomorphism of
for a l l
i s non-negative on some
f
IM(f)
1 5
belonging t o
Z
n
i f and only i f
such t h a t
Z[M]
( a ) i s equiva-
xcz; t h u s t h e n e g a t i o n of
l e n t t o t h e n e g a t i o n of Zn
i s isomorphic
C(X)/M
IR, and t h e r e f o r e archimedean.
i s h y p e r - r e a l , then (2)
: ftMj.
Z(X)
E
(1) Every archimedean f i e l d i s isomorphic t o a sub-
Proof.
(b)
c o n t a i n s a member of
.
Also, M ( If
Z [ M ] : hence
1)
L\: n
i f and only i f
( a ) is equivalent t o ( c )
completing t h e p r o o f . The next r e s u l t r e l a t e s Hewitt-Nachbin
completeness t o
t h e uniform s t r u c t u r e completeness r e l a t i v e t o t h e u n i f o r m i t y @(X)
.
13.4
I t appears a s 1 5 . 1 4 of Gillman and J e r i s o n . THEOREM (Gillman and J e r i s o n )
bin space, -
then
X
is
. If
i s a Hewitt-Nach-
X
complete i n t h e uniform s t r u c t u r e
3
I t w i l l f i r s t be e s t a b l i s h e d t h a t i f
proof.
3
2 - u l t r a f i l t e r on ( X , @ ( X ) ) then
@(X)
i s a Cauchy
has t h e c o u n t a b l e i n t e r s e c L
tion property.
so t h a t
Now, l e t
M
d e n o t e t h e maximal i d e a l
3
3 = Z[M] by 7 . 7 , and suppose t h a t
the c o u n t a b l e i n t e r s e c t i o n p r o p e r t y .
Then
archimedean. ment
M(f).
[xfX :
Hence, C ( X ) / M
n ] belongs t o
f o r each z e r o - s e t a point
pn
Z
in
belonging t o
is a h y p e r - r e a l C(X)/M
i s non-
c o n t a i n s an i n f i n i t e l y l a r g e e l e -
T h e r e f o r e , f o r each
1 f (x) 1 2
f a i l s t o have
M
maximal i d e a l by 7 . 4 i n which c a s e t h e f i e l d
Z-[a]
3 2
ncB
,
Z [MI = 3
the zero-set by 13.3 ( 2 c ) .
nclN
and f o r each
such t h a t / f ( p n )1
lows t h a t , r e l a t i v e t o t h e pseudometric
#f
in
'n
-
Thus,
there e x i s t s
2 n.
~t f o l -
@(X) , t h e
Z-
.
146
H E W I T T - N A C H B I N SPACES AND RELATED SPACES
5 c a n n o t c o n t a i n a z e r o - s e t of f i n i t e Qf-diame t e r . Hence, 5 i s n o t a Cauchy Z - f i l t e r . Therefore, i f 3 i s a Cauchy Z - u l t r a f i l t e r on ( X , @ ( X ) ) , t h e n 3 h a s t h e
ultrafilter
countable i n t e r s e c t i o n property. complete i t f o l l o w s t h a t
3
i n t h e uniform s t r u c t u r e
@(X).
Since
is fixed.
i s Hewitt-Nachbin
X
Hence
i s complete
X
The f o l l o w i n g theorem and i t s c o r o l l a r y w i l l e s t a b l i s h
p X , LX, and
an i m p o r t a n t r e l a t i o n s h i p between
~t ap-
yX.
p e a r s i n Gillman and J e r i s o n ( 1 5 . 1 3 ) . 13.5
THEOREM (Gillman and J e r i s o n ) .
Let
2 Tychonoff
X
space. completion
(1) @(X)
is
completion
(2) @*
(XI
of
(.,x,@(;X)
is
X
i n t h e uniform s t r u c t u r e
X
i n t h e uniform s t r u c t u r e
).
of
(PX,@+(PX)
1.
The uniform s p a c e ( ~ x , @ ( i l X ) )i s complete by 1 3 . 4 .
Proof.
Moreover, X t u r e on
i s dense i n
@(X) b e c a u s e
is
X
VX
and t h e r e l a t i v e uniform s t r u c X
is
C-embedded i n
t h e completion i s unique, t h i s i m p l i e s t h a t
Since
LIX.
is precisely
yX
The proof of ( 2 ) f o l l o w s s i m i l a r l y s i n c e e v e r y
(JX,@(JX)).
compact Hausdorff space i s c o m p l e t e . a Tychonoff s p a c e
(By 1 2 . 5 , s i n c e
@(X) i s an a d m i s s i b l e s t r u c t u r e .
.
is
X
It is
a l s o t h e unique a d m i s s i b l e s t r u c t u r e by 1 2 . 1 0 (1) ) 13.6
COROLLARY (Gillman and J e r i s o n )
.
L2t
X
b e a Tychonoff
space. (1) The s p a c e
i s Hewitt-Nachbin complete i f and
X
only i f i t i s complete i n t h e uniform s t r u c t u r e @(XI
.
The space
(2)
X
compact i f and only i f it i s com-
p l e t e i n t h e uniform s t r u c t u r e Proof.
( X , @ ( X ) ) i s complete,
If
y (X,@ ( X ) ) = 13.5(1).
(ux, @ ( u X ) )
*
@ (X).
i t follows t h a t ( X , @ ( X ) ) =
where t h e l a s t e q u a l i t y f o l l o w s by
Thus, X = uX ( u p t o homeomorphism) s o t h a t
Hewitt-Nachbin complete. entirely similar.
X
The p r o o f o f s t a t e m e n t ( 2 ) i s
is
COMPLETENESS AND UNIFORM SPACES
147
The n e x t s e v e r a l r e s u l t s a r e of a t e c h n i c a l n a t u r e and
w i l l b e used t o e s t a b l i s h t h e main t o o l ( 1 3 . 9 ) f o r p r o v i n g t h e Nachbin-Shirota Theorem. They a p p e a r i n Gillman and J e r i s o n ( 1 5 . 1 7 and 1 5 . 1 8 , r e s p e c t i v e l y ) . THEOREM (Gillman and J e r i s o n )
13.7
uniform s p a c e , and l e t
-e x i s t sets
E
Let
>
( X , & ) b e a Hausdorff
given.
0
There
( n c m , xcx) w i t h t h e f o l l o w i n q p r o p e r t i e s :
Z
n, x (1) The union
(2) Each s e t
U [Zn,x : n c l N , ~ E X : is Z
-less than
(3)
and
dc&
.
For each
n,x
X.
is -
d - c l o s e d and of
tha
family [Zn,x : XCX)
d-diameter
c. nclN,
is
d - a -
Crete. Proof.
Recall t h e usual conventions t h a t
d [ @ , A ] = OD
>
o f the s e t
X,
r
f o r every and l e t
8 =
rclR.
5.
x:
the element
and
be a w e l l - o r d e r i n g
Let
n
For e a c h
S(x,n) = { z : d(x,z) For e a c h f i x e d
d ( @ )= 0
6
-
and
x, d e f i n e
6 ;).
n , w e now proceed by t r a n s f i n i t e i n d u c t i o n on define
Z(x,n) = rz : d[Z(y,n),z]
2;6 ,
z
for a l l
y < x
and
s(x,n)j .
t
Thus, i f w e l e t
then z(x,n) = s(x,n)
n
n
c(y,n).
Y<X
I n order t o e s t a b l i s h t h a t each s u f f i c i e n t t o prove t h a t e a c h i n t e r s e c t i o n of suppose t h a t every t.
If
>
E
p
0
x
z(x,n) i s
C(y,n) i s
d-closed sets i s E
there is a point
i s any p o i n t i n
z
E
(12.4 (3) )
d [ C ( y , n ) , x ] = 0.
in
Z ( y , n ) then
it is
d-closed s i n c e t h e
d-closed
c d C ( y , n ) so t h a t
d-closed
C(y,n) with
.
Hence Then f o r
d(x,zE)
<
148
SPACES AND RELATED SPACES
HEWITT-NACHBIN
Hence x b e l o n g s t o C ( y , n ) . I t f o l l o w s t h a t c d C ( y , n ) c C(y,n) so t h a t C ( y , n ) i s d-closed. Therefore Z ( x , n ) i s a d - c l o s e d s e t and hence i s a l s o a z e r o - s e t by 12.4(1).
Set
.
= Z(x,n) C l e a r l y t h e d - d i a m e t e r of z s a t i sf ies 'n,x n,x 6 dlZn,xl 2 6 < E , and, f o r Y < x , d [ z n , x , Z n , y l 2;. This e s t a b l i s h e s s t a t e m e n t s ( 2 ) and ( 3 ) . T o prove (l), l e t z t x
be a r b i t r a r y .
element that
x
By t h e w e l l - o r d e r i n g of
in
0
d(xo,z)
<
y < xo
for a l l c h o i c e of
x
w E S(y,n).
. 0
.
it i s the case t h a t
y
z c Z(xo,n).
d(y,z)
let
xo
w
nem
Choose
b.
We w i l l show t h a t
For each
1d(y,Z) -
I t follows t h a t
y < xo.
- a n
6
there is a l e a s t
X
<
d(xo,z)
f o r which
2
6
so
Now
by t h e
Z ( y , n ) so t h a t
t
Thus,
d(w,Z)
all
X
d [ Z ( y , n ), z ]
Therefore
2
d(y,w)
z
t
2
a
-
6
(6
- -1n6
so that
6
= -n
-
z E c(y,n) for
Z ( x o , n ) c o m p l e t i n g t h e proof of
t h e theorem. Observe t h a t f o r each f i x e d
n , e v e r y union of sets
(see 12.4(3) and 1 2 . 4 ( 1 ) ) . I n p a r t i c u l a r , f o r each n , t h e set U 'Z : X E x j is a zero-set. n,x Now, t h e union of t h i s c o u n t a b l e f a m i l y of z e r o - s e t s i s x by 1 3 . 7 ( 1 ) . Hence g i v e n any Z - u l t r a f i l t e r 3 w i t h t h e countab l e intersection property, there e x i s t s ktm such t h a t i n 13.7 is a zero-set
'li,X
{z
: xcX: b e l o n g s t o 5 ( i . e . , i f a c o u n t a b l e union of k,x z e r o - s e t s b e l o n g s t o a r e a l Z - u l t r a f i l t e r 3, then a t l e a s t I,
one of them b e l o n g s t o
a).
By r e l a b e l i n g t h e non-empty
w e have o b t a i n e d t h e n e x t t e c h n i c a l r e s u l t . 13.8
COROLLARY
( G i l l m a n and J e r i s o n )
d o r f f uniform s p a c e , l e t
5
be a r e a l
-with t he
dc19
Z-ultrafilter
on
and X.
followins properties: (1) The union U [Za : a&)
E
.
z
k, x
Let (X,&) be a Haus-
>
0
&given,
There e x i s t s e t s belongs
to
3.
and l e t
Za ( a 4 )
149
COMPLETENESS AND UNIFORM SPACES
(2)
Each
d - d i a m e t e r less than
i s of
Za
t.
(3)
The
(4)
The union of any subfamily i s a z e r o - s e t .
family i Z a :
aEG)
is
d-discrete.
The f o l l o w i n g r e s u l t p r o v i d e s t h e main t o o l f o r e s t a b I t appears a s 1 5 . 1 9 i n
l i s h i n g t h e Nachbin-Shirota Theorem. Gillman and J e r i s o n .
Let
LEMMA (Gillman and J e r i s o n ) .
13.9
uniform s p a c e .
be a Hausdorff
(X,f?)
I f f o r each p s e u d o m e t r i c
dc6
every
Crete subspace i s Hewitt-Nachbin complete, and i f
on
Z-ultrafilter
then
is 2
5
Proof.
Z-filter.
By t h e p r e v i o u s c o r o l l a r y , f o r each
E
and l e t
Za
S = :sa
d-discrete subset of
zs
Define
i f and o n l y i f
U (Za
empty s e t does n o t belong t o Moreover, i f
E
U Za c U Za. S,EE sacE'
i t follows t h a t
SS
E
and
E'
E'
b Za sacE
Since
belong t o
Zs,
: sa t E )
(
u za) n
(
sa€E
belongs t o
Now, i f
zs
then
Za
u za)
belongs t o
U Za)
sac E '
u
s a c E nE '
sa E S \ Z ) ) .
E
is S,
C
5. The
S.
3 by d e f i n i t i o n ,
fl (
U
2-filter.
Finally, Za)
if
E
b e l o n g s t o 3.
saeE'
za.
i t i s e a s y t o show t h a t ~t f o l l o w s t h a t E n E '
Ss,
then
are disjoint, =
is a
s i n c e i t d o e s n o t belong t o S s a t i s f i e s E C E l , then
by d e f i n i t i o n . (
0
C
Ss.
Z
C S
and
Z
#
However, by 1 3 . 8 ( 1 ) i t i s t h e c a s e t h a t Observe t h a t
E
Z - f i l t e r on
SaEE
S i n c e the sets
S
for
U Za b e l o n g s t o 3 s i n c e 5 is a S-EE'
belongs t o
El
>
F
choose a p o i n t
a s follows:
is a
3,
and
U
Therefore,
acG
I t is clear that
Zs
zs
Next, i t w i l l be shown t h a t
3.
and
and t h e r e f o r e , by h y p o t h e s i s , S
X
a Hewitt-Nachbin s p a c e . E E
ad).
:
dt8
: a E G j of z e r o - s e t s i n
d - d i s c r e t e family ( Z a
Using t h e axiom of c h o i c e , f o r each
sa
2
with the countable i n t e r s e c t i o n property,
X
cauchy
there e x i s t s a X.
d - 2 -
5
U (Za
Since
: sa E
3
is a
U ( Z a : sa U (Za
S ) = ( U (Za : sa
E
: s
a
2)
E E
S)
# 3. E
Z ) ) U (U (Za
5. :
Z - u l t r a f i l t e r and t h e r e f o r e prime,
HEWITT-NACHBIN SPACES AND RELATED SPACES
150
U TZQ : sa
and s i n c e that 7
S'
U
cZa
:
sa
Since
S
belongs to
S\Z]
t
is a
T h e r e f o r e , ZS
set
Z,
s
belongs t o of
5.
i t follows
Hence, S\Z
Z - u l t r a f i l t e r on
belongs t o
by 6 . 8 ( 3 ) .
S
i s Hewitt-Nachbin complete i t f o l l o w s t h a t
there e x i s t s a point f o r e , (s,]
5
Z ) does n o t belong t o
E
a 5,
E
S
satisfying
by 6 . 8 ( 2 ) .
d - d i a m e t e r l e s s than
F.
sa
n
E
Hence, 5
Ss.
There-
c o n t a i n s the
This concludes t h e
proof o f t h e lemma.
I n 8.18 i t was observed t h a t t h e r e q u i r e m e n t f o r a d i s c r e t e s p a c e t o b e Hewitt-Nachbin complete i s q u i t e weak: namely, a d i s c r e t e s p a c e f a i l s t o b e Hewitt-Nachbin complete i f and o n l y i f i t i s o f measurable c a r d i n a l . p r e c e d i n g lemma, i n o r d e r t h a t r e a l
Moreover, by t h e
Z - u l t r a f i l t e r s b e Cauchy
Z - f i l t e r s i n a uniform s p a c e , w e need o n l y e x c l u d e m e a s u r a b l e cardinals.
These o b s e r v a t i o n s pave t h e way t o t h e f o l l o w i n g
i m p o r t a n t r e s u l t due t o T . S h i r o t a (1951 and 1954) and L . Nachbin (1950 and 1 9 5 4 ) .
13.10
THE NACHBIN-SHIROTA THEOREM.
X
s p a c e i n which e v e r y c l o s e d d i s c r e t e subspace ble cardinal. -
if x
Proof.
Then
2 Tvchonoff nonmeasura-
i s Hewitt-Nachbin complete i f and only
X
a d m i t s a complete Hausdorff uniform s t r u c t u r e . Suppose t h a t
structure
8.
x
For each
a d m i t s a complete Hausdorff uniform dcrD, e v e r y
d - d i s c r e t e subspace
S
i s a c l o s e d d i s c r e t e subspace ( 1 2 . 4 ( 2 ) ) t h a t h a s nonmeasurable c a r d i n a l , and hence i s Hewitt-Nachbin complete by 8.18. f o r e , by 1 3 . 9 e v e r y
Z - u l t r a f i l t e r on
i n t e r s e c t i o n p r o p e r t y i s a Cauchy Z-ultrafilter i s fixed.
X
with t h e countable
Z-filter.
Therefore, X
There-
H e n c e e v e r y such
i s a Hewitt-Nachbin
space, Conversely,
if
X
i s a Hewitt-Nachbin s p a c e t h e n
a d m i t s t h e complete s t r u c t u r e
@(X) by 1 3 . 4 .
X
This concludes
t h e proof o f t h e theorem. W e remark t h a t t h e proof o f t h e n e c e s s i t y i n t h e above
theorem d i d n o t r e q u i r e t h e c o n d i t i o n imposed on t h e s u b s p a c e s . However, t h a t i s n o t s u r p r i s i n g s i n c e e v e r y c l o s e d subspace of
151
COMPLETENESS AND UNIFORM SPACES
a Hewitt-Nachbin s p a c e i s Hewitt-Nachbin complete, and i f i t
i s a l s o d i s c r e t e , then by 8 . 1 8 i t m u s t be o f nonmeasurable cardinal. 13.11
This o b s e r v a t i o n y i e l d s t h e following r e s u l t .
COROLLARY
(Gillman and J e r i s o n )
.
complete Hausdorff
uniform s p a c e (X,&) i s a Hewitt-Nachbin s p a c e i f and o n l y i f e v e r y c l o s e d d i s c r e t e subspace
of
X
i s a Hewitt-Nachbin
space.
I n t h e d i s c u s s i o n immediately f o l l o w i n g C o r o l l a r y 8 . 1 9 , i t was p o i n t e d o u t t h a t e v e r y c a r d i n a l number l e s s t h a n o r e q u a l t o a nonmeasurable c a r d i n a l i s a g a i n a nonmeasurable cardinal.
Hence,
i n a s p a c e o f nonmeasurable c a r d i n a l i t y i t
i s immediate t h a t e v e r y c l o s e d d i s c r e t e subspace h a s nonmeas u r a b l e c a r d i n a l y i e l d i n g a n o t h e r c o r o l l a r y t o t h e NachbinS h i r o t a Theorem. 13.12
COROLLARY.
A
Tvchonoff s p a c e
X
of
nonmeasurable
c a r d i n a l i s a Hewitt-Nachbin s p a c e i f and o n l y i f
-a
X
admits
complete Hausdorff uniform s t r u c t u r e . Now i t i s known t h a t i f
space, then
X
i s a paracompact Hausdorff
X
a d m i t s t h e uniform s t r u c t u r e ( i n t h e s e n s e of
Tukey) c o n s i s t i n g o f a l l neighborhoods o f t h e d i a g o n a l (see Kelley,
1 9 5 5 , Problem 6L, page 2 0 8 ) .
i n that structure.
I n fact, X
i s complete
S i n c e e v e r y member of any u n i f o r m i t y on
X
i s a neighborhood o f t h e d i a g o n a l ( K e l l e y , Theorem 6 , page 1 7 9 ) , i t f o l l o w s t h a t whenever
X
space the u n i v e r s a l uniformity
11 ( X ) i s c o n t a i n e d i n t h e u n i -
i s a paracompact Hausdorff 0
f o r m i t y c o n s i s t i n g of a l l neighborhoods o f t h e d i a g o n a l . 11 0
(X)
Now
i s t h e l a r g e s t a d m i s s i b l e uniform s t r u c t u r e ( s e e Gillman
and J e r i s o n , 1 5 G . 4 ) .
Thus t h e uniform s t r u c t u r e c o n s i s t i n g of
a l l neighborhoods o f t h e d i a g o n a l a s s o c i a t e d w i t h a paracomp a c t Hausdorff s p a c e i s p r e c i s e l y t h e u n i v e r s a l u n i f o r m i t y LO(X).
T h e r e f o r e , e v e r y paracompact Hausdorff s p a c e i s com-
plete i n the structure
Lio(X).
This o b s e r v a t i o n coupled w i t h
t h e Nachbin-Shirota Theorem y i e l d s t h e f o l l o w i n g i m p o r t a n t
r e s u l t due t o M . K a t z t o v (195lA, Theorem 3 ) .
152
SPACES AND RELATED SPACES
HEWITT-NACHBIN
13.13
THEOREM ( K a t z t o v )
.
~fx
is 2
paracompact Hausdorff
s p a c e such t h a t e v e r y c l o s e d d i s c r e t e s u b s p a c e measurable c a r d i n a l , t h e n
X
of
X
h a s non-
i s a Hewitt-Nachbin s p a c e .
I t f o l l o w s from 1 3 . 1 3 t h a t e v e r y paracompact Hausdorff
s p a c e of nonmeasurable c a r d i n a l i s Hewitt-Nachbin c o m p l e t e . S . Mrdwka (1964) h a s e s t a b l i s h e d a n a l t e r n a t i v e proof t o V
K a t e t o v ' s Theorem which d o e s n o t depend on t h e Nachbin-Shirota Theorem. Mrdwka shows d i r e c t l y t h a t a s p a c e s a t i s f y i n g t h e v h y p o t h e s i s o f K a t e t o v ' s Theorem must f u l f i l l c o n d i t i o n ( 3 ) o f V
8 . 8 and hence b e Hewitt-Nachbin c o m p l e t e .
Katetov's original
proof t o 13.13 d o e s n o t u s e t h e uniform s p a c e c o n c e p t e i t h e r , b u t appeals d i r e c t l y t o 8 . 8 ( 4 ) . V
The f o l l o w i n g c o r o l l a r y i s a l s o due t o K a t e t o v (1951A, Corollary 3 ) . 13.14
COROLLARY ( K a t g t o v )
.
Every m e t r i z a b l e s p a c e o f non-
measurable c a r d i n a l i s a Hewitt-Nachbin s p a c e . Every m e t r i c s p a c e i s paracompact H a u s d o r f f .
Proof.
The
r e s u l t i s now immediate from 1 3 . 1 3 . W e comment t h a t t h e 1972 p a p e r of M .
R i c e c o n t a i n s an-
o t h e r proof o f 13.14 t h a t d o e s n o t depend on t h e uniform s p a c e concept. V
K a t e t o v ' s r e s u l t h a s some i n t e r e s t i n g a p p l i c a t i o n s . example, i t was p o i n t e d o u t i n 7 . 1 5 ( 4 ) [O,n)
For
t h a t t h e o r d i n a l space
i s c o u n t a b l y compact and pseudocompact, b u t n o t H e w i t t -
Nachbin complete.
Moreover, s i n c e e v e r y r e g u l a r second counta-
b l e space i s paracompact, i t f o l l o w s from 1 3 . 1 3 t h a t the o r d i n a l s p a c e [O,n)
c a n n o t be second c o u n t a b l e :
i n fact, it is
n o t even L i n d e l o f . One of t h e i m p o r t a n t r e s u l t s c o n c e r n i n g a uniform s p a c e ( X , B ) i s t h a t i t i s compact i f and o n l y i f i t i s complete and
t h e union o f a f i n i t e number of sets o f E
f o r each p s e u d o m e t r i c
dcB
d - d i a m e t e r less than
and e a c h p o s i t i v e
E.
The
following i s t h e analogue t o t h a t r e s u l t i n t h e c a s e o f H e w i t t Nachbin completeness and i s Theoram 1 5 . 2 1 o f Gillman and Jerison.
The proof w i l l r e f e r t o t h e c o n s t r u c t i o n of
c u s s e d a t t h e end o f S e c t i o n 1 2 (see page 1 4 2 ) .
yX
a s dis-
COMPLETENESS AND UNIFORM SPACES
If
THEOREM (Gillman and J e r i s o n ) .
13.15
153
d Hausdorff
(X,;Q)
uniform s p a c e , then t h e f o l l o w i n q s t a t e m e n t s
The
(1)
For each
(2)
s e tin
i s a Hewitt-Nachbin s p a c e , c a r d i n a l of e v e r y
dcr9 X
For e v e r y
(3)
yX
completion
is
Every
(4)
and
dcrS
i
>
0,
&a
X
nonmeasurable
d-diameter less than o r
E .
on
Z-ultrafilter
intersection propsrty Proof.
d-discrete
nonmeasurable.
union o f z e r o - s e t s of ----equal t o
equivalent:
with the
X
is 2
Cauchy
I t w i l l be shown t h a t c o n d i t i o n
countable
Z-filter.
(2) i s equivalent t o
each o f t h e o t h e r c o n d i t i o n s . ( 2 ) implies ( 3 ) :
Suppose t h a t f o r some
derP
and
t
Z
0,
X
i s n o t t h e union o f any nonmeasurable c o l l e c t i o n of z e r o - s e t s d - d i a m e t e r l e s s than o r e q u a l t o
of
c.
r
Let
b e an index-
i n g s e t of measurable c a r d i n a l t h a t i s w e l l - o r d e r e d , and l e t
trarily. x
r.
d e n o t e t h e f i r s t element of
yl
Choose
x
Y1
Using t r a n s f i n i t e i n d u c t i o n , f o r each
in y E r
X
arbi-
choose
i n t h e complement o f
Y
The s e t (x
Y
:
(3) implies
(2):
Let
of gauge
6
>
By ( 3 ) , X
sets of
Y E T ] i s measurable and 0.
dcr9
d-discrete.
and l e t
d-discrete set
be a
S
i s a nonmeasurable union of z e r o 76 . H e n c e , each o f
d-diameter less than o r e q u a l t o
t h e s e s e t s c o n t a i n s a t most one p o i n t of
Therefore,
S.
S
i s of nonmeasurable c a r d i n a l . (2) implies ( 4 ) :
space o f
X.
Let
Then
dcB
and l e t
S
be a
d - d i s c r e t e sub-
i s d i s c r e t e and, s i n c e by h y p o t h e s i s
S
i s of nonmeasurable c a r d i n a l , i t f o l l o w s from 8.18 t h a t
S
S
is
The c o n c l u s i o n i s now immediate from
a Hewitt-Nachbin s p a c e . 13.9. ( 4 ) implies
6
>
0.
(21:
Let
S
be a
I t w i l l be shown t h a t
d - d i s c r e t e set i n S
from which i t f o l l o w s by 8.18 t h a t cardinal,
Since
S
is
X
of gauge
i s a Hewitt-Nachbin s p a c e S
d-discrete i n
i s of nonmeasurable X,
S
is
C-embedded
154
SPACES AND RELATED SPACES
HEWITT-NACHBIN
in
X
CX
d e n o t e t h e c o l l e c t i o n o f a l l Cauchy
by 1 2 . 4 ( 4 ) .
and l e t
Hence, c l u x S = VS
by 8 . 1 1 .
X
d e n o t e t h e c o m p l e t i o n of
yX
~ 1 ,= ~2.5 ~c s;x,
'JS c c X .
i t f o l l o w s from t h e h y p o t h e s i s t h a t
be a neighborhood of p i n cX 6 meter i s l e s s than y I t follows t h a t U
and l e t
U
in
of
p
to
clcx(U
CX
n
p i ;S,
whose
d -dia-
C
contains a t
Since
p t c l j x S c c l c x S , e v e r y neighborhood m u s t i n t e r s e c t U n S. Therefore, p belongs
S).
follows t h a t
Let
ns
.
most one p o i n t .
x
a s d i s c u s s e d i n Sec-
S i n c e i t h a s been e s t a b l i s h e d t h a t
tion 1 2 .
Next, l e t
2 - u l t r a f i l t e r s on
p
Because t h e p o i n t s of
P, S c
U
E
a r e closed, it
S
T h e r e f o r e , s3 ;
S.
c S
so t h a t
S
i s a Hewitt-Nachbin s p a c e . ( 2 ) implies (1): L e t
yX.
t h e composition s u b s e t of every
yX
d'
belong t o t h e uniform s t r u c t u r e on
I t w i l l b e shown t h a t e v e r y
i s of nonmeasurable c a r d i n a l .
.(X
d ' - d i s c r e t e s u b s e t of
Z - u l t r a f i l t e r on
w i t h t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y i s a Cauchy Thus, l e t
f i l t e r and hence f i x e d .
set of
yX
a point
s
s2
and
of
in
S c X
where X x X.
d
be a
T
yX, w e
i s dense i n
X
by c h o o s i n g , f o r each p o i n t d' (s,t)
<
$.
Hence,
i n a s s o c i a t i o n with t h e p o i n t s
S
Z-
d ' - d i s c r e t e subtcT,
s1
if tl
and
it follows t h a t
d ' (s1,s2) 2
Thus
Since
0.
satisfying
X
belong t o T,
>
b
o f gauge
may c o n s t r u c t a s e t
t2
Then by 8.18
i s Hewitt-Nachbin complete
from which i t f o l l o w s by 1 3 . 9 t h a t e v e r y yX
d'-discrete
'5b
and
S
is
d - d i s c r e t e of gauge
i s t h e r e s t r i c t i o n of t h e p s e u d o m e t r i c By h y p o t h e s i s , t h e c a r d i n a l i t y o f
S
d'
-36 ' to
i s nonmeasurable,
and by c o n s t r u c t i o n IT1 = I S . I t follows t h a t every
Z - u l t r a f i l t e r on
c o u n t a b l e i n t e r s e c t i o n p r o p e r t y i s a Cauchy fixed.
Therefore, yX
(1) i m p l i e s ( 2 ) :
in
X.
Let
Then each p o i n t
yX
with the
Z - f i l t e r and hence
i s Hawitt-Nachbin c o m p l e t e . dE;B ptS
and l e t
S
be a
d-discrete set
can b e i d e n t i f i e d w i t h i t s
COMPLETENESS AND UNIFORM SPACES
a s s o c i a t e d Cauchy neighborhood dy-discrete i n associated with
S
dy
.
by 12.4(2).
Hence, S
d e n o t e s t h e p s e u d o m e t r i c on
a s discussed i n Section 12.
d
yX
is closed i n
IJ ( p )
Z-ultrafilter
yX, where
is yX
155
Since
yX
Therefore,
i s a Hewitt-Nach-
b i n s p a c e by h y p o t h e s i s , i t f o l l o w s from 8.10(4) t h a t Hewitt-Nachbin s p a c e .
Therefore, S
is a
S
i s a d i s c r e t e Hewitt-Nach-
b i n s p a c e s o t h a t i t i s of nonmeasurable c a r d i n a l by 8.18. T h i s c o n c l u d e s t h e proof o f t h e theorem. Observe t h a t i f
i s a complete Hausdorff uniform
(X,P)
space, then t h e i m p l i c a t i o n ( 2 ) i m p l i e s
(1) i n 1 3 . 1 5 i s simply
t h e N a c h b i n - S h i r o t a Theorem. R e c e n t l y H . Buchwalter and J . Schmets ( 1 9 7 3 ) have s t u d i e d t h e Hewitt-Nachbin completion and, more g e n e r a l l y , Hewitt-Nachbin s p a c e s i n t h e c o n t e x t of f u n c t i o n a l a n a l y s i s .
I n t h a t theory
Cc(X)
denotes the algebra
C ( X ) w i t h t h e com-
p a c t open topology, and t h e Nachbin-Shirota
Theorem t r a n s l a t e s
i n t o the following:
The Tychonoff
and o n l y (The s p a c e
if
Cc(X)
space Cc(X)
X
is
i s Hewitt-Nachbin complete
if
bornoloqical.
i s b o r n o l o q i c a l i f and o n l y i f e a c h s e m i -
norm t h a t i s bounded on t h e bounded s e t s of
Cc(X)
i s continu-
Thus one i s l e d t o compare b o r n o l o g i c a l l o c a l l y convex
ous.)
t o p o l o g i c a l v e c t o r s p a c e s and Hewitt-Nachbin t o p o l o g i c a l spaces.
I n t h e Buchwalter-Schmets t h e o r y t h e e l e m e n t s o f
VX
comprise t h e s e t of m u l t i p l i c a t i v e l i n e a r f u n c t i o n a l s on t h e a l g e b r a C ( X ) which a r e u n i t a r y ( i. e . , = 1 f o r such a
(L)
linear functional
14).
Then
uX
IR
c o n s i d e r e d a s a subspace of
becomes a t o p o l o g i c a l s p a c e T h i s approach h a s t h e
a d v a n t a g e o f b r i n g i n g t o g e t h e r r e s u l t s i n g e n e r a l topology and functional analysis.
I n t h e i r 1 9 7 1 p a p e r J . Schmets and M. DeWilde markedly s t r e n g t h e n e d t h e N a c h b i n - S h i r o t a Theorem.
They showed t h e
following :
The Tychonoff
and o n l y (The s p a c e
if
Cc(X)
space Cc ( X )
X
i s Hewitt-Nachbin complete
if
is u l t r a b o r n o l o q i c a l .
i s u l t r a b o r n o l o q i c a l i f and o n l y i f each
HEWITT-NACHBIN SPACES AND RELATED SPACES
156
semi-norm t h a t i s bounded on t h e convex compact s e t s o f
Cc(X)
I n t h e i r 1974 p a p e r , D . G u l i c k and F . G u l i c k shed f u r t h e r l i g h t on t h e Nachbin-Shirota Theorem and i t s i s continuous.)
relatives.
They mention t h a t t h e c o l l e c t i o n of theorems under
i n v e s t i g a t i o n began w i t h E . H e w i t t , who proved i n 1950 (Theorem 2 2 ) t h a t
X
i s Hewitt-Nachbin complete i f and o n l y
i f e v e r y semi-norm which i s bounded on a l l order-bounded s e t s of
Cc(X)
i s continuous.
sub-
T h i s was followed by t h e s i m u l -
t a n e o u s e s t a b l i s h m e n t o f t h e Nachbin-Shirota by L . Nachbin and T . S h i r o t a .
Theorem i n 1954
Next o c c u r r e d t h e Schmets-
DeWilde theorem i n 1971 which was a l s o e s t a b l i s h e d by H . BuchWalter i n h i s 1971A p a p e r , a l t h o u g h i n a d i f f e r e n t f o r m u l a t i o n . (Buchwalter proved t h a t
i s Hewitt-Nachbin complete i f and
X
only i f
C c ( X ) i s t h e i n d u c t i v e l i m i t o f t h e Banach s p a c e s [EH : H E # ) , where 51 i s t h e c o l l e c t i o n of a l l b a l a n c e d , con-
vex, p o i n t w i s e c l o s e d , e q u i c o n t i n u o u s and p o i n t w i s e bounded s ubse t s of
C ( X ) , and where
EH
i s t h e span o f
H,
f o r each
I n t h e i r 1974 p a p e r , t h e G u l i c k ' s prove t h a t t h e Nach-
HE#.)
b i n - S h i r o t a Theorem i s n o t e x a c t l y s t r o n g e r t h a n t h e H e w i t t Theorem, b u t t h a t t h e Schmets-DeWilde Theorem i s g e n u i n e l y s t r o n g e r t h a n H e w i t t ' s Theorem and t h e Nachbin- S h i r o t a They a l s o e s t a b l i s h t h e e q u i v a l e n c e of t h e theorems
Theorem.
For f u r t h e r d e t a i l s w e
o f Schmets-DeWilde and o f Buchwalter.
r e f e r t h e r e a d e r t o t h e 1971A and 1971B p a p e r s by H . BuchWalter,
t h e 1971 p a p e r by J . Schmets and M. DeWilde,
t h e 1973
p a p e r by Buchwalter and Schmets, and t h e 1974 p a p e r by D . G u l i c k and F . G u l i c k . The Hewitt-Nachbin completion denote the algebra of s u b s e t s o f Z ( X ) of a l l z e r o - s e t s i n
additive set function in
3(X,IR),
can a l s o be o b t a i n e d
I n t h a t approach w e l e t
a s a s p a c e o f measures. tion
uX
m
on
g e n e r a t e d by t h e c o l l e c -
X
X.
3(X,IR)
A (O,l]-valued f i n i t e l y
3(X,lR)
such t h a t f o r e a c h
A
m ( A ) = sup(m(Z) : Z E Z ( X ) , Z c A ) is a ( 0 , l ) -
measure on
Z(X,IR).
denoted by
Mo(X,IR).
The c o l l e c t i o n of a l l such measures i s The vaque topoloqy f o r
g e n e r a t e d by t h e neighborhood systems
Mo(X,IR )
is t h a t
ALMOST REALCOMPACT AND
m 6 Mo ( X , I R ) , f 0 i s homeomorphic t o p X .
where
*
E
C (X)
,
cb- SPACES
and
E
>
2X.
Mo ( X , IR) Mo(X,IR)
Then
0.
Mo(X,IR) of
The subspace
c o n s i s t i n g o f t h e countably a d d i t i v e members of homeomorphic t o
157
Mo(X,IR)
is
For f u r t h e r d e t a i l s concerning t h i s
approach we r e f e r t h e r e a d e r t o t h e 1 9 6 1 paper of V . Varadarjan and t h e 1 9 7 4 paper of G . Bachman, E . Beckenstein, and L . Narici. Section 14:
Almost Realcompact and
cb-Spaces
I n t h i s s e c t i o n we w i l l i n v e s t i g a t e s e v e r a l c l a s s e s o f spaces t h a t a r e c l o s e l y r e l a t e d t o t h e Hewitt-Nachbin s p a c e s . The f i r s t of t h e s e i s the c l a s s o f almost realcompact spaces f i r s t introduced by 2. FrolTk i n h i s 196lA and 1 9 6 1 B p a p e r s . (Although we have n o t used t h e term "realcompact" f o r Hewitt/
Nachbin spaces i n t h i s book we a r e r e t a i n i n g F r o l i k ' s o r i g i n a l terminology of "almost realcompact
.'I)
Unlike t h e Hewitt-Nach-
b i n s p a c e s , an almost realcompact space need n o t s a t i s f y t h e Tychonof f s e p a r a t i o n p r o p e r t y
.
A n a r b i t r a r y t o p o l o g i c a l space
X
is said
t o be almost realcompact i f f o r every u l t r a f i l t e r
3
of open
14.1
DEFINITION.
-
3 = ( c l F : F E Z ) has t h e c o u n t a b l e i n t e r s e c -
s e t s such t h a t
tion property i t i s the case t h a t
-5
i s fixed.
Before we r e l a t e t h e almost realcompact s p a c e s t o t h e Hewitt-Nachbin
s p a c e s , i t w i l l be u s e f u l t o c h a r a c t e r i z e a l -
most realcompactness i n terms of c e r t a i n c o l l e c t i o n s of open c o v e r i n g s on t h e t o p o l o g i c a l space
X.
T h i s i n t u r n w i l l pro-
v i d e a s i m i l a r c h a r a c t e r i z a t i o n f o r Hewitt-Nachbin complete/
n e s s and prompts t h e f o l l o w i n g d e f i n i t i o n due t o F r o l i k . 14.2
DEFINITION.
Let
a =
(u)
be a non-empty c o l l e c t i o n o f
open c o v e r i n g s of a t o p o l o g i c a l space of s u b s e t s of each
UEa
there e x i s t s e t s
The c o l l e c t i o n ever
63
i s s a i d t o be an
X
i s an
a
AEU
X.
A f i l t e r base
K3
a-Cauchy f a m i l y i f f o r and
BGR
i s s a i d t o be complete i f
satisfying
n
#
@
B
C
when-
a-cauchy f a m i l y .
W e remark t h a t many of t h e r e s u l t s t h a t f o l l o w w i l l b e
A.
HEWITT-NACHBIN SPACES AND RELATED SPACES
158
concerned w i t h some s p e c i f i c f a m i l y o f open c o v e r i n g s t h a t For example, t h e Greek l e t t e r
w i l l be s u i t a b l y named.
I1yI1
w i l l be used t o d e n o t e t h e c o l l e c t i o n of a l l c o u n t a b l e open c o v e r i n g s of a space
and l a t e r on i n t h e s e q u e l w e w i l l
X,
u s e the n o t a t i o n rlB(Q)tl t o r e f e r t o another p a r t i c u l a r family
of open c o v e r i n g s .
Thus, we w i l l c o n s i d e r l'y-Cauchy'l and
"R(Q)-Cauchy" f a m i l i e s i n c o n n e c t i o n w i t h d e f i n i t i o n 1 4 . 2 . /
The f o l l o w i n g r e s u l t s a r e found i n t h e 1963 p a p e r o f F r o l i k . 14.3 X
THEOREM ( F r o l f k )
i s an
. An
5
ultrafilter
a-Cauchy f a m i l y i f and o n l y i f
open cover
Uca.
Proof.
5
If
i s an
t h e r e e x i s t sets
and
FEZ,
A
i s an u l t r a f i l t e r
F F ~such t h a t
5
Conversely, i f
t h e r e e x i s t s an open c o v e r f o r each
n3 #
U
C
f a i l s t o be
F.
A
a-Cauchy and
AEU
Therefore s i n c e
5, whence
cannot belong t o
UEa
Then
A.
such t h a t f o r e a c h
Uca
does n o t c o n t a i n A
F
of
f o r every
a-Cauchy f a m i l y , t h e n f o r each
AEU
n 5.
21
belongs t o
o f open s u b s e t s
21
n
5
5 =
a.
1 4 . 4 LEMMA ( F r o l l k ) , y d e n o t e t h e c o l l e c t i o n of a l l c o u n t a b l e open c o v e r i n q s of a s p a c e X . An u l t r a f i l t e r 5
o f open s u b s e t s of X is the countable i n t e r s e c t i o n Proof.
5
Let
be a :
ism).
X\cl
:
3
fact that
U
n
by 14.3 t h e r e e x i s t s a FA
n
A =
a.
Then
FA
T h e r e f o r e , FA c X \ c l A to that
3.
3
5
FA j?
y-Cauchy f a m i l y .
Let
I(
n
5 =
Then
a.
Hence, f o r 5 such t h a t i s an open s e t .
belonging t o
since
FA
which implies t h a t
Furthermore, s i n c e
a.
This c o n t r a d i c t s the
such t h a t
cl A =
: icm] =
h a s t h e countable i n t e r s e c -
is not a
Ucy
n
[ c l Fi
by 1 4 . 3 .
3:
there e x i s t s a set
AcU
has
so t h a t t h e r e e x i s t s a s e t
Ucy
is a f i l t e r .
t i o n p r o p e r t y , and t h a t
n
with
5
Conversely, suppose t h a t
each
5
property.
Then
belonging t o
Fi
-
y-Cauchy f a m i l y and suppose t h e r e e x i s t s
icm) in
a sequence I F i U = {X\cl Fi
y-Cauchy i f and o n l y i f
c l ( X \ c l A) c X b
X\cl A
belongs
it is t h e c a s e
ALMOST REALCOMPACT AND
cb- SPACES
-
has the countable i n t e r -
3
This c o n t r a d i c t s t h e p r o p e r t y t h a t
159
s e c t i o n p r o p e r t y . T h e r e f o r e , 5 i s y-cauchy. /
The n e x t r e s u l t i s due t o F r o l i k
(196l.A) and p r o v i d e s a
u s e f u l c h a r a c t e r i z a t i o n of a l m o s t r e a l c o m p a c t n e s s i n t e r m s o f t h e c o l l e c t i o n of a l l c o u n t a b l e open c o v e r i n g s on a s p a c e . The r e s u l t w i l l l a t e r be u t i l i z e d t o e s t a b l i s h t h a t e v e r y Hewitt-Nachbin s p a c e i s a l m o s t r e a l c o m p a c t . 14.5
y
s p a c e and l e t coverinqs
(1) (2)
of If If
Proof.
The f o l l o w i n q s t a t e m e n t s a r e t r u e : is complete, then x i s a l m o s t r e a l c o m p a c t . is c o m p l e t e l y r e q u l a r and a l m o s t r e a l c o m p a c t , y is c o m p l e t e .
X.
y X
3
(1) L e t
-
f o r which
3
-
3
5
3
Let
i s f i x e d by t h e completeness of
be a
containing of
y-Cauchy f a m i l y from which i t
must b e a
Go
3 , and l e t
b e an u l t r a f i l t e r o f open s u b s e t s
by 1 4 . 4 ,
G
Qo
Go
and
are
X
i s assumed t o b e a l m o s t r e a l c o m p a c t , belonging t o
3.
If
p
#
x.
cl G
t h e open s e t
X\cl
Moreover, ( X \ c l G ,
f o r some
GEG,
X\Z]
is
Since
Go
p
#
and hence n o t t o
y-Cauchy s o t h a t
f i n i t e i n t e r s e c t i o n s so t h a t
i s f i x e d whereby
p y
X
there
PEZ c X \ c l X
G.
and
c l ( X \ Z ) , i t follows t h a t
This c o n t r a d i c t s t h e p r o p e r t y t h a t
Hence,
i s contained i n
p
satisfying
Z
Since
belongs t o
i s a c o u n t a b l e open c o v e r of
y.
does n o t belong t o
Q
then
p
By t h e complete r e g u l a r i t y o f
G.
t h e r e f o r e belongs t o
-3
Hence,
there e x i s t s a point
We w i l l e s t a b l i s h t h a t
e x i s t s a z e r o - s e t neighborhood
o t h e r hand,
I t is
y-Cauchy.
h a s t h e countable i n t e r s e c t i o n p r o p e r t y .
p
Q.
G.
t h a t i s g e n e r a t e d by t h e open s u b s e t s o f
X
There-
Q be an u l t r a f i l t e r
y-cauchy f a m i l y , l e t
easy t o v e r i f y t h a t both
X\Z
y.
i s almost realcompact.
fore, X (2)
d e n o t e a n u l t r a f i l t e r o f open s u b s e t s o f
h a s t h e countable i n t e r s e c t i o n p r o p e r t y .
According t o 1 4 . 4 follows t h a t
b e an a r b i t r a r y t o p o l o g i c a l
X
d e n o t e t h e c o l l e c t i o n o f a l l c o u n t a b l e open
then X
.
THEOREM (Froll/k)
X\cl G
Q
belongs t o
G.
On t h e
must b e l o n g t o
i s c l o s e d under
n3
as claimed.
i s complete by d e f i n i t i o n .
I n 1 6 . 1 3 we w i l l p r e s e n t an example o f an a l m o s t r e a l -
160
HEWITT-NACHBIN SPACES AND RELATED SPACES
compact space t h a t f a i l s t o be a Hewitt-Nachbin
space.
Next
spaces i n t e r m s of com-
we w i l l c h a r a c t e r i z e Hewitt-Nachbin
A few d e f i n i t i o n s w i l l be appropri-
p l e t e f a m i l i e s of c o v e r s . ate. 14.6
be an a r b i t r a r y t o p o l o g i c a l space.
X
f F C ( X ) define the s e t
elf)
Let
Let
DEFINITION.
For each
= {Cn(f) : n c m ! ,
and l e t
I t i s easy t o v e r i f y t h a t
i f and only i f
M C X
E
C ( X ) i s bounded on a s e t
i s contained i n a s e t
M
Cn(f) for
The next d e f i n i t i o n i s due t o Froll/k (196lA) and
ncm.
some
f
C n ( f ) = i x : If (x) 1 < n ) . 63 = ( h l f f ) : f E c f x ) ) .
provides a notion of "completeness" f o r c o l l e c t i o n s of continu-
ous real-valued f u n c t i o n s .
This new notion of completeness
w i l l then be r e l a t e d t o t h a t a s s o c i a t e d with a family of open
coverings ( a s given i n 1 4 . 2 ) and u l t i m a t e l y t o Hewitt-Nachbin completeness. 14.7
Let
DEFINITION.
A collection
be an a r b i t r a r y t o p o l o g i c a l space.
X
of continuous r e a l - v a l u e d f u n c t i o n s on
b
3
i s s a i d t o be complete i n case whenever f i l t e r base on zero-set i n
3
X
such t h a t f o r each
on which
THEOREM ( F r o l l / k ) .
14.8
f
Let
is a z e r o - s e t
fc-Q there e x i s t s a
i s bounded,
then
3 5
# #.
be a Tychonoff space and l e t
X
.
Q c C(X) The c o l l e c t i o n b ous f u n c t i o n s i f and only i f -
X
2 complete family
of
continu-
R ( & ) = [ S ( f ) : f c Q ) i s a com-
p l e t e family of open c o v e r s . Proof.
5
let
Suppose t h a t
W(B) i s a complete family of covers and
denote a z e r o - s e t f i l t e r base on
i s bounded on some a s s o c i a t e d s e t of
e s s a r i l y r e l a t e d t o the z e r o - s e t
X
f o r which
f E Q
3 ( t h i s s e t i s n o t nec-
Z(f) i t s e l f ) ,
I t follows
from the remark immediately following D e f i n i t i o n 1 4 . 6 t h a t f o r each
fcQ
3
there e x i s t s a s e t Zf c C n ( f ) .
Cn(f)
in
R(Q)
Therefore, 3
and a s e t
is a R(6)Cauchy family. Since B ( K ) i s complete by assumption, i t follows t h a t fl 7 = n 3 # fl thereby e s t a b l i s h i n g t h e completen e s s of Q . Conversely, suppose t h a t Q c C ( X ) i s a complete family Zf
E
such t h a t
ALMOST REALCOMPACT AND
cb- SPACES
3
o f c o n t i n u o u s r e a l - v a l u e d f u n c t i o n s and l e t Cauchy f a m i l y .
Note t h a t
B ( f ) and
*
Moreover, by 1 4 . 2 f o r e a c h
@ ( a ) t h e r e e x i s t sets C n ( f )
@ ( f )b e l o n g i n g t o
F c C n ( f ) c (x : ' f ( x ) 1
F E ~s a t i s f y i n g
a(&)-
denote a
i s a f i l t e r b a s e t h a t may con-
3
t a i n sets o t h e r than z e r o - s e t s . open c o v e r
16 1
n).
E
Let
d e n o t e t h e z e r o - s e t f i l t e r b a s e c o n s i s t i n g o f a l l zero-
5
s e t s c o n t a i n e d i n t h e f i l t e r g e n e r a t e d by
5.
5*
Then
non-empty because i t c o n t a i n s t h e s e t ( x : I f ( x ) 1 5*
s a t i s f i e s t h e c o n d i t i o n t h a t f o r each
a set i n
5b
since
&
i s complete by a s s u m p t i o n .
n
If
a".
set
2 E
p
on which
i s bounded.
f
fE6 p
Also
n
3* # @
be a p o i n t i n
c l F f o r some F c 3 , t h e n t h e r e e x i s t s a z e r o Z ( X ) such t h a t ppZ, cl F c 2 , and Z E a* contrary E ,!
t o t h e assumption t h a t
p
E
n a*.
Hence, p
n3
belongs t o
B(K) i s a complete f a m i l y o f c o v e r s .
so t h a t
n).
there e x i s t s
Therefore, Let
is
This concludes
the proof. Our n e x t o b j e c t i v e w i l l be t o e s t a b l i s h t h a t complete f a m i l y o f c o n t i n u o u s f u n c t i o n s whenever
is a
C(X) X
is a
Hewitt-Nachbin s p a c e . THEOREM ( F r o l f k )
14.9
.
family
of
c o n t i n u o u s f u n c t i o n s on t h e t o p o l o q i c a l
space
X,
then e v e r y
ble intersection (2)
If
X
t i o n s on
(1) L e t X,
E
and l e t
of
C(X)
is a
continuous f u n c t i o n s .
be a complete f a m i l y of c o n t i n u o u s func-
3
be a
b l e intersection property. hence e v e r y
Z - u l t r a f i l t e r w i t h t h e counta-
property is fixed.
i s a Hewitt-Nachbin s p a c e , then
complete f a m i l y Proof.
I f t h e r e e x i s t s a complete
(1)
Z - u l t r a f i l t e r w i t h t h e counta-
By 6 . 1 8 ( 1 ) e v e r y
f
E
fcE, i s bounded on some z e r o - s e t i n
C(X),
3.
and
Since
Q
n
5 # 6 by d e f i n i t i o n . Z - f i l t e r b a s e on X such t h a t each f E C ( X ) i s bounded on some z e r o - s e t o f 3. L e t 1( d e n o t e a Z - u l t r a f i l t e r c o n t a i n i n g 5 . By 6 . 1 8 ( 2 ) , I r h a s t h e c o u n t a b l e i s complete, (2)
Let
5
be a
i n t e r s e c t i o n p r o p e r t y and hence i s f i x e d by t h e Hewitt-Nachbin completeness o f
X.
p l e t e by d e f i n i t i o n .
Hence,
n 3# @
so that
C ( x ) i s com-
16 2
HEWITT-NACHBIN
SPACES AND RELATED SPACES
W e w i l l now summarize t h e p r e v i o u s t h r e e r e s u l t s due t o
Frol
R e c a l l t h e d e f i n i t i o n o f t h e c o l l e c t i o n of 63 = [ B ( f ) : f c C ( X )
open c o v e r i n g s
A
THEOREM ( F r o l f k ) ,
14.10
given i n 14.6.
Tychonoff space
Nachbin s p a c e i f and o n l y i f t h e c o l l e c t i o n
i n 14.6, space
X
63,
is a H e w i t t -
as d e f i n e d
i s a complete c o l l e c t i o n of open c o v e r i n q s Combining t h e r e s u l t s o f
Proof.
X
of
X.
(1) and ( 2 ) i n 1 4 . 9 , t h e
i s Hewitt-Nachbin complete i f and o n l y i f
a complete f a m i l y of c o n t i n u o u s f u n c t i o n s .
C(X)
is
The r e s u l t i s now
immediate from 1 4 . 8 .
I n t h e n e x t r e s u l t , which i s a l s o due t o Froll/k (1961A), t h e c o n c e p t o f Hewitt-Nachbin c o m p l e t e n e s s i s r e l a t e d t o t h a t of a l m o s t r e a l c o m p a c t n e s s . 14.11
then Proof.
(Frollk).
COROLLARY
X
If
X
a Hewitt-Nachbin s p a c e ,
i s almost realcompact.
Let
y
d e n o t e t h e c o l l e c t i o n of a l l c o u n t a b l e open
63 C y , where 63 i s d e f i n e d a s i n 1 4 . 6 . 3 i s a y-Cauchy f a m i l y . Then f o r e v e r y 2ky t h e r e e x i s t s e t s AclI and F E Z w i t h F c A . I n part i c u l a r , such i s t h e c a s e f o r e v e r y s e t B ( f ) i n 63. S i n c e 63
coverings of
X.
Then
Now, suppose t h a t
i s complete by 14.10, i t f o l l o w s t h a t i s complete s o t h a t
X
n 5 # 8.
Therefore,
y
i s a l m o s t r e a l c o m p a c t by 1 4 . 5 ( 1 ) .
I n (1961A) F r o l l k s t a t e s t h a t h e was u n a b l e t o produce an example o f an a l m o s t r e a l c o m p a c t s p a c e t h a t f a i l s t o b e a Hewitt-Nachbin s p a c e .
W e w i l l p r o v i d e such a n example i n
1 6 . 1 2 o f t h e n e x t c h a p t e r based on an example due t o S . Mro/wka
i n h i s 1958D p a p e r (see a l s o 1 6 . 4 ) and u t i l i z i n g one o f t h e
r e s u l t s on c o n t i n u o u s mappings ( 1 6 . 1 1 ( 1 ) ) t h a t w i l l b e e s t a b lished a t that t i m e . / I n h i s 1963 paper F r o l i k e s t a b l i s h e s a v a r i e t y o f i n t e r e s t i n g p r o p e r t i e s of a l m o s t r e a l c o m p a c t s p a c e s .
For example,
a r b i t r a r y i n t e r s e c t i o n s of almost realcompact spaces a r e a l m o s t realcompact: a r b i t r a r y t o p o l o g i c a l p r o d u c t s of a l m o s t
ALMOST REALCOMPACT AND
16 3
cb- SPACES
realcompact spaces a r e almost realcompact; and c l o s e d subspaces of r e g u l a r almost realcompact spaces a r e almost r e a l compact.
I n a d d i t i o n , C . Liu and G . S t r e c k e r (1972) p r e s e n t
a c o n s t r u c t i o n f o r an almost r e a l c o m p a c t i f i c a t i o n of a Hausd o r f f space
which i s contained i n t h e Katztov
X
H-closed
e x t e n s i o n , and which h a s an e x t e n s i o n p r o p e r t y s i m i l a r t o t h a t of the Hewitt-Nachbin completion
vX.
Next we t u r n our
a t t e n t i o n t o a n o t h e r important c l a s s of t o p o l o g i c a l s p a c e s . The study of "cb-spaces" was i n i t i a t e d i n t h e 1959 n o t e of J . G . Horne.
Since t h a t time both J . Mack ( 1 9 6 5 ) , and
Mack with D . Johnson (1967), have made v a l u a b l e c o n t r i b u t i o n s toward t h e understanding of t h e s e s p a c e s .
The
cb-spaces a r e
a c l a s s of countably paracompact spaces t h a t emphasize t h e r o l e of r e a l - v a l u e d continuous f u n c t i o n s .
Closely a s s o c i a t e d
with t h e s e spaces i s t h e l a r g e r c l a s s of "weak
cb-spaces,"
although both c l a s s e s t u r n o u t t o be i d e n t i c a l i n the presence of normality (Horne, 1959 and Mack, 1 9 6 5 ) .
We w i l l now d e f i n e
and study t h e spaces i n q u e s t i o n . 14.12
c a l space
X
A real-valued function
DEFINITION. X
f
on a topologi-
i s s a i d t o be l o c a l l y bounded i f each p o i n t i n
has a neighborhood on which the f u n c t i o n i s bounded.
function
i s s a i d t o be lower semi-continuous
f
upper semi-continuous) i n c a s e the s e t ( x : f ( x ) tively,
(x
:
f(x)
<
b ] ) i s open f o r every
bcIR.
The
(respectively,
>
b ) (respec-
The space
i s s a i d t o be a
cb-space i f f o r each l o c a l l y bounded r e a l -
valued f u n c t i o n
h
on
X
X
t h e r e e x i s t s a continuous f u n c t i o n
g E C ( X ) such t h a t g 2 h . The space X i s s a i d t o be a weak cb-space i f f o r each l o c a l l y bounded, lower semi-continuous function g
E
h
c(x)
on
X
such t h a t
t h e r e e x i s t s a continuous f u n c t i o n g
2
h.
I t i s c l e a r from t h e d e f i n i t i o n t h a t every
a weak
cb-space.
cb-space i s
Moreover, the following r e s u l t s a r e known
t o be t r u e and although t h e p r o o f s a r e omitted h e r e , an approp r i a t e r e f e r e n c e i s c i t e d f o r each r e s u l t .
164
14.13
HEWITT-NACHBIN SPACES AND RELATED SPACES
The
THEOREM.
followinq statements a r e t r u e .
(Horne-Mack, 1965) .
cb- s p a c e i s c o u n t a b l y
Every
paracompact and e v e r y normal and c o u n t a b l y paracom-
pact space i s a (Mack, 1 9 6 5 ) .
cb-space.
A
c o u n t a b l y compact s p a c e i s a
cb-
space. (Mack, 1 9 6 5 ) .
-a
A c l o s e d subspace o f a
c b - s p a c e is
cb-space.
(Mack, 1965) .
A
c o m p l e t e l y r e q u l a r pseudocompact
s p a c e i s c o u n t a b l y paracompact i f and o n l y i f i t i s
-a
( e q u i v a l e n t l y , i f and o n l y i f i t i s
cb-space
c o u n t a b l y compact) (Mack, 1 9 6 5 ) .
---i f it i s both space. (Mack, 1965)
.
A space i s a
c b - s p a c e i f and o n l y cb-
c o u n t a b l y paracompact and a weak
. The t o p o l o q i c a l
product
of fi
cb-
s p a c e and a l o c a l l y compact, paracompact Hausdorff space i s a
cb-space.
(The example g i v e n a t t h e
end of S e c t i o n 3 i n Mack’s 1965 p a p e r s u f f i c e s t o show t h a t a l o c a l l y compact and c o u n t a b l y paracompace s p a c e need n o t b e a (Mack, 1 9 6 5 ) .
-a
An
cb-space.)
e x t r e m a l l y disconnected space i s
cb-space i f and o n l y i f i t i s c o u n t a b l y p a r a -
compact. (Mack- Johnson, 1967)
-a -weak
. The t o p o l o s i c a l
product
of
cb-space and a l o c a l l y compact, paracompact
Hausdorff s p a c e i s a weak (Mack- Johnson, 1967) .
&
compact s p a c e i s a weak (Mack-Johnson, 1 9 6 7 ) .
cb-space. c o m p l e t e l y r e q u l a r , pseudocb-space.
The t o p o l o q i c a l
product
of
any c o l l e c t i o n of s e p a r a b l e , complete metric s p a c e s is a 14.14
EXAMPLE. Let
n
weak
cb-space.
A weak
cb-space t h a t f a i l s t o be a
cb-space.
d e n o t e t h e f i r s t u n c o u n t a b l e o r d i n a l (see Ex-
I N * = IN U [ w ) d e n o t e t h e o n e - p o i n t comIN. The Tychonoff p l a n k i s d e f i n e d a s t h e
ample 7 . 1 5 ) and l e t p a c t i f i c a t i o n of
16 5
cb- SPACES
ALMOST REALCOMPACT AND
space
I t i s w e l l known t h a t
i s pseudocompact b u t n o t c o u n t a b l y
T
compact ( s e e Gillman and J e r i s o n , 8 . 2 0 ) .
Therefore, T
However, T T
i s a weak
fails
cb-space by 1 4 . 1 3 ( 4 ) .
t o be e i t h e r countably paracompact o r a
cb-space by 1 4 . 1 3 ( 9 )
.
Note a l s o t h a t
f a i l s t o be Hewitt-Nachbin complete s i n c e i t i s pseudocom-
p a c t , b u t n o t compact.
(For f u r t h e r information concerning
t h e Tychonoff plank s e e Problem 8J of Gillman and J e r i s o n . ) The p r e v i o u s l y s t a t e d r e s u l t s i n d i c a t e t h e r e l a t i v e p o s i t i o n of t h e
cb-spaces i n t h e c l a s s of
cb- and weak
countably paracompact s p a c e s . cb- and weak
Useful c h a r a c t e r i z a t i o n s of t h e
cb-spaces have been e s t a b l i s h e d by Mack (1965)
and Johnson (1967) which a r e a l s o i n t e r e s t i n g i n comparison with t h e c h a r a c t e r i z a t i o n of normal and countably paracompact spaces given i n 8 . 1 4 .
Moreover, i t w i l l be e v i d e n t from t h e s e
r e s u l t s t h a t t h e normal and c o u n t a b l y paracompact spaces a r e p r e c i s e l y t h e normal 14.15
THEOREM. X
cb- s p a c e s .
(1)
is a
2
(Mack).
a r b i t r a r y t o p o l o q i c a l space
cb-space i f and o n l y i f f o r e v e r y d e c r e a s -
sequence [ F n : n c m } of c l o s e d s u b s e t s w i t h empty -
t i o n such t h a t ---
Fn
C
(Mack and J o h n s o n ) . X
i s a weak
X
Zn
An
X
sequence ( Z n
w i t h empty i n t e r s e c -
€or every
nc I N .
a r b i t r a r y t o p o l o g i c a l space
cb-space i f and only i f f o r e v e r y -&c
c r e a s i n q sequence ( F n : nem s e t s of --
X
i n t e r s e c t i o n t h e r e e x i s t s a sequence
( Z n : n c m ] of z e r o - s e t s of
(2)
of
1 of
r e q u l a r c l o s e d sub-
with empty i n t e r s e c t i o n t h e r e e x i s t s a : n e m ) of z e r o - s e t s of
i n t e r s e c t i o n such t h a t
Fn
C
Zn
X
w i t h empty
f o r every
ncm.
Comparing 8 . 1 4 t o 1 4 . 1 5 (l), i t i s e a s i l y seen t h a t i n t h e presence of normality t h e c o n d i t i o n t h a t a space be countab l y paracompact i s e q u i v a l e n t t o i t s b e i n g a
cb-space.
The
n e x t r e s u l t i s due t o N . Dykes (1969) and g e n e r a l i z e s F r o l f k ' s
166
SPACES AND RELATED SPACES
HEWITT-NACHBIN
r e s u l t t h a t every normal, countably paracompact and almost realcompact space i s Hewitt-Nachbin complete. 14.16 Proof.
cb-space, then
3
Let
@ = [U
@'
Let
Z - u l t r a f i l t e r on
0.
=
For i f
p t x , then
X,
G.
p
F i r s t observe
f o r some
X\Z
E
Q'
lection (Ai
: i c l N ) of
i s almost realcompact.
which i m p l i e s t h a t Therefore,
X
n ( c l Ai
satisfying Set
n
Tn =
: iElN) =
(Ai
:
1
i
Then I T n : nc3N ) i s a d e c r e a s i n g sequence of open sets
n
such t h a t weak
( c l Tn : ntlN ) =
a.
Moreover, s i n c e
is a
X
cb-space by 1 4 . 1 5 ( 2 ) t h e r e e x i s t s a sequence ( Z n
of z e r o - s e t s of
f o r every
3
belongs t o
Thus, Z n
Zn.
n
and
nElN
meets every member of
Now, c l Tn t r u e of
such t h a t t h e r e g u l a r c l o s e d s e t
X
c l Tn c Z n
satisfies
(Zn
i s a Hewitt-Nachbin
:
ncm)
c l Tn
: nEN) =
a.
so t h a t t h e same h o l d s 3 f o r every n c m and 3
f a i l s t o have the c o u n t a b l e i n t e r s e c t i o n p r o p e r t y . X
Zt3.
with
U
However, p j! c l [ X \ c l U] ,
@ I .
z c u).
Next observe t h a t t h e r e e x i s t s a c o u n t a b l e subcol-
because
n).
Set
X.
t h e r e e x i s t s an open s e t
Thus, Z c X \ c l U
belongs t o
U
n TI,
space.
: U i s open and t h e r e e x i s t s z c 3 with
C X
p t U c c l U c X\Z. p j!
Tychonoff almost realcom-
i s a Hewitt-Nachbin
be an open u l t r a f i l t e r c o n t a i n i n g
n 3'
that
is 2
X X
be a f r e e
By t h e r e g u l a r i t y of X\cl
If
THEOREM ( D y k e s ) .
p a c t weak
Therefore,
space completing t h e p r o o f .
The n e x t r e s u l t i s found i n t h e 1967 paper of Mack and Johnson.
I t r e l a t e s t h e weak
space
X
t o i t s Hewitt-Nachbin completion
14.17
THEOREM (Mack and Johnson)
c b - p r o p e r t y f o r a Tychonoff
i s a weak
.
If
uX.
5 Tychonoff weak
X
cb-space.
cb-space, then
EX
Proof.
be a l o c a l l y bounded lower semi-continuous
Let
h
f u n c t i o n on
vX.
Then t h e r e s t r i c t i o n
ed and lower semi-continuous on function
f
e x t e n s i o n of
E
C ( X ) such t h a t
Then
f
X.
2 hlX. u f -h
h(X
Thus,
i s l o c a l l y bound-
there e x i s t s a
Let
fv
denote t h e
i s an upper semi-continu-
f
to
uX.
ous f u n c t i o n on
uX
t h a t i s non-negative
on t h e dense subspace
ALMOST REALCOMPACT AND
Hence,
X.
fId
2
h
16 7
cb- SPACES
completing t h e p r o o f .
The example o f t h e o r d i n a l s p a c e [O,Q) p r e s e n t e d i n 7 . 1 5 i s s u f f i c i e n t t o e s t a b l i s h t h a t t h e normal and c o u n t a b l y paracompact s p a c e s ( i n f a c t , even c o u n t a b l y compact!) a r e n o t n e c e s s a r i l y Hewitt-Nachbin
I n f a c t , s i n c e [0, aZ)
complete.
normal and c o u n t a b l y paracompact i t i s a according t o 14.16,
[O,
n)
cb-space.
is
Therefore,
cannot be almost realcompact because
i t f a i l s t o b e a Hewitt-Nachbin s p a c e . The f o l l o w i n g c h a r t summarizes t h e v a r i o u s r e l a t i o n s h i p s t h a t have been e s t a b l i s h e d i n t h i s c h a p t e r f o r Hausdorff topol o g i c a l spaces.
A l l s p a c e s a r e assumed t o be a t l e a s t r e g u l a r Hausdorff
COMPLETELY REGULAR P SEUDOCOMPA CT
+
EXTREMALLY D ISCONNECTED
NORMAL, COUNTABLY PARACOMPACT
ALMOST REALCOMPACT, WEAK cb- SPACE
cb- SPACE
WEAK
cb- SPACE
COUNTABLY PARACOMPACT S PACE
+
WEAK
cb-
(Tychohof f s p a c e s )
I a
--t
b
every
a
HEWITT- NACHBIN SPACE
space i s a
b
space.
168
HEWITT-NACHBIN
SPACES AND RELATED SPACES
Before c l o s i n g t h i s c h a p t e r i t i s worthwhile t o c o n s i d e r b r i e f l y s e v e r a l c l a s s e s o f t o p o l o g i c a l s p a c e s t h a t have rec e i v e d a t t e n t i o n r e c e n t l y and which a r e a s s o c i a t e d w i t h t h e s t u d y of Hewitt-Nachbin s p a c e s . A weakening o f t h e n o t i o n o f paracompactness h a s been
d e f i n e d by D . Burke i n h i s 1969 p a p e r . c a l space has a
H e d e f i n e s a topologi-
X
t o b e subparacompact i f e v e r y open c o v e r o f
X
o - l o c a l l y f i n i t e closed refinement.
I t is clear that
e v e r y r e q u l a r paracompact s p a c e i s subparacompact.
Moreover,
e v e r y c o l l e c t i o n w i s e normal subparacompact s p a c e i s paracom-
pact. I f w e l e t h d e n o t e t h e c l a s s of Tychonoff s p a c e s which a r e e i t h e r subparacompact o r metacornpact (where a s p a c e X i s metacompact i f e v e r y open c o v e r of
X
h a s a p o i n t f i n i t e open
r e f i n e m e n t ) , then P . Zenor e s t a b l i s h e s t h e f o l l o w i n g r e s u l t
i n h i s 1972 p a p e r . 14.18
W e omit t h e n o n - t r i v i a l
THEOREM ( Z e n o r ) .
proof.
A normal Hausdorff s p a c e
X
is a
Hewitt-Nachbin space i f and o n l y i f t h e c a r d i n a l i t y of each d i s c r e t e s u b s e t of
--
X
is
nonmeasurable
and
X
can b e embed-
ded a s 2 c l o s e d subspace i n t h e p r o d u c t of a c o l l e c t i o n ~
members
of
of
h.
The c l a s s o f
P-spaces due t o K . Morita (1962) is im-
p o r t a n t i n s t u d y i n g t h o s e s p a c e s whose p r o d u c t s w i t h metric s p a c e s a r e normal. t i o n of t h e
( W e a r e o m i t t i n g t h e complicated d e f i n i -
P-space h e r e and r e f e r t h e i n t e r e s t e d r e a d e r t o
Definition V I . 5 ,
page 250, of t h e 1968 Nagata t e x t . )
known t h a t e v e r y c o u n t a b l y compact s p a c e i s a
It is
P-space
(Nagata, page 250) and e v e r y normal paracompact (Nagata, page 2 5 1 ) . almost realcompact
P-space i s c o u n t a b l y T h e r e f o r e , e v e r y normal and
P-space i s a Hewitt-Nachbin s p a c e .
Another i n t e r e s t i n g c l a s s o f t o p o l o g i c a l s p a c e s , a l s o due t o M o r i t a , i s t h e c l a s s of
M-spaces; t h o s e s p a c e s t h a t
can be c o n t i n u o u s l y mapped o n t o a metric s p a c e v i a a map t h a t i s a l s o c l o s e d and " f i b e r - c o u n t a b l y compact'' ( f o r a d e f i n i t i o n
of t h i s mapping see 1 5 . 2 ( 1 ) i n t h e n e x t c h a p t e r ) .
Every
m e t r i z a b l e s p a c e and e v e r y c o u n t a b l y compact s p a c e i s a n
M-
cb- SPACES
ALMOST REALCOMPACT AND
M-space i s a
s p a c e (Nagata, page 2 9 6 ) , and moreover e v e r y s p a c e (Nagata, page 2 9 6 ) . that
an
169
M-space need n o t be Hewitt-Nachbin c o m p l e t e .
over, t h e Sorgenfrey space
E
w
M-space.
More-
p r e s e n t e d i n 7 . 1 2 p r o v i d e s an
example o f a Hewitt-Nachbin s p a c e t h a t i s a to be an -
P-
Note t h a t example 7.15 d e m o n s t r a t e s
The r e a s o n t h a t
E
Y,
P-space y e t f a i l s
is a
P-space can b e
shown d i r e c t l y from t h e d e f i n i t i o n (see Example V I I . 4 , page 299, of N a g a t a ' s t e x t f o r t h e d e t a i l s ) .
The f a c t t h a t i t a l s o
M-space i s a consequence o f t h e r e s u l t t h a t
f a i l s t o be an
M- spaces produce
c o u n t a b l e p r o d u c t s of paracompact Hausdorff paracompact Hausdorff that
E
dorf f .
is also a
w
M-spaces
n
p
in
( s t ( p , l l n ) : nelN ),
X
p s p a c e of A . f o r which t h e r e
X
: n c m ] of open c o v e r s of
t h a t for each p o i n t
sets,
M-space i s t h e
These are t h e s p a c e s
Arhangelskii (1963).
(an
Observe
c b - s p a c e because i t i s paracompact Haus-
A close r e l a t i v e t o the
i s a sequence
(Nagata, page 2 9 9 ) .
X
in
PX
t h e i n t e r s e c t i o n of the s t a r
is contained i n
X.
For t h e p a r a -
compact Haukdorff s p a c e s , t h e c o n d i t i o n f o r b e i n g a
i s e q u i v a l e n t t o t h a t o f b e i n g an
M-space.
Hence, E
example o f a Hewitt-Nachbin s p a c e t h a t f a i l s t o b e a Moreover, e v e r y m e t r i c s p a c e i s a
is a
k-space
such
pspace,
pspace
i s an
CL
pspace.
and e v e r y
p-space
( A r h a n g e l s k i i ( 1 9 6 3 ) , Theorem 7 and C o r o l l a r y 9 ) .
F i n a l l y , w e mention t h e n o t i o n o f a
q - s p a c e due t o E .
S i n c e t h e s e s p a c e s w i l l come t o p l a y a p a r t
Michael ( 1 9 6 4 ) .
i n t h e s t u d y o f Hewitt-Nachbin c o m p l e t e n e s s and c o n t i n u o u s mappings t o b e i n v e s t i g a t e d i n t h e n e x t c h a p t e r , w e w i l l prov i d e a formal d e f i n i t i o n h e r e . 14.19
DEFINITION.
A point
p
sequence I N i belongs t o
of
X
Let
X
be a n a r b i t r a r y t o p o l o g i c a l s p a c e .
i s s a i d t o be a
q-point i f there e x i s t s a
: i c I N ) o f neighborhoods o f
Ni
and t h e
xi
p
such t h a t i f
are a l l distinct,
xi
then t h e
sequence ( x : i c m ) h a s an a c c u m u l a t i o n p o i n t i n X. I f i e v e r y p o i n t i n X i s a q - p o i n t , then X i s c a l l e d a qspace. I t i s c l e a r t h a t every f i r s t c o u n t a b l e space i s a
q-
170
HEWITT-NACHBIN SPACES AND RELATED SPACES
space.
More g e n e r a l l y , every
also a
q-space
[O,hl)
p s p a c e and every
(Michael, 1 9 6 4 ) .
M-space i s
Note t h a t the o r d i n a l space
of 7 . 1 5 i s an example of a f i r s t countable (hence
q-)
space t h a t f a i l s t o be a Hewitt-Nachbin space. However, the space E of 7 . 1 2 i s an example of a q-space t h a t i s a l s o k
Hewitt-Nachbin complete ( i n f a c t , any m e t r i c space o f nonmeasurable c a r d i n a l would provide such an example, b u t observe
E
that
k
countable)
.
i s n o t m e t r i z a b l e because it f a i l s t o be second
Michael introduced t h e n o t i o n of a
q-space i n h i s 1964
paper i n o r d e r t o e s t a b l i s h t h a t every continuous and closed s u r j e c t i o n from a paracompact Hausdorff space onto a f i r s t countable space s a t i s f i e s t h e property t h a t t h e boundary of t h e i n v e r s e image of each p o i n t i n t h e range space i s compact. I n t h e next c h a p t e r a s i m i l a r r e s u l t due t o N . Dykes (1969) w i l l be e s t a b l i s h e d except t h a t t h e domain space w i l l be given
t o be a Hewitt-Nachbin
space and t h e range a
q-space.
The following c h a r t provides a summary o f t h e s e l a s t
results.
A l l spaces a r e assumed t o be a t l e a s t r e g u l a r Hausdorff c
(paracom-
L
a
+
,
b
&
every
E
a
-
,
space i s a
b
space.
Chapter 4
AND
HEWITT-NACHBIN COMPLETENESS
A topological property
P
CONTINUOUS MAPPINGS
i s s a i d t o be i n v a r i a n t ( r e -
s p e c t i v e l y , i n v e r s e i n v a r i a n t ) under a mapping age ( r e s p e c t i v e l y , i n v e r s e image) under property
a l s o has property
P
P.
f
f
i f t h e im-
of a s p a c e w i t h
The purpose of t h i s chap-
t e r i s t o i n v e s t i g a t e t h e i n v a r i a n c e and i n v e r s e i n v a r i a n c e o f Hewitt-Nachbin completeness under v a r i o u s c l a s s e s of c o n t i n u ous mappings.
Unlike t h e p r o p e r t y o f compactness,
the continu-
ous image of a Hewitt-Nachbin s p a c e need n o t b e Hewitt-Nachbin complete.
I n f a c t , an example w i l l b e provided showing t h a t
such i s n o t t h e c a s e even i f t h e mapping happens t o b e a p e r f e c t mapping ( a l s o c a l l e d a " p r o p e r mapping" o r a " f i t t i n g mapping" by M . Henriksen and J . I s b e l l
(1958)).
However, Z.
Froll/k (196lA) h a s shown t h a t Hewitt-Nachbin c o m p l e t e n e s s i s i n v a r i a n t and i n v e r s e i n v a r i a n t under a p e r f e c t mapping whene v e r t h e domain i s a l s o normal and c o u n t a b l y paracompact. A s w e have a l r e a d y s e e n i n t h e development o f p r e c e d i n g
c h a p t e r s , e v e r y compact s p a c e i s paracompact and e v e r y paracomp a c t Hausdorff s p a c e of nonmeasurable c a r d i n a l i s Hewitt-Nachb i n complete.
I f a p e r f e c t map i s d e f i n e d a s a c o n t i n u o u s
c l o s e d s u r j e c t i o n f o r which t h e i n v e r s e images of p o i n t s a r e compact (and t h e r e f o r e
C-embedded i n t h e c a s e t h a t t h e domain
i s a Tychonoff s p a c e by 4 . 8 ( 3 ) ) then i t i s w e l l known t h a t compactness i s b o t h i n v a r i a n t and i n v e r s e i n v a r i a n t under p e r f e c t mappings.
Moreover, Henriksen and I s b e l l (1958) have
shown t h a t paracompactness i s a l s o b o t h i n v a r i a n t and i n v e r s e i n v a r i a n t under a p e r f e c t map whenever t h e domain s p a c e i s Tychonoff.
I n h i s 1966 p a p e r H . L. S h a p i r o h a s d e f i n e d t h e
n o t i o n of a p a r a p r o p e r map (which w e s h a l l l a t e r r e f e r t o a s a " p a r a p e r f e c t mapft) a s a c o n t i n u o u s c l o s e d map f o r which t h e i n v e r s e images o f p o i n t s a r e paracompact and
P-embedded.
(For p u r p o s e s of c l a r i t y , w e p o i n t o u t t h a t S h a p i r o d e f i n e s t h e p r o p e r t y of paracompactness f o r r e g u l a r s p a c e s , b u t d o e s not include the
T1-separation
s e t s of a Tychonoff s p a c e a r e
property. )
Because compact sub-
P-embedded t h e r e i n , i t f o l l o w s
t h a t f o r Tychonoff s p a c e s e v e r y p e r f e c t map i s
-a
paraproper
COMPLETENESS AND CONTINUOUS MAPPINGS
172
surjection.
However, i t i s c l e a r t h a t a p a r a p r o p e r map need
n o t be p e r f e c t by c o n s i d e r i n g a map from a paracompact, noncompact s p a c e o n t o a o n e - p o i n t s p a c e . paraproper, b u t not p e r f e c t .
Such a map i s indeed
The main r e s u l t of S h a p i r o ' s
paper i s t h a t paracompactness i s b o t h i n v a r i a n t
and
inverse
i n v a r i a n t under a p a r a p r o p e r mappinq whenever t h e domain s p a c e
is r e q u l a r . O n e might wonder i f i t would b e p o s s i b l e t o d e f i n e a
n o t i o n of a " r e a l p r o p e r " o r " r e a l p e r f e c t " map s u b j e c t t o t h e f o l l o w i n g two c o n d i t i o n s :
e v e r y p a r a p r o p e r map must b e r e a l -
p r o p e r , and t h e p r o p e r t y of Hewitt-Nachbin c o m p l e t e n e s s m u s t b e i n v a r i a n t and i n v e r s e i n v a r i a n t under any suc'.? r e a l p r o p e r map.
S c h e m a t i c a l l y w e would t h e n have t h e f o l l o w i n g :
+-I
PARACOMPACT
COMPACT
f
HEW1 TT- NACHBIN
c 1
4 I
REALPROPER
t
where t h e downward arrow d e n o t e s i n v a r i a n c e , and t h e upward arrow denotes inverse invariance, o f the t o p o l o g i c a l
4
property indicated.
I t t u r n s o u t t h a t such a d e f i n i t i o n f o r
" r e a l p r o p e r maps" i s n o t p o s s i b l e . c l a s s of maps d i d i n f a c t e x i s t .
For suppose t h a t s u c h a Then, a c c o r d i n g t o o u r f i r s t
c o n d i t i o n , e v e r y p e r f e c t map would b e l o n g t o t h a t c l a s s .
How-
e v e r , i n 1 6 . 4 an example i s g i v e n f o r which t h e p e r f e c t image of a Hewitt-Nachbin s p a c e f a i l s t o be Hewitt-Nachbin c o m p l e t e . Hence, t h e second c o n d i t i o n s t a t e d above i s v i o l a t e d . D e s p i t e t h e f a c t t h a t one c a n n o t s u p p l y a c l a s s of mapp i n g s s u b j e c t t o t h e two c o n d i t i o n s s p e c i f i e d above, t h e r e a r e n e v e r t h e l e s s many i n t e r e s t i n g and u s e f u l c l a s s e s o f mappings under which t h e p r o p e r t y o f Hewitt-Nachbin c o m p l e t e n e s s i s i n variant o r inverse invariant.
I t i s t h e i n t e n t of t h i s chap-
t e r t o i n v e s t i g a t e t h e s e mappings and t h e i r r e l a t i o n s h i p t o Hewitt-Nachbin s p a c e s .
The c h a p t e r i t s e l f i s s u b d i v i d e d i n t o
five sections.
The f i r s t o f t h e s e d e f i n e s the v a r i o u s c l a s s e s of mappings under i n v e s t i g a t i o n and e s t a b l i s h e s t h e i r i n t e r relationships.
The n e x t t h r e e s e c t i o n s d e a l w i t h t h e e f f e c t
o f t h e s e c l a s s e s of mappings on t h e p r o p e r t y of Hewitt-Nachbin
173
SOME CLASSES OF MAPPINGS
completeness.
These s e c t i o n s a r e a r r a n g e d i n such a way a s t o
proceed from t h e s t r o n g e s t c l a s s o f mappings t o t h e w e a k e s t
I n s o d o i n g t h e r e a d e r w i l l become aware o f t h e i n -
class.
c r e a s i n g l y s t r o n g e r c o n d i t i o n s t h a t need be imposed on t h e domain and/or r a n g e s p a c e s i n o r d e r t o p r e s e r v e t h e i n v a r i a n c e o r i n v e r s e i n v a r i a n c e o f Hewitt-Nachbin c o m p l e t e n e s s .
The
f i n a l s e c t i o n i n v e s t i g a t e s t h e p r e s e r v a t i o n o f Hewitt-Nachbin completeness i n t h e c o n t e x t o f t h e
i n C h a p t e r 1.
E-compact s p a c e s s t u d i e d
W e a l s o p r o v i d e a c h a r t summarizing t h e r e s u l t s
o f t h i s c h a p t e r f o r p u r p o s e s of a q u i c k and e a s y r e f e r e n c e t o the r e s u l t s obtained. Some C l a s s e s of Mappinqs
Section 15:
I n t h i s s e c t i o n w e w i l l d e f i n e and i n v e s t i g a t e s e v e r a l of t h e c l a s s e s of mappings t o b e c o n s i d e r e d i n c o n n e c t i o n w i t h t h e p r e s e r v a t i o n o f Hewitt-Nachbin c o m p l e t e n e s s . The f o l l o w i n g c o n c e p t w i l l b e needed i n o u r s t u d y . 15.1
A non-empty
DEFINITION.
subset
s a i d t o b e r e l a t i v e l y pseudocompact function tion
f IS
f
E
C(X)
of a s p a c e
S
in
X
X
is
i f every continuous
s a t i s f i e s the c o n d i t i o n t h a t t h e restric-
i s bounded.
I t i s immediate t h a t e v e r y pseudocompact s u b s p a c e , and
hence e v e r y c o u n t a b l y compact subspace, compact.
i s r e l a t i v e l y pseudo-
Moreover, by c o n s i d e r i n g a pseudocompact subspace
t h a t f a i l s t o be compact, i t i s e v i d e n t from 7 . 1 4 t h a t a r e l a t i v e l y pseudocompact subspace need n o t be Hewitt-Nachbin complete. The f o l l o w i n g d e f i n i t i o n s p e c i f i e s most o f t h e c l a s s e s of mappings t h a t w i l l b e under i n v e s t i g a t i o n . 15.2
space
Let
DEFINITION. X
f
i n t o t h e space
(1) The mapping
tively, pact,
b e a mapping from t h e t o p o l o g i c a l Y.
f
i s s a i d t o b e fiber-compact
fiber-pseudocompact,
(respec-
fiber-countably
:om-
f i b e r - r e l a t i v e l v pseudocompact, f i b e r - p a r a -
compact, o r f i b e r - H e w i t t - N a c h b i n ) c o n t i n u o u s and t h e f i b e r
i n case
f
is
f - l ( y ) i s compact (respec-
174
COMPLETENESS AND CONTINUOUS MAPPINGS
t i v e l y , pseudocompact, c o u n t a b l y compact, r e l a t i v e l y pseudocompact, paracompact, o r Hewitt-Nachbin
i n t h e range o f f . The mapping f i s s a i d t o be z e r o - s e t p r e s e r v i n g i n c a s e f i s c o n t i n u o u s and f o r e v e r y z e r o - s e t Z i n X t h e image f ( Z ) i s a z e r o - s e t i n Y. The mapping f i s s a i d t o be z - c l o s e d ( o r a zmap) i n c a s e f i s c o n t i n u o u s and f o r e v e r y zeros e t Z i n X t h e image f ( Z ) i s c l o s e d i n Y. The mapping f i s s a i d t o b e z - o p e n i n c a s e f complete) f o r e v e r y p o i n t
(2)
(3)
(4)
y
i s c o n t i n u o u s and f o r e v e r y c o z e r o - s e t neighborhood o f a z e r o - s e t 2 i n X t h e image f ( H ) i s a
H
cl f ( Z ) i n
neighborhood of (5)
The mapping
Y.
i s s a i d t o be p e r f e c t i n c a s e i t i s
f
a f i b e r - c o m p a c t and c l o s e d s u r j e c t i o n . (6)
The mapping f i s s a i d t o b e p a r a p e r f e c t i n c a s e i t i s a f i b e r - p a r a c o m p a c t and c l o s e d s u r j e c t i o n such t h a t t h e f i b e r every
f-I(y) is
P-embedded f o r
Y.
in
y
S i n c e e v e r y z e r o - s e t i s a c l o s e d s e t w e have immediately t h e f i r s t two s t a t e m e n t s of t h e f o l l o w i n g r e s u l t .
Let
bs
2 mappinq from t h e t o p o l o q i c a l space X i n t o t h e space Y . (1) If f & a c o n t i n u o u s c l o s e d mappinq, t h e n i t i s 15.3
THEOREM.
f
z- c l o s e d .
(2) (3)
If If
f
i s z e r o - s e t p r e s e r v i n q , then i t i s
X
is 2
then i t i s --Proof
Tychonoff s p a c e and
f
2
z-closed.
z--,
open.
The p r o o f s o f s t a t e m e n t s (1) and (2) a r e t r i v i a l so
we establish ( 3 ) .
Thus l e t
xsG.
y = f ( x ) f o r some p o i n t e x i s t s a cozero-set
H
b e an open s e t i n Since
f ( H ) i s a neighborhood of
Z
x
X
X
and l e t
i s Tychonoff t h e r e
Then by 3 . 6 ( 3 ) Z ( X ) such t h a t x E Z C H , and
such t h a t
t h e r e e x i s t s a zercj-set p l e t e s t h e argument.
G
E H C G.
in c l f ( Z ) by a s s u m p t i o n .
T h i s com-
SOME CLASSES OF MAPPINGS
175
A d d i t i o n a l r e l a t i o n s h i p s between t h e above c l a s s e s of mappings w i l l b e e s t a b l i s h e d a s t h i s s e c t i o n p r o g r e s s e s .
We
p o i n t o u t t h a t no p a r t i c u l a r s e p a r a t i o n p r o p e r t i e s a r e b e i n g imposed on t h e t o p o l o g i c a l s p a c e s o t h e r t h a n t h o s e s p e c i f i c a l -
l y s t a t e d w i t h i n t h e theorems o r d e f i n i t i o n s t h e m s e l v e s . The f o l l o w i n g r e s u l t s a r e due t o R . L . B l a i r (1964) and p r o v i d e c h a r a c t e r i z a t i o n s o f t h e v a r i o u s c l a s s e s o f mappings given i n t h e preceding d e f i n i t i o n . 15.4 a -
THEOREM ( B l a i r ) .
Tychonoff s p a c e
statements
are
If
is 2
f
onto a
X
c o n t i n u o u s s u r j e c t i o n from
T -space
1
Y
then t h e f o l l o w i n q
equivalent:
The mapping f i s f i b e r - c o m p a c t . I f {Fa : a & ] i s any f a m i l y pf c l o s e d on x, t h a t forms 3 -base f o r 2 filter --
If
iZa
subsets
of
X
then
z e r o - s e t s of : a c G j i s a n y f a m i l y pf ---
X
on X, t h e n t h a t forms 2 -base f o r 2 filter -f (
n za)
n
=
a 4
f(za).
a&
z e r o - s e t s of X I f [ z a : a d i ) i s a n y f a m i l y of --on x, t h e n n za t h a t forms 3 -base f o r 5 filter --
-
aEG
gj
n
only i f
f ( z a ) = gj.
acG
(1) i m p l i e s ( 2 ) : L e t {Fa : a d ) be a f a m i l y o f c l o s e d s u b s e t s of X t h a t forms a b a s e f o r a f i l t e r on X. I t s u f Proof.
n
f i c e s t o show t h a t
f (F,)
y E
n f(~,).
Then
n
c f(
a&
F ~ .) Suppose t h a t
adi
f-l(y)
n
F~
# gj
adi.
f o r every
Since
adi
n
a c G ] i s a family o f c l o s e d s u b s e t s o f f - l ( y ) w i t h t h e f i n i t e i n t e r s e c t i o n p r o p e r t y , t h e compactness o f Therefore, f-'(y) i m p l i e s t h a t f - ' ( y ) n ( n Fa) # #. (f-l(y)
F~
:
a&
The i m p l i c a t i o n s ( 2 ) i m p l i e s (3), and ( 3 ) i m p l i e s ( 4 ) , a r e b o t h immediate.
( 4 ) i m p l i e s (1):
Let
ytY, and l e t
S = f
-1( y )
.
W e w i l l show
176
COMPLETENESS AND CONTINUOUS MAPPINGS
that
i s compact by e s t a b l i s h i n g t h a t e v e r y
S
i s fixed.
3
Let
be a
Z - f i l t e r on
t h e i n c l u s i o n mapping from Then
ZE;C~ i f and o n l y i f
Hence, f - l ( y )
n
y c
[f(z)
:
nZ # ZEG).
Choose any p o i n t e x i s t s a zero-set
nz
Hence, A c S
@
X.
into
S
i-’(Z)
G
Let
n G.
E
in
Z
If X
so t h a t (S
x#A
#
= i (5).
f o r some
n
;Ci
# @. then t h e r e
and
x#Z.
A c Z
n
I t follows t h a t
E
5.
Ac5,
such t h a t Z)
S
denote
which i m p l i e s t h a t
Zc;Ci
T h e r e f o r e , by assumption,
x
i
belongs t o
= Z fl S
f o r each
Z - f i l t e r on
and l e t
S
5.
Z E ~ .
x E n F. T h i s i s a c o n t r a d i c t i o n s o t h a t xtA f o r e v e r y A E ~ . T h e r e f o r e , 5 i s f i x e d which c o n c l u d e s t h e proof o f t h e theorem. xiZ
But
and
Observe t h a t t h e Tychonoff p r o p e r t y f o r t h e domain s p a c e
x
i n t h e h y p o t h e s i s o f t h e p r e v i o u s theorem was needed o n l y
f o r the implication (4) implies (1). The n e x t r e s u l t w i l l p r o v i d e s e v e r a l i m p l i c a t i o n s f o r f i b e r - c o u n t a b l y compact mappings t h a t a r e a n a l o g o u s t o t h o s e
i n t h e p r e v i o u s theorem on f i b e r - c o m p a c t mappings. u t i l i z e the following f a c t :
whenever
c o u n t a b l y compact s u b s e t of a s p a c e
It w i l l
i s a non-empty
S
then f o r e v e r y zero-
X,
s e t sequence
( Z n : n f m ) such t h a t ( Z n n S : n c m ) h a s t h e i n f i n i t e intersection property, it i s the case that
S r?
( I?
Zn)
i s non-empty.
The f o l l o w i n g lemma w i l l a l s o b e
nclN u s e f u l and i s Problem 6 F . 4 o f Gillman and J e r i s o n . 15.5
of the
LEMMA
(Gillman and J e r i s o n )
Tychonoff s p a c e
T.
. Let
X
be a dense s u b s e t
Then t h e f o l l o w i n q s t a t e m e n t s
are
equivalent: (1) The s p a c e
compact.
T
(2)
Every
Z - f i l t e r on
(3)
Every
2-ultrafilter
If
X
has a c l u s t e r p o i n t i n
on
X
T.
has a l i m i t point in
i s a f i b e r - c o u n t a b l y compact s u r j e c t i o n from a t o p o l o q i c a l s p a c e X o n t o a s p a c e Y , the f o l l o w i n q s t a t e m e n t s a r e t r u e : (1) (Zn : n c m ) & d e c r e a s i n q sequence of z15.6
THEOREM ( B l a i r ) .
f
T.
SOME CLASSES OF MAPPINGS
x,
empty z e r o - s e t s of
If
(2)
(Zn
1 is 5
: nEm
empty z e r o - s e t s o f
then
177
f (
n
Zn)
Ti f ( Z n ) . nE7N
=
ncm
d e c r e a s i n q sequence of non-
n
then
X,
Zn = @
only i f
nEm
a.
n f(zn) = nEIN
x is
If
(3)
Tychonoff space, then
compact f o r each
x is
If
(4)
cluxf-
1
is
(y)
ycY.
a Hewitt-Nachbin
s p a c e , then t h e mappinq
i s f i b e r - compact.
f
(1) L e t [Zn : n c m ] be a d e c r e a s i n g sequence of zerox . I f y E fl f f z , ) , then f - l ( y ) n Zn # @ f o r
Proof. s e t s of
ncIN each
nEm.
Hence,
section property. foiiows t h a t
n
[f-l(y)
n
(
:
n E m ) has the f i n i t e i n t e r -
f - I ( y ) i s countably compact, i t
Since
f-l(y)
Zn
a.
n zn) #
ncm
Thus, y
E
f (
n zn).
ncm
Statement ( 2 ) i s an immediate consequence of ( 1 ) . -1 ( 3 ) L e t Y E Y and s e t S = f ( y ) . I t w i l l be shown t h a t every Let
2 - u l t r a f i l t e r on 3
be a
s i o n mapping from i s a prime
S
has a l i m i t i n let
i
and l e t
Q
2 - u l t r a f i l t e r on S
into
2 - f i l t e r on
S,
X,
X.
:
= i
Q (3).
I t w i l l be shown t h a t
countable i n t e r s e c t i o n p r o p e r t y . sequence ( Z n
clUxS ( s e e 1 5 . 5 ) . denote the inclu-
G
Then
Q
has the
Consider any ( d e c r e a s i n g )
n c m ) of z e r o - s e t s i n
n
such t h a t
a.
Zn =
nEm
For each
i
nEIN,
so t h a t
integer
nEm.
3
( Z n fl S ) belongs t o
Zn fl S
# @.
by t h e d e f i n i t i o n of
Therefore, f - I ( y )
I t follows t h a t
y
n
E
n
Zn
# @
f o r every
f ( Z n ) so t h a t
ncm ri Zn # @ by ( 2 ) . T h e r e f o r e , Q i s embeddable i n a Z - u l t r a n c IN f i l t e r on X w i t h t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y by 6 . 1 6
so t h a t p
Q
has a c l u s t e r p o i n t
c l U p f o r some
in
p
uX
by 8 . 5 ( 5 ) .
A E ~ ,then t h e r e e x i s t s a z e r o - s e t
Z(uX) such t h a t c l U 2 c Z and pkZ. S i n c e A E ~and A c s n z i t i s t h e c a s e t h a t ( s n z ) E 3. Then (z n contrary t o the f a c t t h a t Thus, p
belongs t o
clu*
Q
has
p
f o r every
If
in
Z
x)
E
as a cluster point. AEB.
I t follows t h a t
q
COMPLETENESS AND CONTINUOUS MAPPINGS
178
p i s a c l u s t e r p o i n t of 3, and hence a l i m i t p o i n t by 6 . 1 2 . C l e a r l y , p E cluxS so the argument i s complete. ( 4 ) I f X i s Hewitt-Nachbin complete, then X = uX s o t h a t -1 1 1 -1 f ( y ) = c l X f - ( y ) = c l u X f - ( y ) . BY ( 3 ) , c l v X f ( y ) is comp a c t f o r each p o i n t y i n Y, and hence f i s fiber-compact. This concludes the proof of the theorem. The next r e s u l t i s due t o B l a i r (1964) and provides a u s e f u l c h a r a c t e r i z a t i o n of zero- s e t p r e s e r v i n g mappings.
statements (2)
If
(3)
If
E
f
3
Z - f i l t e r on
* i s2
5 fq3).
is 5
Let
1( f ( Z ) )
= f-
follows t h a t
f-l(Z')
Conversely, i f follows t h a t
E
3.
3
Z.
on
then
= f
Z E
# (a),
f(5) = X,
is a zero-set i n
Since
Hence, Z '
X,
f(3)
Z - f i l t e r on
be a
f ( Z ) = Z' 3
then
X,
Z-ultrafilter
f (3) i n which case
Then, f - ' ( Z ' )
argument
then t h e followinq
Y,
i s zero-set preservinq.
mappinq
(1) i m p l i e s ( 2 ) :
Proof.
2 continuous s u r j e c t i o n from
f
onto a space
X
are e q u i v a l e n t :
(1)
f (Z)
If
THEOF?EM ( B l a i r ) .
15.7
a t o p o l o q i c a l space -
belongs t o
and l e t Y.
3, i t
f H (5) by d e f i n i t i o n .
f # ( a ) , then
Z' belongs t o f - l ( Z ' ) E 3. I t f(f-'(Z')) = Z' belongs t o . f (5) completing t h e
.
The i m p l i c a t i o n ( 2 ) implies ( 3 ) i s immediate. ( 3 ) implies ( 1 ) : Let
set
3 = (Z
E
Z(X)
: Z'
Z1
c
be a non-empty z e r o - s e t of
z).
Then
3
is a
X,
2 - f i l t e r on
and
x.
Q be a Z - u l t r a f i l t e r c o n t a i n i n g 3. By ( 3 ) , f ( Q ) = # f ( G ) , and s i n c e Z ' belongs t o G i t follows t h a t f ( Z ' ) E Therefore, f ( Z 1 ) is a z e r o - s e t i n Y by t h e d e f i n i t i o n
Let
(G) .
This concludes the p r o o f .
We s h a l l next p r e s e n t a c h a r a c t e r i z a t i o n of pings.
z-open map-
The following t e c h n i c a l lemma w i l l be u s e f u l i n t h e
proof of t h a t r e s u l t .
I t i s Theorem 3 . 1 2 of Gillman and J e r i -
son and i s the p r i n c i p a l t o o l f o r e s t a b l i s h i n g Urysohnls Lemma. We omit t h e s t r a i g h t f o r w a r d proof.
SOME CLASSES O F MAPPINGS 15.8
Let
LEMMA (Gillman and J e r i s o n ) .
--
IR.
b e any
Ro
t o p o l o q i c a l s p a c e , and l e t real l i n e
179
be an a r b i t r a r y
X
d e n s e subset o f t h e
Suppose t h a t t h e open s e t s
r
defined, f o r a l l
u ur
=
rtRo
X
are
such t h a t
Rot
t
of
Ur
x,
n ur
Id,
=
r c Ro
and -
cl Then t h e --
ur
us
c
r
whenever
<
s.
equation f(x) = inf(r
defines
f
R,
E
ur),
: x E
a s a continuous function
XEX,
on
X.
The f o l l o w i n g c h a r a c t e r i z a t i o n i s due t o B l a i r 15.9
-a
If
THEOREM ( B l a i r ) .
Tychonoff s p a c e
X
followinq statements
(1) (2)
Proof.
f
&&'continuous
then t h e
Y,
equivalent:
The mappinq f is z-open. If A and B are c o m p l e t e l y s e p a r a t e d s u b s e t s of X , then f ( A ) and Y\f(X\B) are c o m p l e t e l y separated --
in
Y.
(1) i m p l i e s ( 2 ) :
Suppose t h a t
p l e t e l y separated s u b s e t s of h E C ( X ) such t h a t and
s u r j e c t i o n from
o n t o a Tychonoff s p a c e
are
(1964).
-1i h
1.
if
h(x) = 1
XEA,
r
For e a c h r e a l number = (XEX : h ( x )
<
r)
Zr = ( X E X : h ( x )
A
r).
Vr
rtlR
define
[ E: f
(Vr)
,
a r e com-
if
r < O
if
O
l
if
r
>
r
1.
i
if
XEB,
[0,1], s e t
E
and
Next, f o r each
B
Then t h e r e e x i s t s a f u n c t i o n
X.
h ( x ) = -1
and
A
l
COMPLETENESS AND CONTINUOUS MAPPINGS
180
<
r
trivial if then because
cl NOW,
s
implies t h a t
or
0
s
>
is
ur
= c f f (Vr) c
rrIR.
defines
cl f (Z,)
f (A)
c f (X\B)
U1
y
f o r every
Y',f(X\B)
( 2 ) implies (1): L e t
in
X.
X\H
:
+ a,
2 1
g(y)
by 1 5 . 8 .
Y
Now,
f o r every y
E
g(y)
Y\f(X\B). Y.
b e a c o z e r o - s e t neighborhood of t h e
choose any p o i n t
x
E
Y\int x
E
If
f(H).
f - l ( y )I
[H
E
such t h a t
Z'
f ( Z l ) and
This implies t h a t
Y.
Z'
.
Then Since
c H.
ZI
=
a.
Therefore,
separated since
and
Z
f(x) = y
belongs t o
f - l ( y ) a r e completely
are d i s j o i n t zero-sets.
X\H
By ( 2 )
Y \ f ( X \ f - l ( y ) ) a r e completely s e p a r a t e d i n
a g a i n , f ( Z ) and But c l e a r l y , y
and
Z
Y\f ( H ) are c o m p l e t e l y Thus, i t i s t h e case t h a t
i n t f ( H ) which is a c o n t r a d i c t i o n . f-'(y)
for
a r e d i s j o i n t z e r o - s e t s t h e y a r e completely s e p a r a t e d .
separated i n
n
Y
ur)
a r e completely s e p a r a t e d i n
H
I t f o l l o w s from ( 2 ) t h a t
H
1,
us.
=
i s open i n
Ur
y c
Suppose t h a t y
there e x i s t s a zero-set and
s
from which i t f o l l o w s t h a t
f ( A ) and
E
Thus, f ( A ) and zero-set Z 1 H n f- ( y )
<
r
0
Then t h e e q u a t i o n
and
Uo
C
This i s
Us.
c f (VS)
a s a continuous f u n c t i o n o n
g
C
z-open i t i s t h e c a s e t h a t
f
g ( y ) = inf[rElR
0
c l Ur
Moreover, i f
1.
i s open by 1 5 . 3 ( 3 ) s o t h a t
f
every
<
r
W e assert that
E
Y\f ( X \ f - l ( y ) ) so t h a t
c l f ( Z ) c i n t f(H)
.
I t follows t h a t
y f
c l f (Z) is a
.
Y.
Thus
z-open mapping
completing t h e proof o f t h e theorem. The p r e c e d i n g r e s u l t s have p r o v i d e d f o r m u l a t i o n s f o r t h e f i b e r - c o u n t a b l y compact mappings,
t h e zero- s e t p r e s e r v i n g map-
p i n g s , and t h e
The n e x t sequence of theorems
z-open mappings.
w i l l e s t a b l i s h some o f t h e r e l a t i o n s h i p s between t h e v a r i o u s c l a s s e s o f mappings under i n v e s t i g a t i o n .
Again w e a t t r i b u t e
these r e s u l t s t o Blair (1964). THEOREM ( B l a i r ) .
15.10
If
p i n s from a t o p o l o q i c a l s p a c e
2-epen
.
i s an open and
f X
i n t o a space
z - c l o s e d mapY,
then
f
&
181
SOME CLASSES OF MAPPINGS
Let
Proof.
be a z e r o - s e t i n
Z'
a c o z e r o - s e t neighborhood o f f ( H ) i s open i n
and
Y.
cl f(Zl) c f (H).
that
15.11
Since
is is
i t follows
z-closed z-open.
i s an open p e r f e c t mapping from a
f
Y , then
i n t o a space
X
is
f
z-open.
Every p e r f e c t mapping i s c l o s e d and hence
Proof.
is
H
f (Z' ) c f (H)
Then
X.
f
Hence, f
If
COROLLARY.
t o p o l o g i c a l space
and suppose t h a t
X
in
Z'
z-closed.
The r e s u l t i s now immediate from t h e theorem.
15.12
THEOREM ( B l a i r ) .
If
Y, then
space Proof.
Let
f
be shown t h a t
A =
n
3
e x i s t s a zero-set A =
n 3
i s the
that contain
in
Z
3
where
On t h e o t h e r hand,
follows t h a t
T1-
X
x
if
F
Z - f i l t e r on
A.
i! 5
such t h a t
a s asserted.
Since
(1) i m p l i e s ( 2 ) , f ( A ) = f ( n 3) = f ( Z ) i s closed s i n c e
n
and
A c Z
is
con-
XPA, then t h e r e
and
x
(f(Z) : Zr5;. f
X
I t i s clear t h a t
i s fiber-compact,
f
It w i l l
X.
Z E ~c o n t r a d i c t i n g t h e f a c t t h a t
each image
z-
onto the
is closed.
s i s t i n g of a l l z e r o - s e t s A c fl 3.
X
b e a non-empty c l o s e d s u b s e t of
A
and
i s a fiber-compact
f
c l o s e d s u r j e c t i o n from t h e Tvchonoff s p a c e
E
xLZ.
It
n 5.
Thus,
by 1 5 . 4 , Moreover,
Hence,
z-closed.
f (A) i s c l o s e d . 15.13 and
THEOREM ( B l a i r ) .
f
i s a f i b e r - c o u n t a b l y compact
z-open s u r j e c t i o n from a Tychonoff s p a c e
noff space Proof, nEN,
If
Y, t h e n
Let
f
Z = Z ( h ) be a non-empty z e r o - s e t i n
o n t o a TychoX.
For e a c h
set Un = { X E X : / h ( x )
I < );1
Zn = ( x t x : l h ( x )
1 2 ;)1.
and
Clearly, Z =
fl
un
n c IN that
X
is zero-set preservinq.
=
n zn. nem
H e n c e , by 15.6(1), i t f o l l o w s
182
COMPLETENESS AND CONTINUOUS MAPPINGS
Now, f o r each sets i n and g, y
and
Y\f ( U n ) ,
E
g =
z
nc m
2-"gn
f (2) = Z(g).
n
IN
is
f
gn (y) = 0
5
gn
Q
Z-open, by 1 5 . 9
y
If
y
y
f ( U n ) we have t h a t
gn(y) = 0
f (2).
E
.
Choose
Y.
(z),
f(z)
gn(y) = 1
if
Then t h e f u n c t i o n
I t w i l l be shown t h a t
C(Y).
ncm
f o r every
ncm.
f ( u n ) f o r every
t
f
E
each n c W
f ( Z ) then
E
y
if
1 for
belongs t o
I t follows t h a t nE
a r e completely s e p a r a t e d
X\Un
a r e completely s e p a r a t e d sets i n
such t h a t
C(Y)
and
Z
Therefore, s i n c e
X.
Y\f(Un) E
nElN,
Since
.
f(z) =
T h i s concludes t h e proof
of t h e theorem. 15.14
If
COROLLARY,
Tychonoff space
X
-zero- s e t p r e s e r v i n g .
i s an open p e r f e c t mappins from a
f
o n t o a Tychonoff space
Y,
then
f
is
The proof f o l l o w s immediately from 1 5 . 1 1 and t h e
Proof. theorem.
15.15
THEOREM ( B l a i r ) .
mal Hausdorff space X and only i f i t i s b o t h -
A
continuous mappinq
into a
T1-space
If
f
15.10.
Conversely,
if
&
Y
f
is
z-open,
f
is
then
f
15.3(3) s i n c e t h e domain i s a Tychonoff s p a c e .
i s a c l o s e d mapping.
be shown t h a t
f
closed s e t i n
X,
the closed sets
A
i t f o l l o w s from 1 5 . 9 t h a t
i s open by I t remains t o be a
Y.
X.
Since
f ( A ) and
f
is a
Y\f(X\B)
However,
= Y\f (X\f-'(y) ) = Y\f ( f - ' ( Y \ ( y ) ) )
and t h i s l a s t s e t c o n t a i n s y
z-open by
and suppose t h a t y t c l f ( A ) \ f ( A ) . Then -1 and B = f ( y ) a r e d i s j o i n t , and hence
a r e completely s e p a r a t e d i n Y\f ( X \ B )
z-open i f
A = cl A
Let
completely s e p a r a t e d i n t h e normal space z-open mapping,
from a nor-
open and c l o s e d .
i s open and c l o s e d , then
proof.
f
y.
belongs t o t h e c l o s u r e of
This c o n t r a d i c t s the f a c t t h a t f (A).
Hence, f ( A ) i s c l o s e d
completing t h e p r o o f , The n e x t r e s u l t r e l a t e s
z - c l o s e d mappings and some of
t h e embedding concepts t h a t have been s t u d i e d i n p r e v i o u s chapters.
SOME: CLASSES OF MAPPINGS
15.16
be a
f
THEOREM.
X
arbitrary.
I f the f i b e r
is
z - c l o s e d mappinq from t h e Tycho-
Y , and l e t
i n t o t h e Tychonoff s p a c e
noff space
C-embedded
fW1(y)
is
z-embedded
in
b e such a z e r o - s e t .
Then
y
p
g
fore,
0
f
(90 f ) [ f - l ( Y )
and
g(x) = 1
belongs t o
1
then i t
Thus l e t
=
for a l l
C(X,IR)
Z
f ( 2 ) is closed i n
f ( Z ) and
Hence t h e r e e x i s t s a c o n t i n u o u s f u n c t i o n g(y) = 0
X,
be -
fV1(y) i s completely
s e p a r a t e d from e v e r y z e r o - s e t d i s j o i n t from i t .
that
yeY
X.
By 1 0 . 4 i t s u f f i c e s t o show t h a t
Proof.
183
g
x
E
C(Y,IR)
in
f(Z).
Y.
such There-
and ( g 0 f ) ( Z ) c (11,
(01.
Because of t h e p r e v i o u s r e s u l t w e see t h a t f o r Tychonoff Y
s p a c e s and
z - c l o s e d mappings, t h e c o n c e p t s o f
C-,
C
-,
and
z-embedding a r e e q u i v a l e n t f o r f i b e r s
f-l(y). The f o l l o w i n g c h a r t p r o v i d e s a summary o f t h e r e l a t i o n -
s h i p s t h a t have been e s t a b l i s h e d i n t h i s s e c t i o n .
If there are
c o n d i t i o n s t h a t a r e r e q u i r e d of t h e domain o r r a n g e s p a c e i n order t h a t a p a r t i c u l a r implication hold,
then t h o s e c o n d i t i o n s
a r e so s p e c i f i e d w i t h an a p p r o p r i a t e a r r o w .
The s e c t i o n w i l l
c o n c l u d e w i t h a v a r i e t y of examples e s t a b l i s h i n g t h a t none of t h e i m p l i c a t i o n s i n t h e c h a r t may b e r e v e r s e d w i t h o u t imposing a d d i t i o n a l c o n d i t i o n s on t h e s p a c e s o r mappings i n v o l v e d .
domain and r a n g e Tychonof €
doma i n
a
b
means e v e r y
a
mapping is a
b
mapping.
184
COMPLETENESS AND CONTINUOUS MAPPINGS
15.17
(1) A closed mappinq t h a t f a i l s t o be f i b e r -
EXAMPLES.
compact. Let
be an uncountable space with t h e d i s c r e t e topology, l e t
X
be a space c o n s i s t i n g of a s i n g l e p o i n t , and d e f i n e the mapping f from X o n t o Y by f f x ) = yeY € o r every p o i n t
Y
Then
XEX.
f
x
f-l(y) =
(2) & Let
i s a c l o s e d continuous s u r j e c t i o n .
z-closed mapping t h a t f a i l s t o be c l o s e d .
* x lN \[ (n,W) ) denote t h e Tychonoff plank a s pre-
T = [O,n]
sented i n 1 4 . 1 4 . T
lN
onto
of
T
in
IN
*
*
.
Let 7
T.
Therefore, r
Next, l e t
denote t h e p r o j e c t i o n mapping from
T
Hence,
i s closed i n
.
However,
i s not compact.
i s open.
However,
The " r i g h t - e d g e "
~ ( [ n X)
[n) x
N
N ) i s not c l o s e d
i s not a closed mappinq.
Z E Z(T).
If
Z
i s compact, then
~ ( 2 i)s
compact and hence c l o s e d . I f 2 i s n o t compact, then Z meets the "top-edge" [ O , n ] x (UJ) of T . This l a s t s t a t e m e n t follows from the f a c t ( s e e Gillman and J e r i s o n , Problem 8 J . 1 ) t h a t every non-compact z e r o - s e t
Z of T c o n t a i n s p o i n t s ( a n , n ) f o r a r b i t r a r i l y l a r g e n and t h e r e f o r e Z meets the t o p edge of T . Hence, i f Z i s not compact, then w belongs t o ~ ( 2 and ) ~ ( 2 is ) closed. Hence, T g z-closed mappinq. Observe t h a t T is z-open by 1 5 . 1 0 . Moreover, s i n c e [o,n) i s countably compact and [O,n] i s compact, i t follows t h a t T i s f i b e r - c o u n t a b l y compact. Hence r i s z e r o - s e t -1 Finally, note t h a t r ( w ) = [ 0 , 0 ) so p r e s e r v i n q by 1 5 . 1 3 . t h a t 7 is z-open and f i b e r - c o u n t a b l y compact b u t n o t f i b e r compact. (For a d d i t i o n a l information concerning t h e Tychonoff plank see Problem 8J of Gillman and J e r i s o n . )
-
Example (2) also s e r v e s a s an example of
(3)
that is --
open and
X
function
z-closed b u t n o t p e r f e c t .
A z e r o - s e t p r e s e r v i n q mappinq t h a t f a i l s t o be open.
(4) Let
mappinq
be a Tychonoff space and l e t f
from
X
into
Y
Y = IR.
d e f i n e d by
The c o n s t a n t
f(x) = 0
for a l l
f a i l s t o be an open mapping. Hence, f cannot be z-open by 1 5 . 3 ( 3 ) . However, i f Z i s a z e r o - s e t i n X, then f ( Z ) = ( 0 )which is a z e r o - s e t i n Y ( s i n c e every c l o s e d s u b s e t of a Note a l s o t h a t f i s an example m e t r i c space i s a z e r o - s e t ) XEX
.
SOME CLASSES OF MAPPINGS
of 2
z - c l o s e d mappinq t h a t f a i l s t o be
&
(5)
185
z-open o r open.
z - c l o s e d mappinq t h a t f a i l s t o be z e r o - s e t pre-
servinq. Let
be a Tychonoff space t h a t fails t o be p e r f e c t l y normal
Y
Hence,
c o n t a i n s a closed subset
Y
zero-set. ping.
Let
Then
and l e t
X = F
d e n o t e t h e i n c l u s i o n map-
i
i s a c l o s e d mapping s i n c e e v e r y c l o s e d s e t i n
i
t h e r e l a t i v e topology on
is a l s o closed i n
F
Y.
Therefore
is n o t z e r o - s e t p r e s e r v i n g i ( F ) f a i l s t o be a z e r o - s e t i n Y .
z-closed.
is
i
t h a t f a i l s t o be a
F
since
However, i
A n open mappinq t h a t f a i l s t o be
(6)
z-open.
T
X = {(x,y) c IR x IR : x > 0 , y > 0 , and xy = 1 ) . L e t d e n o t e t h e p r o j e c t i o n mapping from X o n t o Y = I??. Then
T
i s an open mapping.
Let
t h e e n t i r e space containing 7
(7)
Z
i s t h e z e r o - s e t c o n s i s t i n g of
then n o t e t h a t
However, t h e image
Z.
neighborhood of ping
X,
If
cl
~ ( x =)
[O,co)
i s a l s o a cozero-set
X
T ( X ) = (0,co) is not a
in
T h e r e f o r e , t h e map-
Y.
f a i l s t o be z-open. A z-open mapping t h a t f a i l s t o be
z-closed.
T h i s example was p o i n t e d o u t t o t h e a u t h o r i n a p e r s o n a l communication from P . Nyikos. plank
T.
This t i m e l e t
Again c o n s i d e r t h e Tychonoff d e n o t e t h e p r o j e c t i o n mapping o f
cp
onto [O,n]. Observe t h a t t h e top-edge o f t h e p l a n k i s t h e zero- s e t of t h e c o n t i n u o u s r e a l - v a l u e d f u n c t i o n g i v e n by
T
g(
(a,n) )
=
1 , ;
and
g((a,w))
for a l l
= 0,
However, t h e image o f t h e top-edge under
i n [O,n].
T h e r e f o r e , cp
is not
cp
a
E
[O,fi]
.
f a i l s t o be c l o s e d
z-closed.
Next i t w i l l be e s t a b l i s h e d t h a t
cp
z-open.
First,
I t s image i s c l o s e d , and s i n c e
i s a compact z e r o - s e t o f T . cp i s an open mapping, e v e r y
c o z e r o - s e t neighborhood o f
i s mapped i n t o a neighborhood
c o n s i d e r t h e c a s e i n which
of
c l q ( Z ) = cp(Z).
If
Z
Z Z
i s n o t compact, then o b s e r v e t h a t x ( w ) o f t h e t o p e d g e . To
2 = Z ( f ) must c o n t a i n a t a i l [a,O)
see t h i s one needs t o t a k e advantage of t h e f a c t t h a t t h e V pX c o i n c i d e s w i t h t h e o n e - p o i n t * c o m p a c t i f i c a t i o n T , and moreover t h a t t h e S t o n e e x t e n s i o n h* of any f u n c t i o n h E C ( T ) i s given b y Stone-Cech c o m p a c t i f i c a t i o n
COMPLETENESS AND CONTINUOUS M A P P I N G S
186
Now, i f the z e r o - s e t edge [ O , n )
x (w],
Z(f)
about the p o i n t ( 0 , ~i ) n
T* = PT
such t h a t
Z ( f ) would have t o be compact.
f*(U)
edge a s claimed.
Now, l e t
H
u
c IR\{oj.
Z(f).
Therefore, the non-com-
Z ( f ) must contain a t a i l
pact zero-set Then
cannot be z e r o a t
i s compact and c o n t a i n s the c l o s e d s e t
Moreover, T\U
Z(f).
f*
I t follows t h a t t h e r e i s an open s e t
t h e point (Q,w).
Then
f a i l s t o contain a t a i l of the t o p
then t h e extension
[a,n)
x ( w ) o f t h e top-
be a c o z e r o - s e t neighborhood of
m u s t be t h e complement of a compact z e r o - s e t H contains a c o f i n i t e
H
by the previous s t a t e m e n t , and hence
s u b s e t of t h e r i g h t - e d g e of t h e plank. [O,R]
of
Z.
under
cp
Hence i t s image i n t o
m u s t c o n t a i n t h e c l o s u r e of the p r o j e c t i o n
This e s t a b l i s h e s the d e s i r e d r e s u l t t h a t
cp
z-
is
open completing t h e example. Two a d d i t i o n a l mappings w i l l be introduced f u r t h e r on i n t h i s c h a p t e r . These a r e t h e "WZ-mappings" due t o T . Isiwata (1967)
and the "hyper-realtl mappings due t o R . B l a i r ( 1 9 6 9 ) .
Both of
t h e s e c l a s s e s of mappings w i l l then be r e l a t e d t o t h e c l a s s e s
of mappings t h a t were under i n v e s t i g a t i o n i n t h i s s e c t i o n . A c h a r t w i l l be provided i n S e c t i o n 18 which summarizes a l l of t h e v a r i o u s r e l a t i o n s h i p s between t h e s e c l a s s e s of mappings. Section 16 :
Perfect Mappinqs
I n t h i s s e c t i o n we w i l l i n v e s t i g a t e t h e i n v a r i a n c e and
i n v e r s e i n v a r i a n c e of Hewitt-Nachbin completeness under t h e s t r o n g e s t c l a s s of mappings t h a t w e r e considered i n t h e prev i o u s s e c t i o n , namely t h e p e r f e c t mappings.
I t , w i l l be e s t a b -
l i s h e d t h a t Hewitt-Nachbin completeness i s i n v e r s e i n v a r i a n t under p e r f e c t mappings ( 1 6 . 2 ) , and i n v a r i a n t under open perf e c t mappings ( 1 6 . 1 0 ) .
I n obtaining these r e s u l t s w e w i l l
a c t u a l l y e s t a b l i s h a number of s t r o n g e r r e s u l t s due t o R . B l a i r (1969), N . Dykes (1969), and 2. Froll'k (1963). For example, i t w i l l be shown t h a t Hewitt-Nachbin completeness i s i n v a r i a n t under any f i b e r - c o u n t a b l y compact and zero- set pre-
187
PERFECT MAPPINGS
serving surjection (16.8).
The f i n a l r e s u l t o f t h e s e c t i o n
w i l l e s t a b l i s h t h a t t h e p e r f e c t image o f a Hewitt-Nachbin cb-space i s a g a i n a Hewitt-Nachbin space
s p a c e i n t o a weak (16.13) .
The a l m o s t r e a l c o m p a c t s p a c e s i n v e s t i g a t e d i n Sec-
t i o n 1 4 w i l l p l a y an i m p o r t a n t p a r t i n e s t a b l i s h i n g t h a t re-
sult.
Moreover, f o r Tychonoff s p a c e s , i t w i l l b e e s t a b l i s h e d
t h a t t h e p r o p e r t y of almost realcompactness is b o t h i n v a r i a n t and i n v e r s e i n v a r i a n t under p e r f e c t mappings ( 1 6 . 1 1 ) The f o l l o w i n g r e s u l t is due t o B l a i r ( 1 9 6 4 ) .
.
Note t h a t
i t i s a r e f i n e m e n t o f F r o l l k ' s r e s u l t c o n c e r n i n g tkie i n v e r s e i n v a r i a n c e o f H e w i t t-.Nachbi.n c o m p l e t e n e s s under p e r f e c t mapW e w i l l s t a t e ttie l a t t e r r e s u l t a s a c o r o l l a r y .
pings. 16.1
If
THEOREM ( B l a i r ) .
2 continuous
f
j e c t i o n from a Tvchonoff s p a c e Y
such t h a t
f-
1
Let
3
YEY,
be a
X
YEY
f # ( 5 ) i s a prime
Then
with
f - l ( y ) and n o t e t h a t otherwise.
5
y
Then
n
S
y
Hence t h e r e
II f # (3) by 7 . 1 3 .
E
# @
Z
Z - f i l t e r on
f o r every
does n o t b e l o n g t o
and
y
j!
belongs t o
f-'(Z1)
Z'
.
3;.
f ( Z ) f o r some
s
t e r on @.
Hence,
S
n
Z E ~ . I t follows t h a t the t r a c e
Since
S
in
Z
E Z(Y) with
Z'
H e n c e , Z c f - ' ( Z l ) which i m p l i e s t h a t ty T h e r e f o r e , Z ' E f (3) and y E Z '
which i s a c o n t r a d i c t i o n . every
S =
Let
Z E ~ . For suppose
which i m p l i e s t h a t t h e r e e x i s t s a z e r o - s e t
f (Z) c Z'
z-embed-
i s a Hewitt-Nachbin s p a c e . X with t h e countable
with t h e countable i n t e r s e c t i o n property.
exists a point
and
Z - u l t r a f i l t e r on
intersection property. Y
onto a H e w i t t -
( y ) i s Hewitt-Nachbin complete
ded f o r e a c h p o i n t --Proof.
X
z-closed sur-
i s non-empty
Z
as
is a
for
Z-ultrafil-
w i t h t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y by 10.10.
i s Hewitt-Nachbin complete t h i s i m p l i e s t h a t
n
ZS
#
T h i s c o n c l u d e s t h e proof o f t h e theorem.
16.2
COROLLARY
(Blair).
Tychonoff s p a c e
X
If
f
is 2
p e r f e c t mappinq from a
o n t o a Hewitt-Nachbin s p a c e
Y,
then
X
- -
i s 2 Hewitt-Nachbin s p a c e .
Proof.
The f i b e r
f-'(y)
i s compact and t h e r e f o r e
C-embedded
i n X f o r every p o i n t ycY. S i n c e C-embedded s u b s e t s a r e a l s o z-embedded t h e r e s u l t i s now immediate from t h e theorem.
188
COMPLETENESS AND CONTINUOUS M A P P I N G S
The n e x t r e s u l t c o n c e r n s t h e t o p o l o g i c a l sum o f H e w i t t Nachbin s p a c e s .
L e t u s r e c a l l the d e f i n i t i o n of t h a t c o n c e p t
from g e n e r a l t o p o l o g y .
L e t [Xa : a t G ) b e a non-empty
X = U (Xa : a & ) .
o f d i s j o i n t t o p o l o g i c a l s p a c e s , and l e t
w e endow
X
family If
w i t h t h e l a r g e s t t o p o l o g y r e l a t i v e t o which each
i n c l u s i o n mapping from
into
Xa
X
r e s u l t a n t t o p o l o g i c a l space
X is continuous, then t h e is called the t o p o l o q i c a l z
o f ( X u : a t G ) and i s d e n o t e d by
(Xu
W e have t h e
: aEG].
following r e s u l t . 16.3
A
THEOREM.
nonmeasurable t o p o l o q i c a l sum o f Hewitt-Nach-
b i n s p a c e s i s a Hewitt-Nachbin s p a c e . Proof.
Let
where each
d e n o t e t h e t o p o l o g i c a l sum
X
[Xa
i s Hewitt-Nachbin complete and
Xu
measurable c a r d i n a l i t y . o l o g y , then by 8.18
I f w e endow
G
i s a Hewitt-Nachbin s p a c e .
G
f(x) = a
by
whenever
x
i s w e l l - d e f i n e d s i n c e t h e c o l l e c t i o n (Xa
Ci
pairwise d i s j o i n t .
i s simply t h e union o f an
G
a p p r o p r i a t e c o l l e c t i o n of t o p o l o g i c a l sum X . Clearly
each o f which i s open i n t h e is z-closed s i n c e G is
Xu's,
d i s c r e t e , and moreover p l e t e f o r each p o i n t f o r each
f
f-'(a)
a d .
a
For suppose t h a t
Z
Z = Z ( h ) f o r some f u n c t i o n
h
tion
g
from
longing t o Xa.
If
G
X
X\Xa
into and
IR
i s open i n
X.
by
Xu,
is
IR
X
Z(Xa)
.
D e f i n e t h e func-
C(Xa).
f o r every
g ( ~ =) h ( x ) whenever g-l(G)
z-embedded i n
is a zero-set in E
g(x) = 0
i s an open s u b s e t of
the r e a l number zero, then
in
i s Hewitt-Nachbin com-
= X
F i n a l l y , Xa
Then
aEG.
Xa.
: a 4 ) is
i s continuous s i n c e t h e i n -
Moreover, f
v e r s e image o f any s u b s e t o f
Define t h e
belongs t o
f :X
Then
-+
h a s non-
G
w i t h t h e d i s c r e t e top-
mapping f
: ~ E G ] ,
x
x
be-
belongs t o
t h a t f a i l s t o contain
= h-'(G).
which i n t u r n i s open i n
X,
On t h e o t h e r hand,
Since
h-l(G)
g-'(G)
i s open
i f z e r o does belong t o G, -1 ( G ) = ( U X p ) U h - l ( G ) which i s a g a i n open i n X. g
then Thus,
@#a
g
i s c o n t i n u o u s and
Z ( g ) fl Xu = Z .
Hence, t h e f u n c t i o n
s a t i s f i e s t h e h y p o t h e s i s of 1 6 . 1 s o t h a t complete.
T h i s concludes t h e p r o o f .
X
f
i s Hewitt-Nachbin
189
PERFECT MAPPINGS
B e f o r e i n v e s t i g a t i n g t h e i n v a r i a n c e o f Hewitt-Nachbin completeness under c o n t i n u o u s mappings, w e p r e s e n t t h e f o l lowing example i l l u s t r a t i n g t h a t t h e p e r f e c t image of a H e w i t t I n (196lA,
Nachbin s p a c e need n o t b e a Hewitt-Nachbin s p a c e .
3 . 3 ) , F r o l i k p r e s e n t s an example of a c o n t i n u o u s , open, and
f i b e r - c o m p a c t image of a Hewitt-Nachbin s p a c e t h a t f a i l s t o be a Hewitt-Nachbin s p a c e .
16.4
The
EXAMPLE.
need n o t ---
p e r f e c t imaqe o f a Hewitt-Nachbin s p a c e
-
be 2 Hewitt-Nachbin s p a c e .
I n h i s 1958D paper S . Mrdwka c o n s t r u c t s t h e f o l l o w i n g example o f a Tychonoff s p a c e t h a t f a i l s t o be Hewitt-Nachbin complete a l t h o u g h i t i s t h e union o f two c l o s e d Hewitt-Nachbin spaces. Let
d e n o t e t h e subspace of t h e Niemytzki s p a c e
h
I?
p r e s e n t e d i n Example 8.23 c o n s i s t i n g o f t h e p o i n t s i n t h e s e t D =
(x,O) : X E I R ) t o g e t h e r w i t h t h e s e t
h ' = ( (x,y) : y
>
0,
x and y a r e r a t i o n a l ) . The s e t D i s a g a i n a c l o s e d d i s c r e t e subspace o f h . Moreover, s i n c e h! is a countable dense s u b s e t o f lish that
t h e arguments used i n 8.23 a g a i n e s t a b -
h,
i s a non-normal Tychonoff s p a c e t h a t i s H e w i t t -
h
Nachbin c o m p l e t e . Let
hl
h2
and
be two homeomorphic c o p i e s o f
where i t i s a g r e e d t h a t t h e p o i n t s of (p,O) f o r
p
E
n,.
For
d i s c r e t e subspace o f graph.
Q
Let
let
h2
a r e ordered p a i r s
Di
denote t h e closed
a s d i s c u s s e d i n t h e p r e v i o u s para-
hi
d e n o t e t h e f a m i l y of a l l r e a l - v a l u e d func-
t i o n s d e f i n e d on hl
i = 1,2
h
D1
which a d m i t a c o n t i n u o u s e x t e n s i o n o v e r
and s a t i s f y one o f t h e f o l l o w i n g two c o n d i t i o n s :
(i) i f
f c Q , then t h e r a n g e of
f
has c a r d i n a l i t y
c;
( i i ) t h e r e e x i s t a t l e a s t two d i s t i n c t e l e m e n t s a and 1 1 p i n f ( D 1 ) such t h a t If- ( a )1 = If- ( P ) 1 = c,
IR.
t h e c a r d i n a l i t y of
Mro/wka t h e n p r o v e s , u s i n g an a d d i t i o n a l lemma, t h a t t h e r e
e x i s t s a permutation
T
of
Now l e t t h e s p a c e
D1
hl
continuous e x t e n s i o n over X
such t h a t f o r every
fo-rr f
be o b t a i n e d from
in
n,
admits no &.
and
h,
by
190
COMPLETENESS AND CONTINUOUS MAPPINGS
i d e n t i f y i n g each p o i n t p E D1 with t h e p o i n t ( n - l ( p ) , O ) i n D2. Then X i s a Tychonoff space under t h i s i d e n t i f i c a t i o n , and moreover h l and h2 a r e closed s u b s e t s of X . Therefore, X
i s the union of two c l o s e d Hewitt-Nachbin subspaces.
W e w i l l o u t l i n e the e s s e n t i a l reason why
X
f a i l s t o be a
Hewitt-Nachbin s p a c e ,
Y
Let
=
clpXD1.
Then Mr6wka proves i n h i s 1958D paper
t h a t there e x i s t s a point
po
with
>
g(po) = 0
restriction
and
g(p)
f = g/D1
g(po) = 0
and
such t h a t i f p
E
g c C(Y)
then t h e
D1,
s a t i s f i e s e i t h e r c o n d i t i o n ( i ) or
condition ( i i ) given above. that
i n Y\D1 0 for a l l
g(p)
Now, suppose t h a t
>
0
for a l l
i = 1,2, denote the r e s t r i c t i o n of
g
pcX. to
g
t
Let
C(pX)
fi,
Then
Di.
such
fl
sat-
i s f i e s ( i ) o r ( i i ), and moreover f l admits a continuous extension over h l (namely, the r e s t r i c t i o n g i n l ) , Hence, fl belongs t o LX by d e f i n i t i o n . The function f 2 s a t i s f i e s the - 1 ( p ) , O ) ) = f l ( p ) o r , i n o t h e r words, equation f 2 ( ( T T f 2 ( ( p , 0 ) ) = f l ( . ? r ( p ) ) . Now, f10 T does n o t belong t o by t h e n a t u r e of T, so t h a t f2 admits no continuous e x t e n s i o n over
h2.
This i s a c o n t r a d i c t i o n however, because
i n f a c t a continuous extension of f 2 over D2. t h e r e does n o t e x i s t a f u n c t i o n g belonging t o that
g(po) = 0
8.9 (3) that
X
and
g(p)
>
0
for a l l
ptx.
g1h2
is
Therefore, C ( p X ) such
~t follows from
cannot be Hewitt-Nachbin complete,
I n summary, a space X has been c o n s t r u c t e d w i t h t h e following p r o p e r t i e s : (1) The space X i s a union of two c l o s e d , H e w 1 t t- Nachb i n non-normal Tychonoff spaces each of which cont a i n s a closed d i s c r e t e subspace. The space X f a i l s t o be a Hewitt-Nachbin
(2)
Next, l e t
Y
denote the t o p o l o g i c a l sum of t h e two
Hewitt-Nachbin spaces
P1
and
b i n complete by 1 6 . 3 .
Let
cp
Y
onto
X
i t s copy i n
space.
D2.
Then
Y
i s Hewitt-Nach-
denote t h e " n a t u r a l mapping" of
which t a k e s each p o i n t from t h e d i s j o i n t union t o X.
Then i t is t r i v i a l t o v e r i f y t h a t
p e r f e c t map from a Hewitt-Nachbin t o be Hewitt-Nachbin complete.
cp
is a
space o n t o a space t h a t f a i l s
Note t h a t t h i s example a l s o
e s t a b l i s h e s t h a t t h e p a r a p e r f e c t imaqe of a Hewitt-Nachbin
191
PERFECT MAPPINGS
space need n o t be Hewitt-Nachbin complete. Mrdwka comments f u r t h e r on the space
X.
1958D paper he assumes t h a t t h e c a r d i n a l
i n o b t a i n i n g p r o p e r t i e s of the space
I n h i s 1970 paper
I n the original
c =
i s regular
X, whereas i n t h e 1970
paper he shows t h a t a s l i g h t m o d i f i c a t i o n i n some of t h e
I n the l a t e r paper he a l s o shows t h a t the above example can be used t o
proofs e n a b l e s t h e omission of t h a t assumption. establish that
notbe
the p e r f e c t
IN-compact.
imaqe of an
IN-compact space need
This concludes the example.
Despite t h e f a c t t h a t Hewitt-Nachbin completeness i s n o t i n v a r i a n t under p e r f e c t mappings, t h e r e a r e a number of i n t a r e s t i n g s p e c i a l c a s e s f o r which i t i s i n v a r i a n t .
The following
lemma, due t o K . Morita (1962, Theorem 1.4), w i l l be u s e f u l
i n e s t a b l i s h i n g one such r e s u l t . 16.5
If
(Morita).
LEMMA
f
&a
continuous
closed sur-
j e c t i o n from a normal and countably paracompact space a t o p o l o q i c a l space -
Y,
then
Y
onto
X
i s normal and countably p a r a -
compact. For purposes of c l a r i t y we p o i n t o u t t h a t Morita does
n o t assume
t h e Hausdorff c o n d i t i o n f o r t h e spaces i n 1 6 . 5 .
We
a l s o mention t h a t E . Michael (1957, C o r o l l a r y 1) proved t h a t every image of a paracompact Hausdorff space under a continuous
closed mapping i s paracompact Hausdorff.
The n e x t r e s u l t
i s a sharpened v e r s i o n of a theorem due t o Frolck (1963, /
Theorem 1 2 ) . 16.6
We w i l l i n c l u d e F r o l i k ' s r e s u l t a s a c o r o l l a r y .
THEOREM.
&.J
X
be a normal Hausdorff, countably
compact Hewitt-Nachbin space. compact c l o s e d s u r j e c t i o n
is 2Hewitt-Nachbin Proof.
Now l e t
If X
f
i s a fiber-countably
o n t o a space
Y,
then
Y
space.
BY 16.5 t h e space
compact.
from
para-
5
Y
i s normal and countably para-
be a z e r o - s e t u l t r a f i l t e r on
the countable i n t e r s e c t i o n p r o p e r t y .
Y
with
Then t h e c o l l e c t i o n Z E ~ i]s a z e r o - s e t f i l t e r base on X w i t h t h e countable i n t e r s e c t i o n p r o p e r t y . We w i l l prove t h a t F can be embedded i n a Z - u l t r a f i l t e r on X w i t h t h e countable
F
= (f
- 1 (2)
:
COMPLETENESS AND CONTINUOUg,, MAPPINGS
192
\
f i l t e r on
be a
Z-ultra-
G, and l e t { Z i : i c I N ) be an a r b i Ir. S i n c e 1~ i s c l o s e d under
containing
X
LL
To t h i s end, l e t
intersection property.
t r a r y countable s u b c o l l e c t i o n of
f i n i t e i n t e r s e c t i o n s w e may assume t h a t !Zi c r e a s i n g sequence o f z s r o - s e t s i n
1i-m
:
i c m : i s a de-
The c o l l e c t i o n [ f ( Z . ) :
X.
1
i s a sequence of c l o s e d s e t s i n Y . For e v e r y i c m 1 Zr5, Z i f - ( Z ) # @, hence f ( Z i ) 9 Z # @. W e claim
and that
i s non-empty.
!f(Zi) : icI”,
?I
For suppose o t h e r w i s e .
Then t h e n o r m a l i t y and c o u n t a b l e paracompactness of p l i e s t h a t t h e r e e x i s t open neighborhoods that
rOi
: icIN
=
fl
by 8 . 1 4 .
of
Oi
Furthermore,
im-
Y
f ( Z i ) such
s i n c e by Ury-
s o h n l s Lemma any two d i s j o i n t c l o s e d s e t s i n a normal s p a c e a r e completely s e p a r a t e d , i t f o l l o w s t h a t t h e r e e x i s t zero-
sets
n
f(zi)
n
Zil
in
Zil
Z # @ f o r every # @ . Hence Z i t
Z
f o r each
icN.
: icN
fl f Z i l
Z ( Y ) such t h a t
1
= @
c Zit
iclN
and
Zc5
c Oi.
:
Since
it i s the case that
belongs t o the
9 rOi
But
(zi)
f
iclN) = @
implies t h a t
3
c o n t r a r y t o t h e assumption t h a t
the countable i n t e r s e c t i o n property. i c I N ] i s non-empty a s c l a i m e d .
Now,
Therefore,
let
y E
and c o n s i d e r t h e c o u n t a b l e c o l l e c t i o n ! f -
1
(y)
n
n
Zi
(f(Zi)
n
Zi
n
[Zi
:
icmj # @
section property. 1~
n
(f-I(y)
n {z : Z c a ]
x
Hence, s i n c e
5
filter
# @
11
so t h a t
n
i s f i x e d which i m p l i e s t h a t
fore,
:
: iclN
: iclN
1,
of
f-l(y).
: ic7N ) h a s t h e f i n i t e i n t e r s e c t i o n prop-
ert-y i t i s t h e c a s e t h a t fore,
has
rf(Zi)
non-empty c l o s e d s e t s i n t h e c o u n t a b l y compact s p a c e Since ! f - l ( y )
5
2-ultrafilter
n
Zi
: itIN
j # @.
There-
has t h e countable i n t e r -
i s Hewitt-Nachbin c o m p l e t e , [f-l(Z) : ZcZ]
# @.
There-
from which i t f o l l o w s t h a t t h e
i s f i x e d and
Y
Z-ultra-
i s Hewitt-Nachbin c o m p l e t e .
This
c o n c l u d e s t h e proof of t h e theorem. 16.7
COROLLARY ( F r o l f k )
. If
X
i s a normal H a u s d o r f f ,
c o u n t a b l y paracompact, Hewitt-Nachbin s p a c e , and i f p e r f e c t mapping from
X
onto
Y,
Y
f
2
i s a Hewitt-Nachbin
space. The f o l l o w i n g r e s u l t is due t o F r o l i k (196U, Theorem 3.1.2)
.
However, Froll/k’ s v e r s i o n assumes t h e h y p o t h e s i s
PERFECT MAPPINGS
193
based on a z e r o - s e t p r e s e r v i n g s u r j e c t i o n t h a t i s f i b e r - r e l a t i v e l y pseudocompact, r a t h e r than t h e f i b e r - coun t a b l y compact /
I n t h e proof F r o l i k u s e s an i n c o r r e c t
c a s e s t a t e d below.
f o r m u l a t i o n t h a t a s u b s e t be r e l a t i v e l y pSeudocompact which e x p l a i n s t h e a l t e r e d version of h i s r e s u l t h e r e . 16.8
If f
THEOREM ( F r o l i k ) .
-o n t o a Tychonoff -
i s a f i b e r - c o u n t a b l y compact
and
zero- s e t p r e s e r v i n q s u r j e c t i o n from a Hewitt-Nachbin s p a c e Proof.
3
Let
space be a
then
Y,
f-l[S] is a on
2 - u l t r a f i l t e r on
f-l[3].
X.
Q
Let
so t h a t
5
# @
f o r each
Now each image
f(zn)
n z p
gj.
ntm
f(Zn) and
Z E ~ ,
Thus, f ( z n ) b e l o n g s
h a s t h e countable i n t e r s e c t i o n property,
y
there e x i s t s a point Zn
has the countable
Moreover, f o r e a c h i n d e x
Y.
f-’(z) n zn # fi Since
Z-ultrafilter
A s i n t h e proof o f 1 6 . 6 w e may assume t h a t
Q.
is a z e r o - s e t i n 3.
Q
We c l a i m t h a t
{ Z n : n e m ] i s a d e c r e a s i n g sequence.
to
be a
: Z E ~ ] . Then
For l e t { Z n : n c l N ) b e a sequence o f
intersection property. zero-sets i n
with the countable
Y
f - l [ S ] = {f-’(Z)
Z - f i l t e r b a s e on
containing
X
i s a Hewitt-Nachbin s p a c e .
Y
i n t e r s e c t i o n p r o p e r t y , and l e t
X
n
t
[f(Zn) : ncm].
Hence,
f-
1
(y)
n
Furthermore, a s { Z n : n c m ) i s de-
nElN.
c r e a s i n g and t h e c o l l e c t i o n i f -
1
( y ) fl Zn : nE’JN ) h a s t h e
f i n i t e i n t e r s e c t i o n p r o p e r t y , the c o u n t a b l e compactness o f f - l ( y ) implies t h a t
f-’(y)
n
(
n
#
Zn)
@.
Therefore,
has
nEm the countable i n t e r s e c t i o n pr oper ty. Nachbin s p a c e t h e r e e x i s t s a p o i n t longs t o that
fl
3
x
since
f (x) E Z
F
f o r every
f-’(Z)
Since
x
E
n
X
Q.
f o r every
is a H e w i t t f ( x ) be-
Then
zt73
which i m p l i e s
Z E ~ . T h i s concludes t h e p r o o f .
The f o l l o w i n g c o r o l l a r i e s f o l l o w immediately from t h e f a c t t h a t e v e r y open p e r f e c t mapping i s
z-open and f i b e r -
c o u n t a b l y compact ( 1 5 . 1 3 ) and hence zero- s e t p r e s e r v i n g (15.14). 16.9
COROLLARY.
If
f
i s a f i b e r - c o u n t a b l y compact
open s u r j e c t i o n from a Hewitt-Nachbin s p a c e
noff space
Y,
then
Y
X
and
z-
o n t o a Tycho-
i s a Hewitt-Nachbin s p a c e .
194
COMPLETENESS AND CONTINUOUS MAPPINGS
16.10
If
COROLLARY.
Hewitt-Nachbin s p a c e
is -
aHewitt-Nachbin
i s an open p e r f e c t mappinq from a
f
o n t o a Tychonoff space
X
Y,
then
Y
space.
p o i n t o u t t h a t V . Ponomarev proved a weaker v e r s i o n
Wle
of 16.10 i n h i s 1959 p a p e r by r e q u i r i n g t h a t t h e s p a c e
X
also be normal. I n 16.7 i t was e s t a b l i s h e d t h a t Hewitt-Nachbin completen e s s i s i n v a r i a n t under p e r f e c t mappings whenever t h e r a n g e s p a c e i s normal Hausdorff and c o u n t a b l y paracompact.
This
r e s u l t h a s been sharpened by N . Dykes i n h e r 1969 p a p e r : r e q u i r e s i n s t e a d t h a t the range be a Hausdorff weak
she
cb-space
(see 14.13(1)). The n e x t r e s u l t i s found i n F r o l f k ' s 1963 p a p e r and w i l l be u s e f u l i n e s t a b l i s h i n g t h e r e s u l t due t o Froll/k's r e s u l t gives the in-
Dykes t h a t was j u s t mentioned.
v a r i a n c e and i n v e r s e i n v a r i a n c e of a l m o s t r e a l c o m p a c t n e s s under p e r f e c t mappings. /
THEOREM ( F r o l i k )
16.11
-and i f
f
is
.
If
are Hausdorff
Y
X
p e r f e c t mapping from
X
spaces
onto
Y,
then t h e
then
Y
i s almost
followinq statements a r e t r u e :
(1)
If
(2)
If
x
i s almost realcompact,
realcompact.
is c o m p l e t e l y
Y
r e g u l a r and a l m o s t realcompact,
i s almost realcompact. Lc be an u l t r a f i l t e r o f open s u b s e t s o f Y L.l = ( c l U : U E ~ h)a s t h e c o u n t a b l e i n t e r s e c t i o n L e t 63 be an u l t r a f i l t e r o f open s u b s e t s o f X then
X
(1) L e t
Proof.
such t h a t property. containing
f-l[L].
I t w i l l be shown t h a t
63
h a s t h e counta-
b l e intersection property. Then t h e r e e x i s t s a sequence
For suppose o t h e r w i s e . (Bi
: i E I N )
t h e family
h
63
in
m
Y.
W
Y = f ( U X\Cl i=1
y
( c l Bi
: iclN
= (Y\f ( c l B i )
i s an open c o v e r o f
and i f
n
such t h a t
1.
: i E l N )
Y \ f ( c l Bi)
=
0.
Define
F i r s t w e w i l l show t h a t
Now,
cn
00
Bi)
=
U f ( X \ C l Bi)
i=1
€or e v e r y
iclN,
3
U Y\f(Cl B i ) , i=l
then
f - l ( y ) meets
PERFECT MAPPINGS
c l Bi
for a l l
1 95
Since we may choose ( B i
i E l N .
t o be
: i E l N )
a d e c r e a s i n g sequence because of t h e f i n i t e i n t e r s e c t i o n prop-
8, t h e compactness of
e r t y of
f-’(y)
insures t h a t
00
n ( f - l ( y ) 0 c l Bi) i=l ( c l Bi
1
: ic3N
# @
/6.
=
c o n t r a r y t o t h e assumption t h a t
c o u n t a b l e open c o v e r i n g of
Y
i f f o r each
Y \ f ( c l Bi)
icN
then f o r each
the s e t
n
:
Ui
/6.
=
1
icm
c
n
1
ui
lcl
-
)€IN.
n
8.
belongs t o Bj
n
that
63
is a f i l t e r .
Ir such c f(c1 Bi),
pr
f o r some index
k
f-I(Y\f(cl B j ) ) = X\f-l(f(cl B j ) ) B.)) c X\cl B. i n which 7 3 This c o n t r a d i c t s t h e f a c t
However, X \ f - ’ ( f ( c l
case
E
Ui
( f ( c 1 B ~ ): i c m } =
3
I t follows t h a t
Ui
Li,
has the countable i n t e r s e c t i o n
Li
Hence, Y \ f ( c l B . ) belongs t o
property.
Next observe t h a t
i t i s the case t h a t
Y
: icmj c
contrary t o the f a c t t h a t
a
f a i l s t o belong t o
This i m p l i e s t h a t
f ( c l B . ) i s closed i n
and s i n c e
n [ui
a s claimed.
t h e r e e x i s t s an element
i6I.N
that [Y\f(cl Bi)]
m is
Therefore, i t follows t h a t
[X\f-l(f(cl B j ) ) ] =
8.
-
Therefore, 8
has the countable i n t e r -
section property a s a s s e r t e d . Now, s i n c e c o n t i n u i t y of
pact
f(xo)
E
.
(2)
Let
n 5.
x
0
E
#
@.
By t h e
( c l f- 1 (U) : U E L ) #
8.
U) : U E L ] which
fl ( f - ’ ( c l
i s almost realcom-
Therefore, Y
denote t h e c o l l e c t i o n of a l l c o u n t a b l e open cov-
y
e r i n g s of
n
f , t h i s implies t h a t
Hence, t h e r e e x i s t s a p o i n t implies t h a t
n
i s almost realcompact,
X
Y.
Since
i s almost realcompact and completely
Y
i s complete by 1 4 . 5 ( 2 ) .
regular, y
I t w i l l be shown t h a t
U ~ U ]: I J E y j i s a complete family of countaf-l[yl = ( [ f - l ( U ) b l e open c o v e r i n g s o f X . To s e e t h i s , suppose t h a t 3 i s a :
f-’[y]-Cauchy UcU
and
which c a s e y
n
family.
Then f o r each
A E ~such t h a t f [5] = ( f (A)
A c f-l(U).
y-Cauchy f a m i l y .
n
( c l f(A) : A E 3 j =
: A E ~ )i s non-empty.
n
( f ( c l A)
p
every
~ € 3 .Hence, f - l ( p )
f - l ( y ) i s compact.
:
n [n
Therefore,
in Since
Therefore, t h e r e e x i s t s a
~ € 3 so ) that
point
E
there e x i s t s e t s
Thus, f ( A ) c U
: A E ~ )i s a
i s complete, t h i s i m p l i e s t h a t ( f ( c l A)
Ucy
f-l(p)
n
( c l A : AES)] #
n5#
so t h a t
cl A
# pr
for
since f-l[y] is
COMPLETENESS AND CONTINUOUS MAPPINGS
196
complete a s a s s e r t e d .
If
y'
that
denotes the c o l l e c t i o n o f a l l
y'
c o u n t a b l e open c o v e r i n g s o f
i s complete s o t h a t
f- 1 [ y ] c
then
X,
X
I t follows
yl.
i s a l m o s t r e a l c o m p a c t con-
c l u d i n g t h e proof o f t h e theorem.
I n h i s o r i g i n a l p a p e r , F r o l c k (1963, page 136) s t a t e d t h a t he d i d n o t know o f an example o f an a l m o s t r e a l c o m p a c t s p a c e t h a t i s n o t a Hewitt-Nachbin s p a c e .
However, u t i l i z i n g
t h e p r e v i o u s r e s u l t 16.11(1) t o g e t h e r w i t h Example 1 6 . 4 , we can now p r o v i d e such an example. 16.12
An almost realcompact space t h a t f a i l s t o be
EXAMPLE.
Hewitt-Nachbin complete.
I n 1 6 . 4 we p r e s e n t e d a s p a c e
t h a t i s t h e union of
X
h, and b,, b u t t h a t Next w e formed f a i l s i t s e l f t o be a Hewitt-Nachbin s p a c e .
two Hewitt-Nachbin non-normal s p a c e s t h e Hewitt-Nachbin s p a c e
n2.
and to and
I t was p o i n t e d o u t t h a t t h e mapping
cp
from
hl
t o i t s copy i n
p2
Y
X
i s a p e r f e c t mapping.
on-
Y
which t a k e s each p o i n t of t h e d i s j o i n t union o f
X
14.11
X
a s t h e t o p o l o g i c a l sum o f
Y
hl
Now, by
i s a l m o s t r e a l c o m p a c t which i m p l i e s b y 16.11(1) t h a t
i s almost realcompact.
Therefore, the space
X
of 16.4 i s
an a l m o s t realcompact s p a c e t h a t i s n o t a Hewitt-Nachbin s p a c e . The n e x t theorem i s one of t h e main r e s u l t s o f t h i s sect i o n and i s the r e s u l t due t o Dykes t h a t was r e f e r r e d t o p r i o r t o the statement of 16.11. 16.13
Let
THEOREM ( D y k e s ) .
Hewitt-Nachbin space
-a -weak
cb-space,
then
Proof.
By 1 4 . 1 1
X
16.11(1)
Y
f
2 p e r f e c t mapping from a
o n t o a Tychonoff s p a c e
X
Y.
If
Y
&
Y i s a Hewitt-Nachbin s p a c e . i s a l m o s t r e a l c o m p a c t , and hence by
i s almost realcompact.
Therefore, Y
is Hewitt-
Nachbin complete by 1 4 . 1 6 . 16.14
COROLLARY.
Nachbin space then Proof.
Y
X
If
f
is 2 p e r f e c t
mappinq from a H e w i t t -
o n t o a pseudocompact Tvchonoff s p a c e
Y,
i s a Hewitt-Nachbin s p a c e . By 1 4 . 1 3 ( 9 )
Y
i s a weak
cb-space.
The r e s u l t i s
197
PERFECT MAPPINGS
now immediate from t h e theorem. I n 1 7 . 2 0 w e w i l l p r e s e n t a r e s u l t due t o B l a i r t h a t i s
B l a i r ' s r e s u l t requires
v e r y s i m i l a r t o Dykes' r e s u l t 1 6 . 1 3 .
the s t r o n g e r c o n d i t i o n t h a t t h e range space be a
cb-space
r a t h e r than weak cb-, b u t t h e mapping f i n h i s r e s u l t need only b e f i b e r - c o u n t a b l y compact and z - c l o s e d r a t h e r t h a n p e r Hence, i f o n e i s i n t e r e s t e d i n a c l a s s o f Tychonoff
fect.
spaces contained w i t h i n t h e c l a s s of
cb-spaces,
then B l a i r ' s
r e s u l t i s p r e f e r a b l e i n t h a t i t demands fewer c o n d i t i o n s t o be imposed on t h e mapping.
O n t h e o t h e r hand,
i f t h e primary
concern i s w i t h a c l a s s of mappings, t h e n Dykes'
r e s u l t is
b e t t e r i n t h a t i t demands a weaker c o n d i t i o n t o be imposed on t h e range s p a c e . The n e x t r e s u l t s a r e Theorems 8.17 and 8.18 o f Gillman and J e r i s o n , r e s p e c t i v e l y .
W e i n c l u d e them h e r e f o r t h e s a k e
of completeness. THEOREM (Gillman and J e r i s o n ) .
16.15
-t i o n s on
2 Tychonoff s p a c e
Y
are e q u i v a l e n t :
(1) For e a c h Tychonoff s p a c e
f i b e r - c o m p a c t mappinq
-i s Hewitt-Nachbin
f o l l o w i n g condi-
f
X,
i f there exists a
from
onto
Y
i s a continuous
Y,
complete.
Every Tychonoff s p a c e o f which
(2)
then x
X
i n j e c t i v e imaqe i s a Hewitt-Nachbin s p a c e . Every subspace
(3)
of
Y
i s a Hewitt-Nachbin
space.
Proof.
For e a c h p o i n t Y E Y , s u b s p a c e Y\[y} i s H e w i t t Nachbin complete. I t i s immediate t h a t (1) i m p l i e s ( 2 ) , and t h a t ( 3 )
implies
(4).
(4)
(2) implies ( 3 ) :
Let
b e an a r b i t r a r y subspace o f
F
Y
and
e n l a r g e t h e topology on
Y
t h e c l a s s o f open s e t s .
I t i s e a s y t o v e r i f y t h a t t h e new
space
X
Y.
F
and
Y\F
to
t h u s o b t a i n e d i s c o m p l e t e l y r e g u l a r and t h e r e l a t i v e
topology on from
by a d j o i n i n g b o t h
F
in
X
i s t h e same a s t h e r e l a t i v e topology
S i n c e t h e i d e n t i t y mapping from
continuous, Therefore, F
(2) i m p l i e s t h a t
X
X
into
Y
is
i s Hewitt-Nachbin complete.
i s Hewitt-Nachbin complete s i n c e i t i s a c l o s e d
198
COMPLETENESS AND CONTINUOUS MAPPINGS
subspace of
X.
( 4 ) implies (1):
Let
and
X
s a t i s f y t h e h y p o t h e s i s of
f
(1). By ( 4 ) Y i s a Hewitt-Nachbin space because i t i s t h e u n i o n of a compact space ( y ] with a Hewitt-Nachbin space Y\{y] (8.13(1)). Therefore, f h a s a continuous e x t e n s i o n f v from
UX
into
Y.
y
Let
be any p o i n t i n
By 8.10(6)
Y.
i s a Hewitt-Nachbin subspace
the i n v e r s e image [f"]-'(Y\(y))
uX. Hence, by 8.13(1) the union [ f u ] - ' ( Y \ ( y ] ) U f-'(y) i s a Hewitt-Nachbin subspace of uX. Since t h i s space l i e s between X and uX i t m u s t be uX i t s e l f b y 8 . 2 ( 2 ) . In o t h e r words, f v sends no p o i n t of uX\x into y . As t h i s holds t r u e f o r every p o i n t Y E Y , i t follows that. uX\X = !d of
concluding t h e proof of t h e theorem. 16 .16
COROLLARY
ous i n j e c t i o n space
Y,
(Gillman and J e r i s o n ) .
from a Tychonoff space
and if every subspace
Because
f
of
&2
f
continu-
o n t o a Tychonoff
i s Hewitt-Nachbin
Y
a-
i s Hewitt-Nachbin complete. i s i n j e c t i v e , i t i s the c a s e t h a t f - 1 ( y )
p l e t e , then e v e r y subspace
Proof.
of
X
If
i s compact f o r each p o i n t
ycY.
X
Since every subspace o f
Y
i s a Hewitt-Nachbin space,by (1) of t h e theorem it follows
that
i s Hewitt-Nachbin complete.
X
space of over,
X.
Since
F be any sub- 1 ( f ( F ) ) . More-
Now, l e t
is injective,
f
F = f
f ( F ) i s Hewitt-Nachbin complete because i t i s a sub-
space of
Y
so that
F
i s Hewitt-Nachbin complete by 8.10(6).
Section 17:
Closed Mappinqs and Hewitt-Nachbin Spaces I n the preceding s e c t i o n i t was observed t h a t t h e prope r t y of Hewitt-Nachbin completeness f a i l s t o be i n v a r i a n t under p e r f e c t mappings ( 1 6 . 4 ) . However, i t was e s t a b l i s h e d t h a t such i s t h e case i f t h e mapping i s a l s o open ( 1 6 . 1 0 ) , o r i f t h e range space i s a Tychonoff weak
cb-space
(16.13)-
In
t h i s s e c t i o n t h e i n v a r i a n c e of Hewitt-Nachbin completeness under closed mappings w i l l be s t u d i e d when s t r o n g e r c o n d i t i o n s a r e imposed on t h e range space t o compensate f o r t h e loss of t h e property of fiber-compactness €or t h e mapping.
One
r e s u l t t h a t w i l l be e s t a b l i s h e d , which i s due t o R . L. B l a i r (1969) , g i v e s t h e i n v a r i a n c e of Hewitt-Nachbin
completeness
199
CLOSED MAPPINGS
under a c l o s e d c o n t i n u o u s s u r j e c t i o n p r o v i d e d t h a t t h e r a n g e i s a f i r s t c o u n t a b l e Tychonoff
cb-space
w i l l b e sharpened by a theorem o f N .
(17.15).
That r e s u l t
Dykes i n 1 7 . 1 4 .
B l a i r ' s n o t i o n o f a " h y p e r - r e a l map" w i l l a l s o b e i n t r o duced, and i t w i l l be e s t a b l i s h e d t h a t Hewitt-Nachbin completen e s s i s i n v a r i a n t under h y p e r - r e a l maps ( 1 7 . 1 7 ( 1 ) ) .
It is
i n t e r e s t i n g t h a t t h e p r o p e r t y of pseudocompactness i s i n v e r s e i n v a r i a n t under h y p e r - r e a l maps ( 1 7 . 1 7 ( 2 ) ) ,
I t w i l l b e shown
t h a t e v e r y f i b e r - c o u n t a b l y compact and zero- s e t p r e s e r v i n g mapping i s h y p e r - r e a l
(17.19)
.
I n (1967, Theorem 7 . 5 ) , T . I s i w a t a p r o v e s t h a t
i s 5closed
if
f
c o n t i n u o u s mapping from a l o c a l l y compact, counta-
bly paracompact, normal Hausdorff s p a c e X o n t o a Tychonoff space Y , then Y i s a Hewitt-Nachbin s p a c e whenever X & a Hewitt-Nachbin --
space.
A proof
f o r t h i s r e s u l t was o b t a i n e d
i n t h e f o l l o w i n g way.
I t was f i r s t e s t a b l i s h e d t h a t a c l o s e d
c o n t i n u o u s mapping
from a Hewitt-Nachbin,
normal s p a c e where
?.
X
f
onto
Y
Z
onto
Y.
Hewitt-Nachbin, Y ; whence
Z
and
$
Therefore,
I)
X
f =
P,
Z
i s normal and counta-
i s a p e r f e c t mapping from a
normal and c o u n t a b l y paracompact s p a c e Y
$ 0
o n t o a nor-
i s a p e r f e c t mapping from
By 1 6 . 5 i t f o l l o w s t h a t
b l y paracompact. to
admits a f a c t o r i z a t i o n
i s a c l o s e d c o n t i n u o u s mapping from
mal Hewitt-Nachbin s p a c e
l o c a l l y compact,
i s Hewitt-Nachbin complete by 1 6 . 6 .
Z
on-
N.
Dykes g e n e r a l i z e s t h e above r e s u l t by r e q u i r i n g o n l y t h a t t h e image s p a c e b e a normal H a u s d o r f f , weak
cb-, k-space.
Isi-
w a t a ' s r e s u l t t h e n f o l l o w s immediately s i n c e e v e r y l o c a l l y compact space i s a
k-space,
paracompact s p a c e i s a weak
and e v e r y normal and c o u n t a b l y cb-space.
The r e s u l t o f Dykes
w i l l be e s t a b l i s h e d a f t e r t h e f o l l o w i n g t h r e e lemmas.
The
f i r s t o f t h e s e i s due t o A . A r h a n g e l s k i i (1966B, Lemma 1 . 2 ) and i s o f a t e c h n i c a l n a t u r e . (Arhangelskii) . J & Y b e 2 Hausdorff k - s p a c e , be a p o i n t - f i n i t e open c o v e r i n q fo L H a u s d o r f f space X , and l e t f @ e g c o n t i n u o u s c l o s e d s u r j e c t i o n from X -0 17.1
let
Y.
LEMMA
N
T a t & &
COMPLETENESS AND CONTINUOUS MAPPINGS
200
D = f y c y : no f i n i t e
c v.
Kt
covers
f-l(y)j
i s d i s c r e t e in
Y. Suppose t h a t some p o i n t
Proof.
point for
i s an a c c u m u l a t i o n
ycY
D1 = D\!y)
Then t h e s e t
D.
F c Y
f o r e , t h e r e e x i s t s a Compact s e t
There-
is not closed.
F fI D1
such t h a t
is
n o t c l o s e d , and hence i n f i n i t e .
L e t fyn : n c N ] b e a s e q u e n c e F n D1 and assume w i t h o u t l o s s of g e n e r a l i t y t h a t t h e p o i n t s a r e d i s t i n c t . S i n c e F i s compact t h i s se-
o f p o i n t s from
quence h a s an accumulation p o i n t yo t h a t b e l o n g s t o F . L e t f o r e a c h n ~ m For each X E X , l e t M ( X ) d e n o t e t h e union of a l l s e t s i n u t h a t c o n t a i n t h e p o i n t x . W e
.
An = f - ' ( y n )
d e f i n e a sequence [ x x1
A1.
in
: ncN
n
I f [ xl,
inductively a s follows:
. . . ,xm-1]
Select
have been o b t a i n e d w e choose
a s any p o i n t b e l o n g i n g t o t h e s e t
Am\
t h i s l a t t e r s e t i s non-empty s i n c e
u
m- 1 U u(xi). i=l
x m
Note t h a t
is point-finite.
I t w i l l n e x t b e e s t a b l i s h e d t h a t t h e sequence { x n : n c m
is discrete.
Consider any p o i n t
x
o n l y c o n s i d e r t h e c a s e i n which empty. I f xm E u ( x ) , a neighborhood of x . that
then
x
x ( x m ) so t h a t
E
xn
satisfying
t h e d i s c r e t e n e s s of {xn : n c m ) i s proved. P = (x
W e need
X.
fl ( x n : ncN )
i s non-
U = u(x )
m
is
I t f o l l o w s from t h e c o n s t r u c t i o n o f
can c o n t a i n o n l y p o i n t s
U
belonging t o
K(X)
nclN] i s c l o s e d .
n
m.
xm Thus,
I t follows t h a t
f(P) = n ( y n : nEm ] i s n o t c l o s e d because yo b e l o n g s t o c l f (P)\f (P) T h i s i s a c o n t r a d i c t i o n , and t h e r e f o r e w e may c o n c l u d e t h a t D :
On t h e o t h e r hand,
h a s no accumulation p o i n t s .
T h i s completes t h e proof o f t h e
lemma.
Some n o t a t i o n w i l l b e u s e f u l t h r o u g h o u t t h e remainder o f If f i s a c o n t i n u o u s mapping from a Tychonoff i n t o a Tychonoff s p a c e Y , l e t f p d e n o t e i t s S t o n e
t h i s chapter. space
X
e x t e n s i o n from
px
into
py.
The next r e s u l t i s found i n I s i w a t a ' s 1 9 6 7 p a p e r . 17.2
LEMMA ( I s i w a t a )
.
If
1
f
j e c t i o n from a Tvchonoff s p a c e
i s a continuous X
z-closed sur-
o n t o a Tychonoff space
Y,
.
CLOSED MAPPINGS
then
cl
f - l ( y ) = [fP]-’(y)
PX Let
Proof.
201
f o r every p o i n t YEY. P -1 1 be a n a r b i t r a r y p o i n t o f [ f ] (y)\clPxf- (y)
p
Then t h e r e i s a f u n c t i o n f o r all
h(x) = 1
x
x n
M =
such t h a t
h E C(PX)
cl
E
PX
0
f - l ( y ) , and
(X E
contain the point E
clPxM.
c l P y f (M)
y.
Hence,
.
Since
Therefore, y
L.
The s e t
1
PX : h ( x )
z - c l o s e d and Y and d o e s n o t
h(p) = 0
On t h e o t h e r hand,
so t h a t
y = f P ( p ) E f P ( c lP f l )c c l f p ( M ) = PY f(M) i s c l o s e d i n Y, c l f(M) n Y = f ( M ) PY
T h i s is a c o n t r a d i c t i o n .
f (M).
E
h ( p ) = 0,
h
i s a z e r o - s e t i n X . Moreover, s i n c e f i s M fl f - l ( y ) = @, t h e image f(M) i s c l o s e d i n p
.
[ f ’ ~ - ~ ( y\ c ) l p X f - l ( y ) i s empty f o r e v e r y p o i n t
.
Therefore, ~
E
Y
completing
t h e argument.
cl f d l ( y ) = PX i n t h e r a n g e a r e c a l l e d ”WZ-map-
Mappings which s a t i s f y t h e c o n d i t i o n [ f P ] - l ( y ) f o r every p o i n t
y
p i n g s ” b y I s i w a t a . These mappings, and t h e i r r e l a t i o n s h i p t o Hewitt-Nachbin c o m p l e t e n e s s , w i l l b e s t u d i e d i n t h e n e x t section. 17.3
Let
LEMMA.
f
b e a c l o s e d c o n t i n u o u s s u r j e c t i o n from a
Tychonoff space
X
zero-set --Crete i n --
and i f
in
PX
o n t o a Wchonoff Z c pX\X,
k-space
then
Y.
fP(Z)
If
Y.
n
Y
z
is a
is dis-
*
( P X ) such t h a t Z = 1 Z ( g ) and 0 g 7 For e a c h n E l N , s e t Un = ( X G X : < n + 2 1 g ( x ) < ;]. C l e a r l y , K = (un : nEm ) i s a p o i n t - f i n i t e open Moreover, by 1 7 . 1 t h e s e t D = ( Y E Y : no c o v e r i n g of X . f i n i t e H ’ c x c o v e r s f - l ( y ) ] i s d i s c r e t e i n Y . To comp l e t e t h e p r o o f i t w i l l b e shown t h a t D = f P ( 2 ) n Y . To see t h i s , l e t ycY. 1 f y p f P ( Z ) , t h e n [ f P ] - ’ ( y ) n Z = 16. S i n c e g must assume i t s infimum on compact s u b s e t s , t h i s i m p l i e s t h a t i n f ( g ( x ) : x E [ fP ] - 1 ( y ) ) = a > 0. T h e r e f o r e , -1 inf(g(x) : x E f ( y ) ] 2 a . Hence, f - l ( y ) can b e covered a f i n i t e s u b f a m i l y 1c’ C % . Therefore, y/D so t h a t D C f p ( Z ) n Y. Conversely, i f y E fp(Z)\D, then t h e r e exists Proof.
If
Z c PXB,
1
.
let
g
be i n
C
-
COMPLETENESS AND CONTINUOUS MAPPINGS
202
an
a
>
0
such t h a t
0
<
<
a
g(x)
< 71
whenever
x
E
f - 1( y ) .
G n f - l ( y ) = @. Moreover, t h e r e i s a p o i n t p t Z such t h a t f P ( p ) = y . H e n c e , prG. B u t p E [ f P 1 - 1 ( y ) = c l P xf - l ( y ) ( 1 7 . 2 ) because f i s a c l o s e d mapping. T h i s i s a c o n t r a d i c t i o n which c o n c l u d e s t h e
Then, G = g - ' [ ( - l , a ) ]
i s an open s e t and
proof. The f o l l o w i n g i d e a s w i l l b e u t i l i z e d i n e s t a b l i s h i n g t h e main r e s u l t 1 7 . 1 0 . 17.4
A map i s s a i d t o be minimal i n c a s e t h e
DEFINITION.
image o f e v e r y p r o p e r c l o s e d s u b s e t o f t h e domain i s a p r o p e r s u b s e t of t h e range s p a c e . 17.5
(1) I t i s shown i n t h e 1967 p a p e r by D .
REMARKS.
Strauss that
if
d p e r f e c t mapping from
f
---
then there exists 5 c l o s e d subspace
restriction
fix, is 2
T o see t h i s ,
let
minimal map o n t o
X
onto
Y,
such t h a t t h e
c X
Y.
d e n o t e any c h a i n of c l o s e d s u b s e t s of
t h a t a r e mapped o n t o
r,
Xo
Y
by
f.
Then
I'
X
is non-empty b e c a u s e
'
n [F : F t r ) i s c l o s e d , I f ycy and F c r , then f - l ( y ) n F # fi. H e n c e , s i n c e f - l ( y ) i s compact, i t f o l l o w s t h a t f - I ( y ) n Xot # @
X
belongs t o
and t h e r e f o r e
and moreover t h e s e t
xO
=
The r e s u l t i s now an immediate
f ( X o t ) = Y.
consequence of Zorn' s Lemma. (2)
T h e n e x t c o n c e p t of a "normal"
(upper o r lower)
semi-continuous f u n c t i o n was f i r s t i n t r o d u c e d by R. P. D i l worth i n h i s 1950 p a p e r .
S i n c e t h a t t i m e the l a t t i c e p r o p
e r t i e s of t h e s e f u n c t i o n s have b e e n s t u d i e d , and t h e r e h a s been some i n v e s t i g a t i o n o f t h e a l g e b r a i c s t r u c t u r e of normal f u n c t i o n s (see K . H a r d y ' s 1970 p a p e r ) .
Moreover, t h e normal
semi-continuous f u n c t i o n s p l a y an i m p o r t a n t and i n t e r e s t i n g r o l e i n t h e s t u d y o f weak
c b - s p a c e s j u s t a s t h e semi-continu-
ous f u n c t i o n s i n t h e s t u d y of
(see J . Mack's 1965 p a p e r , Theorem 1) For example, i n (1967, Theorem 3 . 1 ) , Mack and Johnson have shown t h a t 2 t o p o l o q i c a l s p a c e X i s a weak cb-space i f and o n l y if g i v e n a p o s i t i v e ( m - v a n i s h i n q ) norcb-spaces
.
--mal lower semi-continuous
function
g
%
X,
there exists
CLOSED MAPPINGS
f E C ( X ) such t h a t
0
<
f (x)
20 3
g ( x ) f o r each
(The
XEX.
cb-space a s g i v e n i n Theorem 1 of t h e
characterization for a
Mack and Johnson p a p e r i s t h e i d e n t i c a l s t a t e m e n t w i t h t h e word "normal" d e l e t e d . )
D i l w o r t h o b t a i n e d two u s e f u l c h a r a c -
t e r i z a t i o n s of normal semi-continuous f u n c t i o n s i n 1 9 5 0 .
For
p u r p o s e s of s i m p l i c i t y , we w i l l t a k e one o f t h e s e c h a r a c t e r i z a t i o n s a s our d e f i n i t i o n . 17.6
A lower ( r e s p e c t i v e l y , u p p e r ) semi-continu-
DEFINITION.
ous f u n c t i o n
f
on an a r b i t r a r y t o p o l o g i c a l s p a c e
t o b e normal i f e v e r y
i
)
0, pcX, and open s e t
p, t h e r e e x i s t s a non-empty open s e t
A c G
X
is said
containing
G
such t h a t
f ( p ) + E (respectively, f ( y ) f ( p ) - E ) whenever w i l l say t h a t f i s normal s e m i - c o n t i n u o u s i f f
f(y)
YEA.
<
We
i s normal
lower s e m i - c o n t i n u o u s o r normal upper s e m i - c o n t i n u o u s .
one o f t h e u s e f u l f e a t u r e s o f normal s e m i - c o n t i n u o u s f u n c t i o n s i s t h a t t h e y a r e determined on dense s u b s e t s .
This
is the c o n t e n t of t h e next r e s u l t . 17.7
If
THEOREM.
f
4
are normal
g
semi-continuous
f u n c t i o n s on an a r b i t r a r y t o p o l o q i c a l s p a c e f = g
D c X,
on t h e d e n s e s u b s e t
such t h a t
X
f (x) = g ( x )
for
x(X I
every p o i n t Proof.
then
W e w i l l prove t h e theorem f o r lower semi-continuous
f u n c t i o n s , t h a t f o r upper s e m i - c o n t i n u o u s f u n c t i o n s b e i n g e n t i r e l y s i m i l a r w i t h t h e obvious m o d i f i c a t i o n s . ptX.
the set p
f ( p ) # g ( p ) assume t h a t
If
U = (x : f(x)
because
f
f ( p ) - $1
\
tain the point D
- g(p)
v c
U
p) such t h a t
i s dense,
r
v n
D
# r
(V
g(x)
a.
Let
=
Hence, l e t
r
>
0.
Then
i s a n open neighborhood o f
i s lower s e m i - c o n t i n u o u s .
t h e r e e x i s t s an open s e t since
f(p)
Since
g
i s normal,
d o e s n o t n e c e s s a r i l y con-
<
g(p) xo c
+ whenever X E V . v n D . Then f ( x0 )
3
g(xo) < g(p) + 3 < f (p) - 5 < f ( x o ) . S i n c e g i s lower semi-continuous and f i s normal, an a n a l o g o u s argument a p p l i e s i f g ( p ) - f ( p ) = r > 0 . Hence, f ( p ) = g (p) f o r every
pcX
=
This is a contradiction.
completing t h e argument.
2 04
COMPLETENESS AND CONTINUOUS MAPPINGS
Given a c o n t i n u o u s mapping from one t o p o l o g i c a l s p a c e t o a n o t h e r and any c o n t i n u o u s r e a l - v a l u e d f u n c t i o n on t h e domain s p a c e which i s bounded i n some way ( t o b e made p r e c i s e f u r t h e r o n ) , t h e r e w i l l be two a d d i t i o n a l mappings t h a t a r e induced on t h e range s p a c e i n r a t h e r a " n a t u r a l way". The f o l l o w i n g d e f i n i t i o n and accompanying lemma (see Mack and Johnson, 1967, Theorem 2 . 1 ) i n t r o d u c e t h e s e mappings and s p e c i f y some of t h e i r p r o p e r t i e s . They w i l l be u t i l i z e d i n e s t a b l i s h i n g 17.10 L e t h be a c o n t i n u o u s mapping from a 17.8 DEFINITION. space X o n t o a s p a c e Y , and l e t f E C ( X ) be such t h a t f i s bounded on each f i b e r h - l ( y ) f o r ycY. D e f i n e t h e followi n g functions : f
i
E
h
(Y) = s U p ( f ( x ) : x
E
1 h- (y)}.
and s f
-1
(Y) = i n f ( f ( x ) : x
(y))
S c h e m a t i c a l l y , a diagram i l l u s t r a t i n g t h e above d e f i n i t i o n would have t h e f o l l o w i n g form:
x-Y
h The f o l l o w i n g lemma w i l l b e u s e f u l , and s p e c i f i e s some o f t h e c h a r a c t e r i s t i c s o f t h e mappings d e s c r i b e d i n t h e p r e v i ous d e f i n i t i o n . 17.9
LEMMA
1 7 . 8,
(Mack and J o h n s o n ) .
With t h e h y p o t h e s i s a s i n
followinq statements a r e t r u e : (1) (2)
If
h i s an open mapping, t h e n f S ( r e s p e c t i v e l y , f i ) i s lower ( r e s p e c t i v e l y , u p p e r ) s e m i - c o n t i n u o u s . If h i s a c l o s e d mapping, t h e n f1 ( r e s p e c t i v e l y , f s ) i s lower ( r e s p e c t i v e l y , u p p e r ) s e m i - c o n t i n u o u s . addition, h minimal and f i b e r - c o m p a c t ,
If,
proof.
then -
fi
i s normal lower s e m i - c o n t i n u o u s .
(1) W e w i l l e s t a b l i s h t h e theorem f o r
fS, the c a s e
CLOSED MAPPINGS
for
205
b e i n g e n t i r e l y s i m i l a r with t h e obvious m o d i f i c a t i o n s .
fi
H e n c e , i t must b e shown t h a t f o r each p o i n t
>
E
t h e r e e x i s t s an open neighborhood
0
yo
and
Y
E
of
U
yo
such t h a t
S
f s ( y ) > f ( y o ) - E (see Dugundji, Chapter 111, Problem 5 , page 9 5 ) . H e n c e , choose a p o i n t x E h- 1 ( y o )
implies t h a t
YEU
such t h a t
>
f(xo)
f S (yo)
5.
-
0
Since
t h e r e e x i s t s an open neighborhood then
XEW
f(x)
>
f(xo) -
open neighborhood o f
E 7
yo.
.
f
i s continuous,
of
W
Since
xo
such t h a t i f h(W) i s a n
i s open,
h
Moreover, i f
y
E
h (W)
,
then
n w it is the h - l ( y ) n w # @. Thus, f o r some x E h-'(y) t case t h a t f ( x ) > f(xo) - 7 > f S ( y ) - E . Hence, f s ( y ) > 0
S
(Yo) - E . ( 2 ) For t h i s p a r t w e w i l l e s t a b l i s h t h e r e s u l t f o r
f
for
fS
let
E
>
fi, t h a t being e n t i r e l y s i m i l a r . L e t y E Y be a r b i t r a r y , - 1 ( y o ) . For e a0c h p o i n t X E F choose 0 , and l e t F = h
an open neighborhood
of
U
5.
x
such t h a t
aEU
implies
Denote by U ' f(x) - 7 < f(a) < f(x) + t h e union of a l l such neighborhoods U a s x r a n g e s o v e r F . S e t 1 V = U (h-'(y) : h- ( y ) c U'). Then V = h [Y\h ( X \ U f ) ] and hence i s an open s u b s e t o f X s i n c e h i s c l o s e d . Next i t i w i l l b e shown t h a t y E h(V) i m p l i e s t h a t f ( y ) > f i ( y o ) - E . For i f y E h ( v ) , t h e n y p! h(X\Ut ) and hence h - l ( y ) n E
(X\Ut)
a
= @.
U(x) where
E
h-l(y) c U'
Thus
E
a E h-'(y).
Now choose
i s a neighborhood o f
U(x)
xcF
Then
on which
f
5 < fi ( a ) < f ( x ) + 7 .
5.
Hence, f ( x ) h-l(y0) implies t h a t f ( x )
v a r i e s by l e s s t h a n Moreover, x
.
5.
5
E
2
f
(yo).
Hence,
f (a) > f (x) 2 f i (yo) Since t h e l a t t e r i n e q u a l i t y holds f o r every p o i n t a E h-'(y), i t follows t h a t f i ( y ) > i f (yo) - E a s c l a i m e d . I f yo b e l o n g s t o h(X\V), t h e n
n
h-l(yo) (X\V) i s non-empty c o n t r a r y t o t h e f a c t t h a t h-'(yo) c V . F i n a l l y , s i n c e yo b e l o n g s t o t h e open s e t Y\h(X\V), and
h(X\V)
i t follows t h a t yo
E
yo
[Y\h(V) ] b e c a u s e
3
Y\h(X\v) c h ( v ) .
E
i n t h(V) so t h a t
fi
be a r b i t r a r y , l e t
E
fi
>
is surjective,
is l o w e r semi-continuous.
Next w e w i l l assume t h a t p a c t , and prove t h a t
h
Therefore,
h
i s minimal and fiber-com-
i s normal.
0 , and l e t
xo
To t h i s e n d , l e t E
yo E Y
h - l ( y 0 ) be such t h a t
COMPLETENESS AND CONTINUOUS MAPPINGS
206
.
i
( y o ) = f (x,) The l a t t e r c h o i c e i s p o s s i b l e because c o n t i n u o u s f u n c t i o n s assume t h e i r infimum on compact s e t s . NOW, l e t f
U
be an open neighborhood of
yo
v
and d e f i n e
=
: f ( x ) < f ( x ) + Ll. ‘Then V i s an open neighbor0 2 xo. S i n c e h i s c l o s e d and m i n i m a l , t h e s e t U ’ = Y‘\h ( X \ V ) i s non-empty and open i n Y . I f y E U ’ , then h-’(y) fl ( X \ V ) = @ so t h a t h - l ( y ) c V . Since v c h - l ( u ) ;X
.-
h-’(U)
hood of
i t follows t h a t then
h-’(y)
f(xo)
+
Hence,
Hence, U’ c U .
c h-l(U).
5. fi
ycU.
Finally, i f
y
i U’,
Hence, x t h - l ( y ) i m p l i e s f ( x ) i E i f (y) f ( x0 ) + 2 < f ( y o )
< +
I t follows t h a t
F.
i s normal by 1 7 . 6 , completing t h e proof o f t h e
lemma. The n e x t theorem i s one o f t h e main r e s u l t s o f t h i s c h a p t e r c o n c e r n i n g t h e i n v a r i a n c e o f Hewitt-Nachbin completeI t a p p e a r s a s Theorem 2 . 4 i n
n e s s under c o n t i n u o u s mappings. N.
Dykes’ 1969 p a p e r .
Lat
THEOREM ( D y k e s ) .
17 . l O
j e c t i o n from a s p a c e k-space
If
Y.
be a c l o s e d c o n t i n u o u s K -
f
o n t o a normal H a u s d o r f f , weak
X
i s a Hewitt-Nachbin s p a c e , t h e n
X
Y
cb-,
is 2
Hewitt-Nachbin s p a c e . Proof.
The theorem w i l l b e proved by e x h i b i t i n g a f u n c t i o n
rh
C ( P Y ) f o r each p o i n t
in it
r (y)
and
>
0
whenever
t h a t t h e Stone extension
q
t
PY\Y
such t h a t
ytY (see 8 . 8 ( 3 ) ) . fP
~ * ( q= ) 0
F i r s t observe
i s a p e r f e c t map from
pX
onto
Y t h a t i s properly contained i n PY. H e n c e by 1 7 . 5 t h e r e e x i s t s a c l o s e d subspace Xo c P X such t h a t fop = f P lXo is a mini-
PY, f o r otherwise
fP(PX) i s a compact s p a c e c o n t a i n i n g
mal p e r f e c t mapping o n t o PY\Y to that
and a p o i n t X
p
E
PY.
Now, s e l e c t a p o i n t
[ f oP ] - 1 ( 4 ) .
p
Since
t h e r e e x i s t s a non-negative f u n c t i o n h(p) = 0
and
h(x)
>
0
for e v e r y
q
from
does n o t belong h
xcx
in
C ( P X ) such
by 8.8(3).
Define t h e f u n c t i o n , hi(y) = inf[h(x) : x Then
hi
E
[fOP]-’(y)
1.
i s a normal lower semi-continuous f u n c t i o n on
PY
CLOSED MAPPINGS
207
Moreover, Z ( hi ) = f P ( Z ( h ) ).
according t o 1 7 . 9 ( 2 ) .
To see
t h i s l a t t e r e q u a l i t y , suppose t h a t y E Z ( h i ) . Then t h e r e i s a p o i n t x F [ f O P J - l ( y )such t h a t h ( x ) = 0 . S i n c e y = f P ( x ) ,
y c f P ( Z ( h ) ) . Conversely, i f
t h i s implies t h a t
then there e x i s t s a point
Furthermore,
h ( x ) = 0.
x c Z ( h ) such t h a t
# @
P Xo
[fP]-’(y)
Next s e t
x
Y of
Yo = f P ( Z ( h ) )
Z(h)
n
[X\f-’(Y0)J
f o r some
y c Y
neighborhoods Yo,
n
U
that
For e a c h p o i n t
V = $3.
the set
= $3.
p
and
of
€-’(y)
: y F Yo]
p
Hence, F
if q
*
y
E
Yo
p
E
f-l(y)
f/F
of t h e
X
is closed a s claimed. I t i s a c o n t i n u o u s closed b i -
E
referred t o a t
does belong t o
g
does not belong t o
PY\Y
g
g(q) = 0 .
and
7
F i r s t we d e f i n e t h e f u n c t i o n
q
then t h e r e e x i s t s a f u n c t i o n
g(y) = 1
if
*
I f the point
pose t h a t
i s closed,
so t h a t Yo i s a Hewitt-Nachbin s p a c e .
t h e b e g i n n i n g of t h e p r o o f . c l PyYo,
Yo
x
and
Next w e w i l l c o n s t r u c t t h e f u n c t i o n a s follows.
is a d i s c r e t e subset
For s i n c e
i s an open neighborhood i n F.
0
choose a p o i n t
Yo
# x y , t h e n t h e r e e x i s t open
follows t h a t the r e s t r i c t i o n j e c t i o n from F o n t o Y
i
r e s p e c t i v e l y , such Y’ f - l ( y ) i s open by t h e d i s c r e t e n e s s of
V
Since
U
i s d i s c r e t e and c l o s e d
Yo
On t h e o t h e r hand,
and i f
t h a t misses
p
point
0
U
(xy
F =
z ( h ) i s a zero-set
Since
y
i s closed.
Moreover, F
X.
n Y.
t h e space
pX\X,
The s e t
E f-’(y).
cl F
C
by 1 7 . 3 .
Y
is a
x c Xo s u c h P -1 O [fo ] ( y ) : = 0 so
t h a t f o B ( x o ) = y . Hence, i n f { h ( x ) : x c i t h a t h ( y ) = 0 . Therefore, y c Z ( h i ) .
in
f oP
since
I t follows t h a t t h e r e i s a p o i n t
surjection.
satisfying
y c fP(Z(h)), y = f P ( x ) and
E C(pY)
such t h a t
On t h e o t h e r hand,
Since
clPyYo.
sup-
i s normal,
Y
Yo i s C -embedded i n Y and hence @Yo = c l PyYo (see 6 . 9 ( a ) i n Gillman and J e r i s o n ) . By t h e Hewitt-Nachbin com-
Yo
p l e t e n e s s of
t h e r e e x i s t s a non-negative f u n c t i o n
Y
go E C
(clPyYo)
go(q) = 0
such t h a t
by 8 . 8 ( 3 ) .
such t h a t g l c l P y k o = ;o i tion h + g. Then h f u n c t i o n on
PY.
= Z ( h i ) fl Y
go(y)
>
g 20.
and
+
g
y E Yo
f o r every
0
NOW, l e t t h e f u n c t i o n
g
and
C(pY) be
E
Next, d e f i n e t h e func-
i s a normal lower s e m i - c o n t i n u o u s
Moreover, h i
+
g
i s p o s i t i v e on
implies t h a t the only points of
Y
Y
because
f o r which
COMPLETENESS AND CONTINUOUS MAPPINGS
208
hi
t a k e s on t h e v a l u e z e r o a r e p o i n t s t h a t belong t o Yo, b u t a t those p o i n t s t h e f u n c t i o n g p r e v i o u s l y c o n s t r u c t e d i i s p o s i t i v e . Also, (h + 9 ) ( 4 ) = 0 . S i n c e Y i s a weak cbT
space, t h e r e e x i s t s a f u n c t i o n
0
<
<
T
+
(hi
T
t e n s i o n of
T*
Let
9) IY ( 1 7 . 5 ( 2 ) ) . from
*
C ( Y ) such t h a t
E
be t h e c o n t i n u o u s ex-
i n t o the r e a l s .
BY
S i n c e normal lower
semi-continuous f u n c t i o n s a r e determined on dense s u b s e t s i * ( 1 7 . 7 ) , the functions h + g - T cannot b e n e g a t i v e on because i t i s p o s i t i v e on T
tion every
*
in
Thus, 0
~*(q) 5 (hi
g
t h e r e e x i s t s a func-
Y.
Therefore, f o r each p o i n t
0.
T
C ( P Y ) such t h a t
E
*
pY\Y
(4) = 0
Hence, by 8.8(3) Y
ycY.
pY
and
T
*
+
9 ) (9) =
>
(y)
for
0
i s Hewitt-Nachbin complete,
completing t h e proof of t h e theorem, The following r e s u l t i s C o r o l l a r y 2.6 of t h e 1969 paper by N . Dykes.
The proof u t i l i z e s a technique employed by K.
Morita and S. Hanai i n proving Theorem 1 of t h e i r 1956 p a p e r .
17.11
THEOREM (Dykes).
If
-
t i o n of 2 Hewitt-Nachbin -space
f
space
i s a c l o s e d continuous s u r j e c onto a Tychonoff weak
X
such t h a t t h e boundary
Y
i s compact -
f o r each p o i n t
6f-’(y)
Y E Y , then
of t h e f i b e r
cbf-l(y)
i s a Hewitt-Nachbin
Y
space, Proof.
ycY
For each p o i n t
d e f i n e an open s u b s e t
L ( y ) of
L = U ( L ( y f : Y E Y ] and set
Let
c l o s e d subspace of
i
s e l e c t a point
X
a s follows:
Xo = X\L.
i s a continuous mapping from
ycY
6f-’(y)
f(p) = y.
such t h a t py
E
Xo
because
# @,
and
Xo
into
Xo
then t h e r e i s a p o i n t On t h e o t h e r hand,
f(py) = y.
i s closed.
if
into p
g
satisfies
is a
complete.
Let
Then
X. Y.
For i f
6f-’(y)
E
af-l(y) =
Moreover, t h e mapping Since
Xo
Then
Xo
g = f o i
f - l ( y ) and
E
Y
and hence Hewitt-Nachbin
X
denote t h e i n c l u s i o n mapping from and
p
g
a,
c Xo
then i s closed
209
CLOSED MAPPINGS
i t follows t h a t
g
-1
T h e r e f o r e , by 1 6 . 1 3
( y ) i s compact f o r e v e r y p o i n t
ycY.
i s Hewitt-Nachbin complete which con-
Y
c l u d e s t h e proof o f t h e theorem. I n S e c t i o n 14 w e i n t r o d u c e d E . M i c h a e l ’ s n o t i o n o f a space.
q-
The n e x t aim w i l l b e t o e s t a b l i s h t h a t Hewitt-Nachbin
completeness i s i n v a r i a n t under a c l o s e d c o n t i n u o u s s u r j e c t i o n provided t h a t t h e r a n g e i s a Tychonoff weak cb- , q- s p a c e A p r e l i m i n a r y r e s u l t w i l l be u s e f u l
(17.14).
I t i s due t o Michael
that fact.
is a
ycY
on
is
X
Proof.
T -space
1
q-point,
bounded on t h e boundary h
>
/ h ( x n )1
i s open,
+
xi
Next p i c k a sequence
a s i n t h e d e f i n i t i o n of a
let
determined.
and choose
z1 = xl.
n
6f-
zi
Vi. E
[Vi
q-point
XEX
n
f
-1
If
function
h
is not
: iEN ) i n
nEIN.
6f-l(y)
Define
1 < 71 ) . h a s a neighborwhere
(Ni)],
Ni
is
(14.19), and s u c h t h a t a l l
T h i s i s e a s i l y done by i n d u c t i o n a s f o l Suppose t h a t
zl,.. . , z k e l
h a v e all been
Define t h e s e t
zk
1( y ) .
Wk\f-’(y). The l a t t e r c h o i c e i s i s open and xk b e l o n g s t o
from t h e s e t
p o s s i b l e because Wk
f o r every
and e v e r y
E Vi,
hood i n t e r s e c t i n g a t most one
f(zi) are distinct.
1
sY ,
.
C ( X ) and t h a t
= (XEX : / h ( x ) - h(xi)
Vi Vi
Sf-’(y)
belongs t o
Choose a sequence ( x i
6f-’(y),
such t h a t / h ( x n + l ) 1
lows:
be a c l o s e d c o n t i n u o u s
o n t o a t o p o l o q i c a l space
X
then e v e r y c o n t i n u o u s r e a l - v a l u e d
Suppose t h a t
bounded on
Then
Let f
LEMMA ( M i c h a e l ) .
17.12
j e c t i o n from a
i n establishing
(1964).
Wk
This
m e n t s . Now, d e f i n e
zk
c l e a r l y s a t i s f i e s a l l of t h e require-
Z = (zi
lows t h a t e v e r y s u b s e t of
: icm).
Z
Since
zi
E
vi
it f o l -
is c l o s e d , and h e n c e so also i s
2lo
COMPLETENESS AND CONTINUOUS MAPPINGS
every subset of
But
f(2).
f ( z i ) belongs t o
Ni
and t h e
f ( Z ) must have an
f ( z i ) a r e a l l d i s t i n c t i n which c a s e accumulation p o i n t . T h i s i s a c o n t r a d i c t i o n , c o m p l e t i n g t h e proof. 17.13
LEMMA
(Dykes)
.
If
i s a closed continuous s u r j e c -
f
-
t i o n of 2 Hewitt-Nachbin s p a c e --
-1(y) 6€
compact f o r e a c h p o i n t
Proof. each
onto a
X
By 1 7 . 1 2 e v e r y
yeY; whence
h
C(X)
E
q-space
Y,
then
ycY. i s bounded on
bf-l(y) for
cl b f - l ( y ) i s compact f o r each
ycY
by
11.25.
The n e x t theorem i s a primary r e s u l t and i s due t o N . Dykes (1969, C o r o l l a r y 3 . 5 ) .
I t g e n e r a l i z e s t h e r e s u l t due t o
Blair t h a t w a s cited i n the introduction t o t h i s section.
We
w i l l state Blair’s result a s a corollary.
If
f
i s a closed continuous s u r j e c t i o n from 2 Hewitt-Nachbin s p a c e X o n t o a Tychonoff, weak cb-, q-space Y , then Y i s a Hewitt-Nachbin s p a c e . 1 P r o o f . By 1 7 . 1 3 t h e boundary 6f- ( y ) i s compact f o r each 17.14
THEOREM ( D y k e s ) .
-- -
Hence, Y
i s Hewitt-Nachbin complete by 1 7 . 1 1 .
point
YEY.
17.15
COROLLARY ( B l a i r ) .
If
f
i s a c l o s e d c o n t i n u o u s E-
j e c t i o n from a Hewitt-Nachbin space space Y
-& a
Proof.
Y
that
X
o n t o a Tychonoff
cb-
s a t i s f i e s t h e f i r s t axiom o f c o u n t a b i l i t y ,
Hewitt-Nachbin s p a c e . Every f i r s t c o u n t a b l e s p a c e i s a
cb-space i s a weak
q - s p a c e and e v e r y
cb-space.
Next w e would l i k e to i n t r o d u c e B l a i r ’ s n o t i o n of a “ h y p e r - r e a l map”. The f i r s t r e s u l t w i l l s t r e s s t h e s u i t a b i l i t y o f t h i s c l a s s of mappings f o r t h e i n v a r i a n c e of Hewitt-Nachbin completeness, and i s due t o B l a i r .
The h y p e r - r e a l mappings
w i l l then be r e l a t e d t o t h e o t h e r c l a s s e s o f mappings t h a t w e r e investigated i n Section 15.
Finally,
the hyper-real map
p i n g s w i l l p r o v i d e us w i t h a d d i t i o n a l r e s u l t s r e g a r d i n g t h e i n v a r i a n c e o f Hewitt-Nachbin c o m p l e t e n e s s under c l o s e d c o n t i n u o u s mappings ( 1 7 . 2 0 and 1 7 . 2 1 )
.
211
CLOSED MAPPINGS
17.16
A c o n t i n u o u s mapping
DEFINITION.
space
i n t o a Tychonoff s p a c e
X
i f t h e Stone e x t e n s i o n
fP (P X \,X )
fP
from a Tychonoff
i s s a i d t o be h y p e r - r e a l
Y
into
PX
satisfies
PY
c PY\-Y.
THEOREM ( B l a i r ) . && f be a h y p e r - r e a l s u r j e c t i o n X onto Y . Then t h e f o l l o w i n q s t a t e m e n t s a r e t r u e :
17.17
from -
(1)
If
X
(2)
If
Y
Proof.
i s a Hewitt-Nachbin s p a c e , then Hewitt-Nachbin s p a c e . i s pseudocompact, t h e n
(1) Suppose t h a t
if a point
w
belongs t o
belongs t o
PY\s;Y.
p
s f (X)
E
PX\JX
Hence, since
x0
there e x i s t s a point
T h e r e f o r e , p c f ( X ) and
.
= PXW,
p
C
f (X)
2
Y
is pseudocompact.
X
Since
f
i s hyper-real,
t h e n t h e image
fP(w)
does n o t b e l o n g t o
BY\>LY
such t h a t
t X
,df ( X )
By 11.1 t h e s p a c e
(2)
from
f
f
.
P (xo) = f ( x ) = p . 0
i s pseudocompact i f and o n l y i f
X
P X = ;X. to
Now, suppose t h a t t h e r e e x i s t s a p o i n t p b e l o n g i n g P PX\vX. Then f ( p ) b e l o n g s t o P Y \ v Y . But PY\uY = # by
assumption so t h a t
PX\;X
must a l s o b e empty c o n c l u d i n g t h e
proof. Although t h e n e x t r e s u l t d o e s n o t c h a r a c t e r i z e t h e c l a s s of h y p e r - r e a l mappings, it d o e s a t l e a s t p r o v i d e a s u f f i c i e n t c o n d i t i o n t h a t a mapping b e hyper- r e a l . 1 7 . 1 8 THEOREM ( B l a i r ) . Let X and Y Tychonoff spaces. If f i s a mapping from X onto Y , t h e n f is hyper-real whenever the f o l l o w i n q two c o n d i t i o n s satisfied: (1) The mappinq f i s f i b e r - c o u n t a b l v compact, and ( 2 ) If ( Z n : n E N ) i s a d e c r e a s i n q sequence of zero-
are
sets i n --
n Proof. on
X
Let
X
such t h a t
n
( f (Zn)
( c l u y f (zn) : n E N 1 = #. p E pX\ux, and l e t 3’ denote the
t h a t converges t o
p.
Hence
3’
c o u n t a b l e i n t e r s e c t i o n p r o p e r t y by 8 . 5 ( 5 ) t h e r e e x i s t s a sequence [Zn that
fl (Zn
: ntN ) =
:
f o r each p o i n t
ncm) = ycY,
8.
:
2-ultrafilter
d o e s n o t have t h e
.
I t follows that
ncm ) of z e r o - s e t s i n
Since
#, t h e n
’3’
such
f - l ( y ) i s c o u n t a b l y compact
it f o l l o w s t h a t
fl [ f ( Z n ) : n c m ) =
fl by
COMPLETENESS AND CONTINUOUS MAPPINGS
212
15.4(2).
@. Now, p E n t o n :clpyf ( Z n )
ncm ) = longs
I t follows t h a t
vY.
n
Hence, by ( 2 ) i t i s the case t h a t
( c l d Y f( Z n )
:
[claXZn : n e m ) and hence f p ( p ) be: n€N 1. Thus f P (p) cannot belong t o f P ( P X \ u X ) c PY\vY concluding t h e proof
of the theorem. The following r e s u l t r e l a t e s the c l a s s of h y p e r - r e a l mappings t o t h a t of the z e r o - s e t p r e s e r v i n g mappings. (Blair).
Let
and
&
Tychonoff spaces.
17.19
COROLLARY
If -
i s a f i b e r - c o u n t a b l y compact and z e r o - s e t p r e s e r v i n q
f
surjection
from
X
onto
Y,
X
then
Y
i s hyper-real.
f
Proof. Suppose t h a t ( Z n : n c m ) i s a d e c r e a s i n g sequence of z e r o - s e t s i n X such t h a t n ( f ( Z n ) : n c m 1 = 6 . S i n c e
n c m ) i s a countable family of z e r o - s e t s i n Y , i t n ( c l v y f ( Z n ) : n c m ) = @. Hence, f i s hyper- r e a l according t o t h e theorem. { f(Zn)
:
follows from 8.5.(3) t h a t
The next r e s u l t r e l a t e s the i n v a r i a n c e of Hewitt-Nachbin completeness under f i b e r - c o u n t a b l y compact and
z-closed m a p
pings by u t i l i z i n g t h e notion of a h y p e r - r e a l mapping.
Note
the s i m i l a r i t y of t h e r e s u l t t o t h a t of Dykes proved i n 16.13. Whereas i n 16.13 t h e mapping i s p e r f e c t and the range i s a weak
cb-space,
t h e next r e s u l t imposes t h e weaker c o n d i t i o n
t h a t t h e mapping be f i b e r - c o u n t a b l y compact and z-closed tog e t h e r with t h e s t r o n g e r c o n d i t i o n t h a t t h e range be a cbspace. other
. THEOREM ( B l a i r )
17.20
and
-a
The two r e s u l t s a r e e v i d e n t l y independent of each
.
Let f
be a f i b e r - c o u n t a b l y compact
z-closed s u r j e c t i o n from a Hewitt-Nachbin
Tychonoff space
Y.
If
2
Y
space
cb-space, then
X
Y
onto
is 2
Hewitt-Nachbin space. I t w i l l be shown t h a t
Proof.
f
i s h y p e r - r e a l from which the
.
r e s u l t w i l l follow immediately from 1 7 . 1 7 (1) {Zn
that
:
Hence,
n t m ) be a decreasing sequence of z e r o - s e t s i n
n
( f ( Z n ) : n6m ) =
6.
let X
i s z-closed, n c m ) i s a d e c r e a s i n g sequence of closed s e t s i n
( f (Zn) : with empty i n t e r s e c t i o n .
Since
such
f
Y
Hence, by 1 4 . 1 5 ( 1 ) t h e r e e x i s t s a
2 13
WZ-MAPPINGS
sequence (Hn f o r each
: n c l N ) of z e r o - s e t s
and
nclN
fi c l u y f ( z n ) c
n
n
{Hn : ncN
clvpn=
6.
in
Y
=
a.
such t h a t
f ( Z n ) c Hn
T h e r e f o r e , by 8 . 5 ( 3 )
H e n c e by 1 7 . 1 8
f
i s hyper-real
completing t h e p r o o f . The f o l l o w i n g r e s u l t i s s i m i l a r t o t h a t proved i n 1 6 . 6 . 17.21
COROLLARY.
If
f
is a
z - c l o s e d and f i b e r - c o u n t a b l y
compact s u r j e c t i o n from a Hewitt-Nachbin s p a c e mal Hausdorff -
c o u n t a b l y paracompact s p a c e
Y,
X
then
onto a nor-
&=
Y
Hewitt-Nachbin s p a c e . Proof.
Every normal and c o u n t a b l y paracompact s p a c e i s a
cb-
space. Observe t h a t t h e p r e v i o u s l y s t a t e d c o r o l l a r y d i f f e r s from 1 6 . 6 by r e q u i r i n g t h e weaker h y p o t h e s i s t h a t t h e mapping be
z - c l o s e d r a t h e r than c l o s e d ,
However, i t i s t h e n assumed
t h a t t h e r a n q e s p a c e b e normal Hausdorff and c o u n t a b l y paracompact r a t h e r t h a n t h e domain s p a c e s i n c e one c a n no l o n g e r take advantage of M o r i t a ’ s r e s u l t 16.5.
Moreover, 1 6 . 6 would
f o l l o w a s a d i r e c t consequence o f 1 7 . 2 1 coupled w i t h 1 6 . 5 . However, o u r approach i s j u s t i f i e d by t h e e x p o s u r e o f t h e embedding c o n s t r u c t i o n of a zero- s e t f i l t e r w i t h t h e c o u n t a b l e intersection property i n t o a zero-set u l t r a f i l t e r w i t h the countable i n t e r s e c t i o n p r o p e r t y f o r t h e p a r t i c u l a r c a s e a s pres e n t e d i n t h e proof o f 1 6 . 6 . S e c t i o n 18 : WZ- Mappinqs I n t h i s s e c t i o n w e w i l l s t u d y t h e i n v a r i a n c e and i n v e r s e i n v a r i a n c e of Hewitt-Nachbin c o m p l e t e n e s s under a w i d e r c l a s s of mappings than t h e c l o s e d mappings; namely, t h e
WZ-mappings
which w e r e f i r s t i n v e s t i g a t e d by T. I s i w a t a i n h i s 1967 p a p e r . One r e s u l t g i v e n i n 18.9 y i e l d s t h e i n v a r i a n c e of H e w i t t - N a c h b i n completeness under an open and c l o s e d c o n t i n u o u s s u r j e c t i o n f o r which t h e boundary of e a c h f i b e r i s compact p r o v i d e d t h a t t h e r a n g e i s a Tychonoff s p a c e .
T h i s result generalizes
what was proved i n 16.10 f o r open p e r f e c t mappings.
The re-
s u l t is similar t o t h a t stated i n 17.11 e x c e p t t h a t t h e
COMPLETENESS AND CONTINUOUS MAPPINGS
214
hypothesis t h a t
f
a l s o b e open r e p l a c e s t h e c o n d i t i o n t h a t
t h e r a n g e b e a weak independent.
cb-space.
The two r e s u l t s a p p e a r t o b e
I t w i l l a l s o be e s t a b l i s h e d ( 1 8 . 1 2 )
that Hewitt-
Nachbin completeness i s i n v a r i a n t under an open and c l o s e d continuous s u r j e c t i o n o n t o a
k-space.
However, Hewitt-Nach-
b i n completeness i s n o t i n v e r s e i n v a r i a n t under an open and closed continuous s u r j e c t i o n o n t o a
To see t h i s
k-space.
l a s t a s s e r t i o n observe t h a t 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 a s s o c i a t e d w i t h an open and c l o s e d subspace Hewitt-Nachbin s p a c e
X
A
o f a non-
o n t o t h e two-point d i s c r e t e s p a c e F i n a l l y , i t w i l l b e shown i n
( O , l ] a f f o r d s a counterexample.
1 8 . 1 5 t h a t Hewitt-Nachbin c o m p l e t e n e s s i s i n v e r s e i n v a r i a n t
under
WZ-mappings f o r which f i b e r s a r e Hewitt-Nachbin com-
p l e t e and
C-embedded.
i s n o t i n v a r i a n t under a
However, Hewitt-Nachbin c o m p l e t e n e s s WZ-mapping f o r which f i b e r s a r e
Hewitt-Nachbin complete and
C-embedded by Example 1 6 . 4 s i n c e
e v e r y p e r f e c t mapping s a t i s f i e s t h o s e c o n d i t i o n s .
The r e s u l t
1 8 . 1 5 i s s i m i l a r t o 1 6 . 1 e x c e p t t h a t i t u t i l i z e s t h e hypothe-
s i s t h a t t h e mapping be a WZ-mapping r a t h e r t h a n z - c l o s e d , and t h a t f i b e r s b e C-embedded r a t h e r than z-embedded. Moreo v e r , t h e r e s u l t s 18.15 and 1 6 . 1 a r e i n d e p e n d e n t b e c a u s e t h e r e exist
WZ-mappings t h a t f a i l t o be
(18.7(1)) and
z-closed
c l o s e d Hewitt-Nachbin s u b s p a c e s t h a t f a i l t o b e
C-embedded
(8.23) . A s i n t h e p r e v i o u s s e c t i o n , whenever
mapping from a Tychonoff space then
fp
f
i s a continuous
i n t o a Tychonoff s p a c e
X
w i l l d e n o t e i t s S t o n e e x t e n s i o n from
BX
into
Y,
BY.
According t o I s i w a t a ( 1 9 6 7 ) w e have t h e f o l l o w i n g d e f i n i t i o n of t h e c l a s s o f maps which w i l l b e of primary i n t e r e s t i n t h i s section.
18.1 D E F I N I T I O N . space
X
ping i f
A continuous s u r j e c t i o n
o n t o a Tychonoff space clgxf
-1
(y) = [f’]-l(y)
Y
f
from a Tychonoff
i s s a i d t o be a
f o r every p o i n t
WZ---
y c ~ .
The f o l l o w i n g two r e s u l t s a p p e a r i n I s i w a t a ‘ s 1967 p a p e r and e s t a b l i s h t h e r e l a t i o n s h i p between
WZ-mappings and some
o f t h e o t h e r c l a s s e s of mappings t h a t have b e e n under i n v e s t i -
WZ- MAPPINGS
215
ga t i o n i n t h i s c h a p t e r .
J &
THEOREM ( I s i w a t a ) .
18.2
and l e t f -the followinq -
(1)
If
(2)
If
and
X
Y
be
Tychonoff s p a c e s ,
2 c o n t i n u o u s s u r j e c t i o n from
X
onto
Then
Y.
statements a r e true: f
is a
z - c l o s e d mappinq, t h e n
f
is 2
WZ-
mappinq.
is a
f
WZ-mappinq and i f
i s normal, t h e n
X
i s a c l o s e d mappinq.
f
The r e s u l t (1) was proved a s Lemma 1 7 . 2 , b u t i t i s
Proof.
r e s t a t e d h e r e i n connection w i t h D e f i n i t i o n 18.1. To t h i s end, l e t
need o n l y e s t a b l i s h ( 2 ) .
X
s e t of
and l e t
y
j o i n t closed sets X.
Y\f(F)
E
.
f - l ( y ) and
Since
b e a c l o s e d sub-
i s normal, t h e d i s -
X
a r e completely s e p a r a t e d i n
F
there i s a function
Hence,
F
Hence, w e
h
E
C ( X ) such t h a t
h ( F ) c il), and 0 2 h 1. S i n c e f i s a 1 * P -1 WZ-mapping, c l P x f - ( y ) = [ f p ] - l ( y ) . Hence, h ( [ f J (y)) c h [ f - l ( y ) ] c (01, [ O ) where
set
M = f
P
*
i s t h e e x t e n s i o n of h o v e r P X . Define t h e 1 [ ( p t PX : h * ( p ) > T ) ] n Y . Then y,kM b e c a u s e h*
h
i s z e r o on [ f P J - ’ ( y ) .
Since
an open s e t c o n t a i n i n g
y
c l y f ( F ) so t h a t
belong t o
fp
i s a c l o s e d mapping, Y b l f ( F ) c M.
and
Thus, y
i s a c l o s e d mapping.
f
is
does n o t T h i s con-
c l u d e s t h e proof o f t h e theorem. P. Zenor i n h i s 1969 p a p e r h a s e s t a b l i s h e d n e c e s s a r y and
s u f f i c i e n t c o n d i t i o n s on a s p a c e mapping b e a z-closed
z - c l o s e d mapping.
i f and o n l y i f
X
X
i n o r d e r t h a t every
Precisely,
WZ-mappinq
WZ-
is
i s a Tychonoff s p a c e w i t h t h e
p r o p e r t y t h a t every closed set i s completely s e p a r a t e d e v e r y z e r o - s e t t h a t i s d i s j o i n t from i t .
from
Moreover, Zenor a l s o
shows t h a t 2 Tychonoff s p a c e i s normal i f and o n l y i f e v e r y z - c l o s e d mappinq i s c l o s e d . noff space
X
9
Finally,
a pseudocompact
Tycho-
c o u n t a b l y compact i f and o n l y i f e v e r y
mappinq d e f i n e d 2
X
is
WZ-
z-closed.
I s i w a t a (1967) f u r t h e r i n v e s t i g a t e s t h e r e l a t i o n s h i p s between c l o s e d , z- c l o s e d , and
WZ-mappings.
r e s u l t s h e r e i n o r d e r t h a t t h e concept o f a
W e include those
WZ-mapping may be
b r o u g h t more s h a r p l y i n t o f o c u s r e l a t i v e t o t h e mappings i n t r o -
COMPLETENESS AND CONTINUOUS MAPPINGS
2 16
duced i n Section 1 5 .
Example 1 8 . 7 ( 1 ) w i l l i l l u s t r a t e t h a t t h e
converse f a i l s t o hold f o r 1 8 . 2 ( 1 ) . n o t e t h a t every closed mapping i s a
With r e f e r e n c e t o 1 8 . 2 ( 2 ) WZ-mapping whether o r n o t
t h e domain i s a normal space. The a u t h o r h a s n o t been a b l e t o f i n d an example of a z-open mapping t h a t f a i l s t o be a WZmapping. The following terminology w i l l be h e l p f u l i n e s t a b l i s h i n g t h e v a r i o u s r e l a t i o n s h i p s under i n v e s t i g a t i o n .
We
remark t h a t I s i w a t a simply r e f e r r e d t o t h e concepts d e f i n e d below a s a s u b s e t o r a mapping p o s s e s s i n g " p r o p e r t y ( * ) . I 1 18.3
A non-empty
DEFINITION.
subset
F c X
i s s a i d t o be
s t r o n q l v p o s i t i v e i f each continuous r e a l - v a l u e d f u n c t i o n h F C ( X ) t h a t i s p o s i t i v e on F s a t i s f i e s i n f ( h ( x ) : x c F ) 0.
A mapping
f
from a t o p o l o g i c a l space
X
Y
onto a space
i s s a i d t o be f i b e r - s t r o n g l y p o s i t i v e i f t h e f i b e r s t r o n g l y p o s i t i v e f o r every ycy. 18.4
>
f-'(y)
is
(1) Every pseudocompact subspace of a topo-
REMARKS.
l o q i c a l space x is s t r o n s l y p o s i t i v e . For suppose F i s a pseudocompact subspace of X t h a t f a i l s t o be s t r o n g l y positive.
Then t h e r e e x i s t s a f u n c t i o n
on
f o r which
F
inf(h(x) : xcF)
h
5 0.
C(X) that is positive
E
Thus, f o r every posi-
x belonging t o 1 ' with 0 < h ( x e ) < t . Then t h e f u n c t i o n r; i s defined and continuous on F, y e t f a i l s t o be bounded t h e r e . This i s a contradiction. ( 2 ) I n Theorem 1 . 5 of h i s 1967 paper I s i w a t a proves t h a t every z e r o - s e t of a pseudocompact Tychonoff space i s strongly positive. t i v e r e a l number
E
there e x i s t s a point
The following r e s u l t s a r e due t o I s i w a t a .
F
Without im-
posing a d d i t i o n a l c o n d i t i o n s on t h e t o p o l o g i c a l spaces i n volved a s i n the c a s e of Z e n o r ' s r e s u l t s , they provide i n f o r mation a s t o when one might e x p e c t a WZ-mapping t o be zclosed. 1 8.5
J& X and Y & Tychonoff spaces. z-closed f i b e r - r e l a t i v e l y pseudocomp a c t mappinq from X onto Y, then f i s f i b e r -
THEOREM ( I s i w a t a ) .
(1)
If
f
is 2
217
WZ-MAPPINGS
stronqly positive.
If
(2)
is a
f
WZ-mappinq from
onto
X
fiber-stronqly positive, then
t h a t is
Y
is 2
f
z-closed
mapping.
(1) Suppose t h a t t h e r e i s a p o i n t y c Y such t h a t - 1 ( y ) i s n o t s t r o n g l y p o s i t i v e . Then t h e r e e x i s t s a non-
Proof.
F = f
negative function
h
C ( X ) such t h a t
E
and a sequence {xn : ncEJ 0.
1
in
h(x)
>
f o r which
F
XCF,
when
0
i n f j h ( x n ) : nElN?=
Now, Z = Z ( h ) i s non-empty b e c a u s e Z ( h ) = fl i m p l i e s t h a t 1 i s unbounded on t h e r e l a t i v e belongs t o C ( X ) However, -
.
l y pseudocompact s u b s e t
I t w i l l s u f f i c e t o show t h a t
F.
f
i s n o t z - c l o s e d by e s t a b l i s h i n g t h a t y E c l f ( Z ) b e c a u s e Z i s a z e r o - s e t and y f f ( Z ) . Hence, suppose t h a t y !I, c l f ( Z ) Then t h e r e e x i s t s a f u n c t i o n g[cl f(Z)]
C
1 for a l l
< L.
go f(x) = 0
and c o n t i n u o u s on t h e o t h e r hand,
g
C ( Y ) such t h a t
E
X,
1 5;
F.
1 5;
and t h e r e f o r e
y f f(Z).
Since
f
Since
h
over
Moreover, t h e p o i n t
PY
Hence, y
*
(P)
C(X)
.
On
n
(V
n
be-
and s u p
(X),
>
0.
*
h*
Let
h (x)
2
it
a
for
Now, t h e s e t
< a/21
does n o t belong t o f P ( M ) . P V = PY\f ( M ) i s an open sub-
Y) c fP(M)
does n o t belong t o
*
y
y
t h a t contains the point f(z)
C
E
Hence
PX.
1 [fP]-'(y) = clPxf- ( y ) .
f P ( M ) i s compact, t h e s e t
s e t of
is positive
f
is fiber-strongly positive,
M = { p E PX : h
i s compact.
h
inf(h(x) : x E f-l(y)] = a
is the case t h a t
d e n o t e t h e e x t e n s i o n of F
0
i s unbounded on t h e r e l a t i v e l y pseudocom-
c l f ( Z ) as desired.
x
g
This contradiction establishes t h a t
2 = Z ( h ) f o r some n o n - n e g a t i v e
a l l points
+
go f(x) =
belongs t o
(2)
pose t h a t
belongs t o
xcF, and
k = h
longs to Let
g ( y ) = 0,
Therefore, g o f
for a l l
Now t h e f u n c t i o n
XEZ.
pact subset
0<
( 1 ) , and
C ( X ) and s a t i s f i e s
g
y.
n
(V
Furthermore,
n Y)
=
pr.
c l f ( 2 ) and t h e r e f o r e Y c l o s e d c o m p l e t i n g t h e proof o f t h e theorem.
f(Z) is
.
2 18
COMPLETENESS AND CONTINUOUS MAPPINGS
18.6
COROLLARY.
(1)
If
f
space tive . (2)
If
f
space
Proof.
Let is 2
Y
X
z - c l o s e d mapping from a pseudocompact
Y, t h e n
onto
X
is 2
is fiber-stronqly
f
a-
WZ-mapping from a c o u n t a b l y compact Y, then
onto
X
Tychonoff spaces.
9
f
z-closed.
(1) T h i s i s immediate f r o m (1) o f the theorem.
Y E Y , f-'(y) i s a c l o s e d s u b s e t of the (2) c o u n t a b l y compact s p a c e X and t h e r e f o r e pseudocompact.
For each p o i n t
Hence, f - l ( y ) i s s t r o n g l y p o s i t i v e and the r e s u l t f o l l o w s from ( 2 ) o f t h e theorem. W e can modify t h e c h a r t p r e s e n t e d i n S e c t i o n 1 5 t o i n -
c l u d e t h e h y p e r - r e a l and
WZ-mappings, and o b t a i n t h e follow-
i n g summary o f t h e v a r i o u s r e l a t i o n s h i p s between t h e mappings W e w i l l assume t h a t a l l of t h e s p a c e s a s s o c i a t e d w i t h t h e mappings of t h i s c h a r t have t h e Tychonoff p r o p e r t y s i n c e o t h e r w i s e t h e hyper- r e a l and WZ-mappings would n o t b e d e f i n e d .
t h a t have been under d i s c u s s i o n i n t h i s c h a p t e r ,
2- 0 PEN I domain normal
1
Z-OPEN
a
+
b
+
FIBER-COUNTABLY I
means e v e r y
a
COMPACT)).~HYPER-
REAL^
t I
mapping i s a
b
tt
n OPEN
+
CLOSED
mapping.
The f o l l o w i n g examples a r e d u e t o I s i w a t a and appear i n h i s 1967 p a p e r .
2 19
WZ-MAPPINGS
(1) A n open
EXAMPLES.
18.7
WZ-mappinq t h a t f a i l s t o b e
z-
closed, Let
d e n o t e t h e Tychonoff p l a n k p r e s e n t e d i n 1 4 . 1 4 , and l e t
T
d e n o t e t h e p r o j e c t i o n mapping o f
cp
is an open y c
mappinq.
T
Onto [ O , n ] . H e n c e , cp -1 cp (y) for
Now e v e r y i n v e r s e image
i s r e l a t i v e l y pseudocompact (see Gillman and J e r i s o n ,
[O,R]
*
it follows t h a t clBTcp-l(Y) = WZ-maPPinq. However, cp i s n o t -1 z - c l o s e d by 1 8 . 6 ( 1 ) because cp (n) f a i l s t o be s t r o n g l y p o s i 8.20). S i n c e PT = [ O , n ] [ c p P ] - l ( y ) . Hence, cp
t i v e and
T
N
X
i s pseudocompact.
An open f i b e r - c o m p a c t mappinq t h a t f a i l s t o b e a
(2)
WZ-mapping.
n
Let
d e n o t e t h e f i r s t u n c o u n t a b l e o r d i n a l , and d e f i n e
x Z = ( (x,y)
where
[o,nl x
=
: x =
n
nate space. x
T - ~ ( X )
i t follows t h a t
[O,R]
E
Now,
Since
[O,n)
x [O,n]
w
and
X
t h e p r o j e c t i o n mapping from
\z
[O,nl
<
n).
y
onto [O,n],
Let
T
denote
t h e f i r s t coordi-
i s compact f o r each p o i n t T
i s an open f i b e r - c o m p a c t mappinq.
X
i s pseudocompact so t h a t
i s t h e union
of a pseudocompact space w i t h t h e compact s p a c e ( ( n , y ) : 0 y
Hence, X i s pseudocompact.
w).
[~']-'(n)
so t h a t
#
However, c l p X T - l ( Q )
f a i l s t o be a
T
WZ-mappinq.
Note a l s o
t h a t a c l o s e d mapping t h a t f a i l s t o b e open p r o v i d e s an example o f a
WZ-mapping t h a t i s n o t open, and hence n o t
Next, l e t space
X
each f i b e r h
b e a c o n t i n u o u s mapping from a t o p o l o g i c a l
h
onto a space
Y,
f - l ( y ) €or
induced mappings
z-open.
fi
and l e t
YEY.
and
fS
f
E
C ( X ) b e bounded on
Recall t h e d e f i n i t i o n of the given i n 1 7 . 8 .
Note t h a t i f
i s b o t h an oper! and c l o s e d mapping, t h e n b o t h
a r e c o n t i n u o u s f u n c t i o n s by 1 7 . 9 .
fi
and
fS
These f u n c t i o n s w i l l b e u s e -
f u l i n p r o v i d i n g p a r t ( 2 ) of t h e n e x t r e s u l t .
ment provides a useful characterization of r e s u l t s a r e found i n I s i w a t a ' s 1967 p a p e r .
The f i r s t s t a t e -
WZ-mappings.
Both
COMPLETENESS AND CONTINUOUS MAPPINGS
220
(1) & mappinq
-if and If
X
and
X
onto
if
(1) Observe t h a t
n
U
WZ-mapping
= h P ( u ) fl Y
f o r every
if hP n (U n
# @
h-’(y)
then
Y,
i s open.
# 0
X)
U c pX
f o r e v e r y open
onto
X
i f and o n l y
because
is
h
For t h e n e c e s s i t y i t s u f f i c e s t o show t h a t
WZ-mapping.
n
is a
Y
PX.
C
i s open i f and o n l y
i f [hP]]-’(y) hP(U)
n x)
h(U
Tychonoff s p a c e s .
Y
i s a c l o s e d mappinq from
h
h Proof.
h
only
open u (2)
a
Let from
THEOREM ( I s i w a t a ) .
18.8
x) .
Y c h(U fl
t o h P ( U ) fl Y . Then, a p o i n t p F [h-’(y) Thus, y c h(U
X)
To see t h i s , suppose t h a t [ hP ] - 1 ( y )
n
.
n
(U
y
belongs
nu # @
so t h a t t h e r e e x i s t s X ) ] by o u r o p e n i n g o b s e r v a t i o n . p E [hP ] - 1 ( y ) \
To prove t h e s u f f i c i e n c y , suppose t h a t
c l P x h - l ( y ) . Then t h e r e i s an open U c PX c o n t a i n i n g p t h a t s a t i s f i e s U n h - l ( y ) = 0. However, h P ( p ) = y so t h a t y E
[hP(u)
n
Y]
.
y E h(U
Thus,
n x)
by a s s u m p t i o n .
There-
f o r e , h-’(y) n (U fl X) # @ which i s a c o n t r a d i c t i o n . ( 2 ) By s t a t e m e n t (1) j u s t e s t a b l i s h e d , i t s u f f i c e s t o prove
i s an open mapping, then hP i s open. Hence, w e want t o show t h a t f o r each p o i n t p c P X and neighborhood U of p , t h e r e e x i s t s an open s u b s e t W of P Y such t h a t hP ( p ) c W c h P ( U ) Now, s i n c e PX i s r e g u l a r and p j! pX\u, that i f
h
.
G1
t h e r e e x i s t open s e t s pX\U
C
G2,
and
n
G1
V =
Thus, t h e f u n c t i o n and
V
satisfies
(x f
0.
G2 =
0
f c C ( P X ) such t h a t Moreover, i f
and f
: f (x)
pX
in
G2
g
p
E
G1,
Then t h e r e e x i s t s a f u n c t i o n
1, f ( p ) = > 01, then
satisfies
clPxV c U.
1, and
f ( p ) = 1 and
Since
denote t h e extension of
( f IX)’
c (0).
f (pX\G1)
p E V c G1 c pX\G2
h
CU.
f(PX\U) c [ O ] ,
i s b o t h open and
c l o s e d by h y p o t h e s i s , the f u n c t i o n ( f IX)’ Let
such t h a t
.
C* (Y)
belongs t o
over
PY.
Then 1
P g o h ( p ) = 1 and moreover t h e s e t W = [ y : g ( y ) > T ] i s open i n P Y . H e n c e , h P ( p ) E W and h P ( c l p x v ) c h P (u) It
.
w i l l b e e s t a b l i s h e d t h a t W c h P ( c l V ) . Suppose t h a t ZEW and z hP ( c l p x V ) . Then s i n c e h PPX ( c l P x V ) i s c l o s e d i n BY, t h e r e e x i s t s an open s e t S C PY s n h P (claxv) = 0. Hence, i f x
such t h a t z E S c W and P -1 E [h ] ( s ) , then h P ( x ) E
s
221
WZ- MAPPINGS
P P h (x) & , h I t follows t h a t f ( x ) = 0.
from which i t f o l l o w s t h a t x
p clpxV.
sup{f(x)
:
i n which c a s e [ g / Y ]( S ) C 1 whenever ycS. g(y) > z
h P (clPxV) and
F
hP
( c l P x v ) . Thus, Therefore,
[hp]-l(S) j = 0
x c
S c W
But
{O].
implies t h a t
This i s a c o n t r a d i c t i o n .
i s open a s a s s e r t e d .
Therefore,
This concludes
t h e p r o o f of t h e theorem. The n e x t theorem i s one of t h e main r e s u l t s o f t h i s secI t o r i g i n a l l y a p p e a r s i n t h e 1967 p a p e r of T . I s i w a t a
tion.
a l t h o u g h o u r proof i s due t o N . Dykes (1969, Theorem 4 . 2 ) and employs a t e c h n i q u e s i m i l a r t o t h a t used i n t h e p r o o f o f 1 7 . 1 0 . A s was p o i n t e d o u t i n t h e i n t r o d u c t i o n t o t h i s s e c t i o n , t h e
r e s u l t p r o v i d e s an i n t e r e s t i n g comparison w i t h 1 7 . 1 1 where t h e
r e s t r i c t i o n i s imposed on t h e r a n g e s p a c e ( i . e , , t h a t i t b e a weak
c b - s p a c e ) r a t h e r than on t h e open p r o p e r t y o f t h e map-
ping. 18.9
THEOREM ( I s i w a t a ) .
ous s u r j e c t i o n noff space Y
X
function
equality
then q
F
Since
of
f-l(y)
i s a Hewitt-Nachbin s p a c e . PY\Y and a p o i n t p E [ fP ] - 1 ( 9 ) .
x
h(x)
>
0
whenever
XEX
i s open and c l o s e d t h e mapping
f
I t follows t h a t
hi
t
c ( ~ Y ) where
and
fP is i h (y) =
[ f P J - ’ ( y ) ] . Now, i f h i ( y ) = 0, t h e n t h e 1 clPxf- ( y ) = [fP]-’(y) t og e t h e r with t h e f a c t t h a t
:
E
i s p o s i t i v e on
Hence, i n t f - l ( y )
X
i m p l i e s that
# 6 because
f - l ( y ) c a n n o t b e compact.
6fm1(y) i s compact.
f [ i n t f - l ( y ) J = ( y } i s open b e c a u s e
Therefore, each
Moreover,
i s an open mapping. Thus Yo = Z ( h i ) fl y and hence C-embedded t h e r e i n . f
y E Z(hi) i s i s o l a t e d .
i s b o t h open and c l o s e d i n
Y
A s i n t h e p r o o f of 1 7 . 1 0 ,
x E f-l(y). Y discrete subset of X a point
af-I(y)
o n t o a Tycho-
Y
such t h a t
E C(PX)
open by 1 8 . 8 ( 2 ) .
h
YEY,
X
i s Hewitt-Nachbin complete by 8 . 8 ( 3 ) t h e r e e x i s t s a h
h(p) = 0. inf[h(x)
i s an open and c l o s e d c o n t i n u -
such t h a t t h e boundary
Select a point
Since
f
from a Hewitt-Nachbin space
compact f o r each Proof.
If
f o r each p o i n t
y
E
Yo
choose
Then F = { x : y E Yo) i s a c l o s e d Y and hence i s Hewitt-Nachbin c o m p l e t e .
222
COMPLETENESS AND CONTINUOUS M A P P I N G S
i s a homeomorphism from i s a Hewitt-Nachbin space.
Moreover, f l F Yo
q
Next observa t h a t the p o i n t First
belongs t o
g
Z(hi) and i
G c Z(h )
meets
Y
(since
f~Y .
clPYyo * Thus, i f
then s o i s
PY,
u n
G
n
G
q . Hence, U
u for
BY), and t h e r e f o r e U m u s t Yo i s C-embedded i n Y by
i s dense i n
Y
contain p o i n t s of
in
containing
so t h a t
Yo
belongs t o
Yo = Z(hi)
q
i s an open neighborhood of
every open s u b s e t
onto
F
Yo.
Since
Yo
t h e f i r s t p a r t of the proof, i t follows t h a t
is also
C-
embedded i n P Y . Thus, c lPyYo = BYo. Therefore, t h e p o i n t q belongs t o BY,. By 8 . 8 ( 3 ) t h e r e then e x i s t s a non-negat i v e function
g
E
C ( P Y ) such t h a t
g(q) = 0
and
g(y)
>.
0
whenever y c Yo. F i n a l l y , t h e f u n c t i o n g + hi is positive on Y and s a t i s f i e s [g + h l ] (9) = 0 . Hence, by 8 . 8 ( 3 ) Y
i s a Hewitt-Nachbin space which completes t h e proof of t h e theorem. The previous r e s u l t a s s e r t s t h a t Hewitt-Nachbin
cornplete-
n e s s i s i n v a r i a n t under an open and closed continuous mapping provided t h a t t h e boundary of each f i b e r i s compact.
One
might wonder i f i t would be p o s s i b l e t o d r o p t h e l a s t condit i o n i n favor of some r e s t r i c t i o n on t h e range space. such s o l u t i o n i s given i n 18.12 below.
One
However, two lemmas
w i l l be u s e f u l i n e s t a b l i s h i n g t h a t r e s u l t .
The f i r s t of
these i s due t o I s i w a t a ( 1 9 6 7 , Theorem 6 . 1 ) and we w i l l omit The second lemma i s due t o
t h e lengthy and t e d i o u s p r o o f . Dykes (1969, Theorem 4 . 3 ) . 18.10
LEMMA ( I s i w a t a ) .
If € i s an open not i s o l a t e d , if -a
function
ever
XEX
then
Z(hi)
h
E
and
Let
X
and
Wz-mappinq from
Y
Tychonoff spaces.
x onto
Y,
if
YEY
is
f - I ( y ) i s not compact, and i f t h e r e e x i s t s
c(PX) such t h a t h(p) = 0
0
h
i 1,
f o r some p o i n t
Z ( P Y ) i s a neiqhborhood
p
of
E
y
h ( x ) > 0 when[f P ] - 1( y ) \ f - ’ ( y ) ,
& I BY.
18.11 LEMMA (Dykes). If f i s an oPen and c l o s e d continuous s u r j e c t i o n from a Hewitt-Nachbin space X o n t o a Tvchonoff -1 k-space Y , then t h e f i b e r f ( y ) is compact f o r every non-
isolated point
ycY.
WZ- MAPPINGS
Proof.
f - I ( y ) f a i l s t o be compact f o r some non-
Assume t h a t
isolated point and s i n c e
f
ycY. is a
f - l ( y ) cannot be c l o s e d i n WZ-mapping c l P x f - 1 ( y ) = [ f P ] - 1 ( y ) Then
i t i s possible t o select a p o i n t
PX,
.
from [ f P ] - ’ ( y ) / x .
p
Hence Since
i s Hewitt-Nachbin complete t h e r e e x i s t s a f u n c t i o n
X
h
223
C ( P X ) such t h a t
E
hood of
in
y
h(x)
the zero-set where
Y
1,
h
Q
By 18.10
h(p) = 0.
Z(hi)
F
whenever
0
n
Z(hi) Z(PY)
.
Moreover, a s i n t h e fP [Z(h)] = Z(hi).
proof of 1 7 . 1 0 , one can e a s i l y show t h a t However, by 1 7 . 3
n
Z(hi)
Hence t h e p o i n t
is discrete.
Y
X I X , and
i s a neighbor-
Y
y
This i s a c o n t r a d i c t i o n .
is isolated.
The n e x t r e s u l t i s C o r o l l a r y 4 . 4 o f Dykes’ 1 9 6 9 p a p e r .
If
THEOREM ( D y k e s ) .
18.12
f
i s an open and c l o s e d c o n t i n u -
-
ous s u r j e c t i o n from a Hewitt-Nachbin
noff
k-space
Proof.
If
open i n
then
Y,
space
o n t o a Tycho-
X
i s a Hewitt-Nachbin space.
Y
i s an i s o l a t e d p o i n t o f Y , t h e n f - l ( y ) i s 1 f- ( y = i n t f - ’ ( y ) . Thus, t h e boundary
y
so t h a t
X
6 f - l ( y ) i s empty and hence compact.
Otherwise, y
i s o l a t e d from which i t f o l o w s t h a t
bf-’(y)
i s non-
i s compact a s a
Tha r e s u l t i s now immediate from 1 8 . 9 .
consequence of 18.11.
F i n a l l y , w e should l i k e t o f o c u s o u r a t t e n t i o n on t h e i n v e r s e i n v a r i a n c e o f Hawitt-Nachbin c o m p l e t e n e s s under mappings.
The f i r s t r e s u l t p r o v i d e s a c h a r a c t e r i z a t i o n o f
Hewitt-Nachbin c o m p l e t e n e s s i n t e r m s o f 18.13
Let
THEOREM ( D y k e s ) .
Tychonoff s p a c e
-i s Hewitt-Nachbin f o r e v e r y ytY. Proof.
WZ-
f
be a
WZ-mappings
WZ-mapping from a
o n t o a Hewitt-Nachbin s p a c e
X
complete i f and o n l y
.
if
Y.
cluXf-’(y)
Then
X
= f-l(y)
The n e c e s s i t y of t h e c o n d i t i o n i s immediate s i n c e
c l o s e d s u b s p a c e s o f a Hewitt-Nachbin s p a c e a r e Hewitt-Nachbin Conversely, l e t
complete. f
cl
to PX
f
= f
P
fv
lux.
f - l ( y ) it i s t h e c a s e t h a t
P -1
[f ]
Then
iiX.
U
(y)
n
ux
=
[f
v -1
1
(y).
d e n o t e t h e unique e x t e n s i o n of Moreover, s i n c e [f’]]-’(y) ~ l ~ ~ f - =~ c (l y f )- l ( y )
I t follows t h a t
PX
=
n
uX =
224
COMPLETENESS AND CONTINUOUS MAPPINGS
ux
=
u i I f U 3 -1 ( y )
: YEY)
= Li I c l , J , f - l ( y )
: Y€Y!
= ii ( f - l ( y ) : Y E Y )
=
Therefore, X
x.
i s Hewitt-Nachbin complete which concludes t h e
proof of t h e theorem. The following lemma i s needed t o e s t a b l i s h t h e main r e s u l t (18.15) concerning t h e i n v e r s e i n v a r i a n c e of H e w i t t Nachbin completeness under 18.14
LEMMA.
-
X
noff space
complete Proof.
and Let
&&
f
WZ-mappings.
&5
onto a space C-embedded -1 S = f (y).
c l u x S = US by 8.11.
in
c o n t i n u o u s s u r j e c t i o n from a TvchoY.
If
X
then
Since
Since
assumption, i t follows t h a t
S
f - l ( y ) i s Hewitt-Nachbin
1
cluxf-
is
(y) = f - l ( y ) .
C-embedded i n
X,
i s Hewitt-Nachbin complete by US = S . The r e s u l t i s now imme-
S
diate. Note t h e s i m i l a r i t y of t h e n e x t theorem t o t h a t s t a t e d
i n 1 6 . 1 i n t h e sense t h a t t h e c o n d i t i o n f o r t h e mapping t o be "2-closed" i n 1 6 . 1 i s r e p l a c e d by t h e weaker c o n d i t i o n of tlWZ-mapping,
b u t t h e 'fz-embeddingfa of each Hewitt-Nachbin
complete f i b e r i n 1 6 . 1 i s r e p l a c e d by t h e s t r o n g e r c o n d i t i o n of "C-embedding.
I'
The two r e s u l t s a r e e v i d e n t l y independent
f o r a r b i t r a r y Tychonof f s p a c e s . THEOREM (Dykes).
18.15
noff space
-
X
-
YEY,
fiber f-l(y) each p o i n t Proof.
f
is a
WZ-mapping from a Tychospace
i s Hewitt-Nachbin complete
then
By 1 8 . 1 4
fore, X
If
o n t o a Hewitt-Nachbin
Y
and
such t h a t t h e C-embedded
i s a Hewitt-Nachbin s p a c e . 1 c l U xf - l ( y ) = f - ( y ) f o r each ycY.
for
X
There-
i s Hewitt-Nachbin complete by 1 8 . 1 3 .
S i n c e every L i n d e l c f subspace of a Tychonoff space
X
is
z-embedded i n
is
C-embedded i f and o n l y i f i t i s completely s e p a r a t e d from
X (10.7(2))
and s i n c e a
z-embedded s u b s e t
E- PERFECT MAPPINGS
225
every z e r o - s e t d i s j o i n t from i t ( 1 0 . 4 ) , t h e f o l l o w i n g c o r o l I t i s C o r o l l a r y 4 . 9 of Dykes'
l a r y may be e a s i l y e s t a b l i s h e d . 1969 p a p e r . COROLLARY (Dykes)
18.16
-a
Tychonoff space
t h a t each f i b e r --X
Then
f-l(y)
is L i n d e l o f
Z
cp
in
such
Y
f o r each p o i n t
y
&
ycY,
then
and
Z
X
z-embedded i n
y
and
f(Z).
The func-
Thus, f - l ( y ) i s
f-'(y).
by
f-'(y).
Hence t h e r e i s a
f(Z).
C ( Y ) that separates
separates
embedded i n
space
i s a z e r o - s e t d i s j o i n t from
f ( Z ) i s a c l o s e d s e t and cpof
z - c l o s e d mapping from
space.
Suppose t h a t
function tion
2
f
f - l ( y ) i s Lindelof i t i s
Since
10.7(2).
If
o n t o a Hewitt-Nachbin
X
i s a Hewitt-Nachbin
Proof.
.
C-
F i n a l l y , s i n c e Lindelof spaces a r e H e w i t t -
X.
Nachbin complete t h e r e s u l t i s immediate from t h e theorem. Section 19 :
E - P e r f e c t Mappinqs
I n t h i s s e c t i o n we w i l l c o n s i d e r a g e n e r a l i z a t i o n of t h e n o t i o n of a p e r f e c t mapping i n connection with t h e p r e s e r v a t i o n of
E-compactness
S e v e r a l of t h e re-
(see Section 4 ) .
s u l t s w e have o b t a i n e d p r e v i o u s l y concerning t h e i n v e r s e i n v a r i a n c e of Hewitt-Nachbin completeness can be e s t a b l i s h e d b y The d e f i n i t i o n o f an " E - p e r f e c t " mapping i s
t h i s approach.
motivated by t h e f o l l o w i n g r e s u l t concerning p e r f e c t mappings. _Let
f
d e n o t e a c o n t i n u o u s s u r j e c t i o n from
t h e Tvchonoff space
X
onto t h e Tychonoff space
19.1
THEOREM.
-
are
equivalent:
(1) The magpinq
If
(2)
L
is 2
f
perfect.
Z-ultrafilter
ycY,
converqes t o a p o i n t point
x
condition
L
be a
point
Z - u l t r a f i l t e r on ycY.
L
then
f p : pX
Let
X
f
f'(L)
converqes t o a
--f
PY
satisfies
the
be a p e r f e c t mapping and l e t
such t h a t
Note f i r s t t h a t i f
x
such t h a t
X
fp(pX\X) c pY\Y.
(1) i m p l i e s ( 2 ) :
then n e c e s s a r i l y
on
fT1(y).
The Stone e x t e n s i o n
(3)
Proof.
E
The f o l -
Y.
belongs t o
L f-
f # (Ir) converges t o a
converges t o a p o i n t
1( y ) .
For i f
Ir
XEX,
converges
COMPLETENESS AND CONTINUOUS MAPPINGS
2 26
x, then x F n Lc so t h a t x E f-'(Z) f o r e v e r y 2 E f # (It). Thus f ( x ) E Z f o r e v e r y Z E f # ( L A ) , and s i n c e f # (Ir) i s a prime 2 - f i l t e r on Y i t f o l l o w s from 6 . 1 2 t h a t f # (11) conv e r g e s t o f ( x ) , Because Y i s a Hausdorff s p a c e , f ( x ) = y to
.
so t h a t x E f - l ( y ) Next w e e s t a b l i s h t h a t
Suppose n o t .
I4
t h a t f o r each Zx
converges.
f a i l s t o have a c l u s t e r p o i n t i n f - l ( y ) s o -1 x E f ( y ) t h e r e i s a z e r o - s e t neighborhood Zx
Then, by 6 . 1 2 , such t h a t
L
f - l ( y ) i s compact i t i s covered by
Since
LA.
j!
a f i n i t e s u b f a m i l y (Zx jy=l, i
and t h e z e r o - s e t
Z
*
n
U Zx
=
i=l i
L b e c a u s e Lc i s a l s o a prime Z - f i l t e r . T h e r e f o r e , by 6 . 8 ( 3 ) t h e r e e x i s t s a z e r o - s e t Z1 C X\Z* with Z1 E LA s i n c e Ir i s a Z - u l t r a f i l t e r . Because f i s a cannot belong t o
c l o s e d mapping and
Z1
i s a neighborhood o f
Il
y.
f - l ( y ) = fi i t follows t h a t Y \ f ( z l ) A l s o f 8 (Ir) c o n v e r g e s t o y by
assumption so t h e r e i s a z e r o - s e t and
Z'
f8(Lc).
L
c Y\f(Z1). But
Hence
n
f-'(Z1 )
(2) implies ( 3 ) :
5
ultrafilter
Let on
X
E
E Lr
f-'(Z')
Z(Y) with
Z'
E
f#(LA)
from t h e d e f i n i t i o n of
which i s a c o n t r a d i c t i o n .
Z1 =
converges t o a p o i n t i n
2'
Thus
f - l ( y ) which p r o v e s ( 2 ) .
p E PX.
Then t h e r e e x i s t s a u n i q u e
such t h a t
j u s t t h e a n a l o g u e of 8 . 4 ( 5 ) f o r
5
converges t o
Z-
p ( t h i s is
P X ; see G i l l m a n and J e r i s o n
f # (3) c o n v e r g a s t o a P p o i n t q i n PY ( i n f a c t , q = f ( p ) a c c o r d i n g t o 6 . 6 ( a ) o f Gillman and J e r i s o n ) . I f q b e l o n g s t o Y t h e n 5 conv e r g e s t o a p o i n t x i n f - I ( q ) by a s s u m p t i o n . S i n c e PX i s
6.G f o r t h e d e t a i l s ) .
Hausdorff, n e c e s s a r i l y
I t follows t h a t
x = p
so t h a t
pcX.
which a r e mapped t o p o i n t s o f
p oi nt s of
PX
p o i n t s of
PXb.
Thus t h e o n l y PY\Y
a r e the
This proves s t a t e m e n t ( 3 ) .
pX i s compact, f P i s a c l o s e d mapping, and t h e i n v e r s e image of e v e r y compact s e t under f p i s c l e a r Therefore l y compact. Also, by assumption, [ f p ] - l ( U ) = X . t h e mapping f = f P IX h a s t h e same p r o p e r t i e s a s f p b e c a u s e i t i s t h e r e s t r i c t i o n of f P t o a t o t a l preimage. T h i s conc l u d e s t h e proof of t h e theorem. ( 3 ) i m p l i e s (1): Now
Motivated b y t h e c o n d i t i o n i n s t a t e m e n t ( 3 ) o f the pre-
E- PERFECT MAPPINGS
227
*
ceding theorem w e n e x t d e f i n e a g e n e r a l i z e d concept of perf e c t mappings.
--
the space
E
Throuqhout
we w i l l assume - s e c t i o n ----
Also, i f
Hausdorff s p a c e .
--
E-completely r e g u l a r Hausdorff spaces and
X
mapping of
pEX
from
into
into
then
Y,
that are
Y
i s a continuous
f
w i l l denote t h e e x t e n s i o n
f*
(see 4.3 ( 2 ))
BEY
and
X
.
The f o l l o w i n g c o n c e p t s a r e
found i n the 1973 paper by J . H . T s a i .
19.2
Let
DEFINITION.
spaces and l e t
f
and
X
be
Y
E-completely r e g u l a r
be a continuous s u r j e c t i o n from
(1) The mapping
i s s a i d t o be
f
i f i t maps each
o n t o Y.
E-closed s u b s e t ( s e e 3 . 7 ) of
t o a c l o s e d s u b s e t of
(2) The mapping
X
E-closed i f and o n l y X
Y.
i s s a i d t o be weakly E-closed i f * - 1 ( y ) f o r each y ~ y . c l p .f-'(y) = [f 1 f
and only i f
E
(3)
The mapping only i f
i s said t o be
f
E - p e r f e c t i f and
c P,Y\Y.
f*(p,x\rc)
I n t e r p r e t i n g t h e above d e f i n i t i o n we s e e t h a t a c l o s e d mapping i s simply an 19.2 (1), where
i s a weakly which i s
i s t h e u n i t i n t e r v a l [0,1]: a
1
z-
I - c l o s e d mapping a c c o r d i n g t o WZ-mapping
I - c l o s e d mapping; and a p e r f e c t mapping i s one
I-perfect.
B l a i r has i n v e s t i g a t e d t h e concept
R.
I R - p e r f e c t mapping i n h i s 1969 paper and c a l l e d i t a
of an
Taking i n t o account t h a t w e always
" r e a l - p r o p e r mapping." have t h e i n c l u s i o n
f-l(y) c c l
f-l(y)
C
[f*]-'(y),
t h e con-
BEX
d i t i o n t h a t a mapping be (a)
f-'(y)
= clp
xf-
1
(Y)
E - p e r f e c t s p l i t s i n t o two e q u a l i t i e s : and
(b)
c l p .f-l(y)
E E Condition ( b ) i s simply t h e c o n d i t i o n t h a t closed.
f
= [f
* -1 3 (Y).
i s weakly
E-
We w i l l i n v e s t i g a t e when c o n d i t i o n ( a ) i s s a t i s f i e d
f u r t h e r on i n t h e s e q u e l .
The n e x t s e v e r a l r e s u l t s r e l a t e t h e
v a r i o u s c l a s s e s of mappings d e f i n e d above and a r e found i n T s a i ' s 1 9 7 3 paper. 19.3
THEOREM ( T s a i ) .
Proof.
Every c l o s e d mappinq
This i s immediate s i n c e every
E-closed.
E-closed s e t i s c l o s e d .
The f o l l o w i n g lemma w i l l be u s e f u l i n e s t a b l i s h i n g t h a t
COMPLETENESS AND CONTINUOUS MAPPINGS
228
every
E-closed mapping i s weakly
If
LEMMA ( T s a i ) .
19.4
E-closed.
is a r e q u l a r s p a c e and i f
E
F c X
E-completely r e q u l a r , then f o r each c l o s e d s u b s e t point
p&F t h e r e e x i s t s an
fyinq
p c int A
Proof.
Since
and
is
X
n
A
E-closed s u b s e t
A
X
C
X
and
satis-
a.
F =
E-completely r e g u l a r , b y 3 . 3 ( b ) t h e r e
e x i s t s a f i n i t e number
n
and a continuous f u n c t i o n
c l n f ( F ) . Since En is regular E f ( p ) and t h e r e a r e d i s j o i n t open neighborhoods U and V of -1 n Define A = f (E \V). Clearly c l f ( F ) , respectively. such t h a t
f E C(X,En)
&
f (p)
En
p
int A
E
A r! F = @
and
which concludes t h e argument. z-
The n e x t r e s u l t g e n e r a l i z e s t h e f a c t t h a t every c l o s e d mapping i s a
WZ-mapping ( 1 8 . 2 (1))
THEOREM ( T s a i )
19.5
E-closed mappinq Proof.
Let
. If
Y.
i n t o the
X
Suppose t h a t ycY
Then t h e r e e x i s t s a p o i n t cl
E- c l o s e d .
E-closed mapping from t h e
r e g u l a r Hausdorff space Hausdorff space
i s a r e q u l a r s p a c e , then every
E
weakly
be an
f
.
E-completely r e g u l a r
i s n o t weakly
f
and a p o i n t
p
BEX
set
of
A
6.
Let
so t h a t
E
such t h a t
pEX
M = A
n
X.
Then
p M
f (M) i s closed i n
M fl f - l ( y ) = @
clp yf(M).
E
By t h e p r e v i o u s lemma t h e r e i s an
f-l(y).
so t h a t
y
i s an
and
f (M)
This i m p l i e s t h a t
f (M), which i s a c o n t r a d i c t i o n ,
y
.
E
E-closed.
[f*] - 1( y ) \
E-closed sub-
n
c l p .f-’(y) E E-closed s u b s e t o f X A
by assumption.
Y E ,’
int A
E
E-completely
=
Now,
On t h e o t h e r hand,
c l p y f ( ~ )n Y = c l f(M) = Y E
The n e x t r e s u l t g e n e r a l i z e s t h e f a c t t h a t t h e i n v e r s e image of a compact space under a p e r f e c t mapping i s compact. (See a l s o 1 6 . 2 which g i v e s t h e i n v e r s e i n v a r i a n c e of H e w i t t Nachbin completeness under p e r f e c t mappings.)
E- PERFECT MAPPINGS
19.6
the
Let
THEOREM ( T s a i ) .
f
be an
229
E - p e r f e c t mapping from
E-completely r e q u l a r Hausdorff s p a c e
p l e t e l y r e q u l a r Hausdorff s p a c e
then
onto the
X
Since
space of
c pEY\Y
f*(p,X/X)
image o f e v e r y
E-compact
i t is c l e a r t h a t the i n v e r s e
subspace o f
i s an
Y
E-compact sub-
X.
R e c a l l from D e f i n i t i o n s 3 . 1 and 4 . 1 t h a t
@ ( E ) and
d e n o t e t h e c l a s s e s of
E-completely r e g u l a r and
spaces, r e s p e c t i v e l y .
I n 4 . 2 ( 4 ) i t was found t h a t i f
8 (El)
a r e two Hausdorff s p a c e s w i t h
E2
R(E2)
E - z -
E-compact,
Y
E-compact.
X
Proof.
If
Y.
i f and o n l y i f
El
= @ (E2),
and
El
then
R(E1)
C
An e q u i v a l e n t f o r m u l a t i o n
R(E2).
E
R(E)
E-compact
o f t h a t r e s u l t i s found i n Mr6wka's 1968 paper a s f o l l o w s , a l though w e o m i t t h e proof h e r e . 19.7
6 (El)
i f f o r each ---
X
pE X
.
THEOREM (Mrdwka)
spaces with
p
into
E
&&
and
El
= @ (E2).
Then
b e t w o Hausdorff
E2
R(E )
C
1
R(E2)
i f and o n l y
--
t h e r e e x i s t s 2 homeomorphism 1 which i s t h e i d e n t i t y on X .
@(E )
X
h
from
2 W e can now r e l a t e weakly
E - c l o s e d mappings t o
E -per-
1
2
f e c t mappings. 19.8
THEOREM ( T s a i )
spaces w i t h
-b e two a
.
@(El) = @(EZ)
E1-completely
weakly
Let
El-closed
and
El
and
R(E1)
d e n o t e two Hausdorff
E2
Let
C R(E2).
and Y f be
X
r e q u l a r Hausdorff s p a c e s and l e t
mapping from
onto
X
Then t h e f o l -
Y.
lowinq s t a t e m e n t s a r e t r u e :
(1)
The mappinq
f
is
E 2 - p e r f e c t i f and o n l y i f
f - l ( y ) f o r each (2)
If
if Proof.
Y
E
R(E2),
X
E
R(E2).
then
f
E 2 - p e r f e c t i f and o n l y
Throughout t h i s p r o o f w e w i l l l e t
n o t e t h e e x t e n s i o n s of
f
from
ycY.
BE X
to
1
fl
*
pE Y 1
pE Y, r e s p e c t i v e l y . 2 (1) Assume f i r s t t h a t
f
is
E2-perfect.
Then
and and
f2
*
de-
pE X 2
to
COMPLETENESS AND CONTINUOUS MAPPINGS
2 30
*
f 2 (BE X\X) 2 which i s c l o s e d i n
Y E Y , f- 1 ( y ) = [ f 2 * ] - 1( y )
Thus f o r each
ycY
Conversely, assume t h a t f o r each i s closed i n
y
Y.
is i n
PE2X.
Let
Since
f
the f i b e r
p c PE X and suppose t h a t f 2 2 i s weakly E - c l o s e d , we have
1
= f-
f2
*
(2)
a r e t h e p o i n t s of Assume t h a t
then
X
is
compact then
is
(PI =
n PE 2 x
(y) c
x.
BE X
t h a t a r e mapped i n t o Y 2 X ; whence f is E2-perfect. E2-compact.
E2-compact by 1 9 . 6 . 8, X = X by 4 . 4 . 2 cl
Hence
Y
*
1
= c l p E, . f - l ( Y )
Thus, t h e only p o i n t s of
f-’(y)
1
PE Lqx
follows immediately t h a t
is
f
X
is
E2-
ycY
(y) = f-l(y).
PE X
f o r each
2 is
E2-perfect.
f
E2-perfect
if
Thus, f o r each
f - l ( y ) = clxf-
f - l ( y ) i s closed i n
If
Conversely,
by
ycY
from which i t
T h i s concludes
the proof. Before we c o n s i d e r i n t e r p r e t a t i o n s of t h e p r e v i o u s r e -
s u l t we c o n s i d e r t h e following concept and i t s consequences. I t g e n e r a l i z e s t h e n o t i o n s of
19.9
DEFINITION.
l o g i c a l space
X.
Let
S
Then
S
C-
and
be a non-empty s u b s e t o f t h e topo-
i s s a i d t o be
i f every continuous f u n c t i o n from t i n u o u s e x t e n s i o n from
X
*
C -embedding.
into
S
into
E-embedded E
X
admits a con-
E.
I n t h e above terminology we s e e t h a t a C-embedded sub* s e t i s the c a s e where E = IR , and a C -embedded s u b s e t corresponds t o t h e c a s e where
E
i s t h e u n i t i n t e r v a l [0,1].
E- PERFECT MAPPINGS
By t h e
Theorem 4.3(1) w e see t h a t e v e r y
E-Compactificatian
c o m p l e t e l y r e g u l a r Hausdorff s p a c e
pEX.
E-compactification
231
is
X
E-
E-embedded i n i t s
The f o l l o w i n g r e s u l t a p p e a r s i n t h e
1 9 7 3 p a p e r by T s a i .
19.10
THEOREM ( T s a i )
--t i o n from t h e
. Let
f
be a c l o s e d continuous s u r j e c -
E-completely r e q u l a r Hausdorff s p a c e
t h e E-completely r e q u l a r Hausdorff s p a c e Y , and 1 be a r b i t r a r y . I f t h e f i b e r f - ( y ) is E-compact dedi n X, then f - l ( y ) i s c l o s e d i n pEX. Proof.
Since
f-l(y) is
onto
X
let
ycY
and
E-embed-
pEf -1( y ) = f - 1( y ) .
E-compact,
s e q u e n t l y i t i s s u f f i c i e n t t o show t h a t
Con-
1
c l p X f - ( y ) = pEf-l(y). E
E-compact b e c a u s e i t i s a c l o s e d s u b s e t
f-l(y) is
NOW, c l
of t h e
E-compact
embedded i n sequently,
X
space
f - l ( y ) i.s
according t o 4. 3 ( 3 ) ,
Moreover,
PEX.
E-embedded i n
it is
E-emhedded i n
f - l ( y ) is
E-
by 4 . 3 ( 1 ) ; con-
f - l ( y ) . However, PEX i s t h e unique E-compact s p a c e
pEf-'(y)
f P 1 ( y ) i s d e n s e and
i n which
since
BEX
cl
E-embedded.
Thus, pEf-
1(y)
=
f - l ( y ) which c o n c l u d e s t h e p r o o f .
cl BEX
Because of t h e p r e v i o u s r e s u l t w e now have a s u f f i c i e n t c o n d i t i o n which y i e l d s t h e e q u a l i t y ( a )
f-l(y) = cl
f-l(y) PEX
demanded f o r a mapping f t o be E - p e r f e c t ; namely, t h a t e a c h 1 Thus w e f i b e r f - ( y ) b e E-compact and E-embedded i n X. see t h a t 2 mappinq
f
&
-----
c l o s e d and each f i b e r i s
domain every
X.
Since every
E - p e r f e c t whenever i t i s weakly E-compact
and
E-closed map i s weakly
C-embedded s u b s e t i s
z-embedded,
E-
E-embedded i n t h e E - c l o s e d and
w e see t h a t e v e r y
z-
c l o s e d mapping f o r which e a c h f i b e r i s H e d i t t - N a c h b i n complete and
C-embedded i s
IR-perfect.
T h i s o b s e r v a t i o n coupled w i t h
1 9 . 6 immediately g i v e s an a l t e r n a t i v e p r o o f t o B l a i r ' s r e s u l t 16.1.
W e a l s o o b t a i n t h e r e s u l t s 1 8 . 1 5 and 18.16 by t h e same
interpretation.
he n e x t r e s u l t w i l l p r o v i d e us w i t h a d d i t i o n -
a l interpretations 19.11
.
THEOREM ( T s a i ) .
-t h e same
Let
E,,
hypotheses a s i n 19.8.
E2,
X, Y ,
and
I f the f i b e r
f
satisfy
f-'(y)
is
E2-
232
COMPLETENESS AND CONTINUOUS M A P P I N G S
compact
and
E2 -pe r f e c t
Proof. f
.
ycY, then
f o r each
X
x
BE
f - l ( y ) i s closed i n
By 1 9 . 1 0
is
in
E2-embedded
is
f
so t h a t by 19.8(1)
2
E2-perfect.
We now formally i n t e r p r e t t h e above r e s u l t s f o r t h e c a s e
.
S e t t i n g E l = [0,1] spaces (when E = IR) i n 1 9 . 8 and 1 9 . 1 1 we immediately o b t a i n t h e fol-
of Hewitt-Nachbin and
E 2 = IR
lowing r e s u l t s . 19.12
COROLLARY.
space
X
Let
2
f
WZ-mappinq from t h e Tychonoff
o n t o t h e Tvchonoff space
---
The f o l l o w i n q
Y.
state-
ments are true : (1)
The
mapping
f-’(y) (2)
&J
f
is
=-perfect
if
I R - p e r f e c t i f and only
i s c l o s e d i n UX f o r each y6Y. Y be a Hewitt-Nachbin s p a c e . Then i f and o n l y i f
X
f
i s a Hewitt-Nachbin
space. (3)
If
f - l ( y ) is Hewitt-Nachbin
-ded i n X f o r feet mapping. -
each
YEY,
complete
then
and
i s an
f
C-embed-
m-per-
Comparing 19.12 w i t h p r e v i o u s l y o b t a i n e d r e s u l t s w e s e e t h a t s t a t e m e n t ( 2 ) of t h e above r e s u l t i s simply a r e s t a t e m e n t of 18.13, and t h a t s t a t e m e n t ( 3 ) coupled with 19.6 g i v e s 18.15. W e a l s o have t h e f o l l o w i n g c o r o l l a r y . 19.13
COROLLARY
Tvchonoff space
-of the
(Tsai) X
Let
&2
f
followinq c o n d i t i o n s h o l d s , then
(2)
(4)
f
If any one
Y.
is
f - l ( y ) i s Hewitt-Nachbin
IR-perfect: complete
z-embedded
&I
The f i b e r
f - l ( y ) i s Hewitt-Nachbin complete
*
C -embedded
(3)
WZ-mappinq from t h e
o n t o t h e Tvchonoff space
(1) The f i b e r
Proof.
.
The space
X
b i n complete The f i b e r
(1)
By 1 5 . 1 6
X
f o r each
X f o r each i s normal and f o r each
f-l(y)
is
and
ycY. ysy. f - l ( y ) i s Hewitt-Nach-
ycY. Lindelof f o r each
f - l ( y ) is
C-embedded i n
s u l t i s now immediate from 19.12 (3)
.
YEY. X.
The re-
E- PERFECT MAPPINGS
(2)
S i n c e every
C*-embedded s u b s e t i s
233
z-embedded s t a t e m e n t
( 2 ) i s immediate from s t a t e m e n t ( 1 ) .
(3)
Every c l o s e d s u b s e t of a normal s p a c e i s
*
C -embedded so
t h i s r e s u l t i s immediate from p a r t ( 2 ) . (4)
Every L i n d e l o f subspace i s
z-embedded
( 1 0 . 7 ( 2 ) ) so t h e
r e s u l t f o l l o w s from s t a t e m e n t ( 1 ) . Comparing 1 9 . 1 3 w i t h p r e v i o u s r e s u l t s w e see t h a t s t a t e -
m e n t (1) g i v e s B l a i r ' s r e s u l t 1 6 . 1 , s t a t e m e n t ( 2 ) g i v e s 1 8 . 1 5 , and s t a t e m e n t ( 4 ) g i v e s 1 8 . 1 6 . The f o l l o w i n g two c h a r t s p r o v i d e a summary o f t h e res u l t s t h a t have been o b t a i n e d i n t h i s c h a p t e r . I n t h e f i r s t c h a r t , which summarizes t h e r e s u l t s r e l a t i n g t o t h e i n v e r s e i n v a r i a n c e o f Hewitt-Nachbin c o m p l e t e n e s s , i t i s assumed t h a t t h e mapping i s a c o n t i n u o u s s u r j e c t i o n , t h a t t h e domain i s a Tychonoff s p a c e , and t h e r a n g e i s Hewitt-Nachbin c o m p l e t e . Any a d d i t i o n a l r e s t r i c t i o n on e i t h e r the mappings o r t h e spaces involved a r e s o i n d i c a t e d . A r e f e r e n c e t o t h e proof o f each p a r t i c u l a r r e s u l t i s a l s o p r o v i d e d . The second c h a r t i s e n t i r e l y s i m i l a r e x c e p t t h a t t h e domain s p a c e i s assumed t o be Hewitt-Nachbin complete and t h e r a n g e space t o be Tychonoff. I t summarizes t h e i n v a r i a n c e of Hewitt-Nachbin c o m p l e t e n e s s
under c o n t i n u o u s mappings.
2 34
m
d
m d
a, rl
a, d
II
h
h
v
I lu
d
d
U
d
4 E
!ii
f
:
X
-f
Y CONTINUOUS SURJECTION
Y TYCHONOFF
X HEWITT- NACHBIN
REFERENCE
16.10
Open perfect
6 f - I ( y ) compact
Open, c l o s e d , Perfect
weak cb- space
16.13
Perfect
p s e ud ocompa c t
16.14
Open, c l o s e d
k- space
18.12
z-open,
I
18.9
f i b e r - c o u n t a b l y compact
16.9 1 7 . 1 7 (1)
Hyper-real ~~
Zero- s e t p r e s e r v i n g , f i b e r c o u n t a b l y compact Closed,
f i b e r - c o u n t a b l y compact
Closed, 6 f -
16.8 normal, c o u n t a b l y p a r a c ompac t
16.6
( y ) compact
weak cb- sDace
I
Closed
normal, weak cb- , k- space weak cb- ,qspace
Closed Closed
f i r s t countab l e , cb- space
z- c l o s e d , f i b e r - c o u n t a b l y compact
cb- space
z-closed,
normal, countab l v paracomDact
f i b e r - c o u n t a b l y compact
I I
17.11 17.10
17.14 17.15 17.20
i
17.21
h) W ul
This Page Intentionally Left Blank
237
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"An example o f a non-normal c o m p l e t e l y r e g u l a r s p a c e , " B u l l . Acad. Polon. S c i . S e r . S c i . Math. Astronom. Phvs. 6 ( 1 9 5 8 1 , 161-163.
1 9 58D
"On t h e unions of Q - s p a c e s , " B u l l . Acad. Polon. S c i . S e r . S c i . Math. Astronom. Phvs. 6 ( 6 ) ( 1 9 5 8 1 , 365- 368.
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"On t h e form of p o i n t w i s e c o n t i n u o u s p o s i t i v e f u n c t i o n a l s and isomorphisms of f u n c t i o n s p a c e s , " S t u d i a Math. 21 ( 1 9 6 2 1 , 1 - 1 4 .
1964
"An e l e m e n t a r y p r o o f o f K a t e t o v ' s theorem concerni n g Q-spaces," Michisan Math. J . 11 ( 1 9 6 4 ) , 6163.
1965
Math. -27
1966
"On E-compact s p a c e s . 11," B u l l . Acad. P o l o n . S c i . S e r . S c i . Math. Astronom. Phvs. 14 (Ll) ( 1 9 6 6 1 , 597-605.
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" F u r t h e r r e s u l t s on E-compact Math. 120 ( 1 9 6 8 1 , 1 6 1 - 1 8 5 .
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1972A
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1972B
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INDEX
Indexing of i n d i v i d u a l s i s f o r c i t e d theorems o n l y . admits (a uniform s t r u c t u r e ) ,
138
admissible uniform s t r u c t u r e , 138
b o r n o l o g i c a l , 3, 155 u l t r a b o r n o l o g i c a l , 155 Bourbaki f i l t e r , 43, 44, 48,
52, 70, 71, 91
No, 92
i n complete uniform space, 141
almost realcompact space, 157 i f and only i f , 159 i n v a r i a n c e and i n v e r s e i n v a r i a n c e o f , 196 necessary c o n d i t i o n f o r Hewitt-Nachbin space, 162 n o t Hewitt-Nachbin comp l e t e , 196 p r o p e r t i e s o f , 162, 163 weak cb-space, 166 Alo and Shapiro, 100, 101,
102, 104, 108, 112, 113 a-Cauchy family, 157
Buchwalter and Schmets, 155 cardinality, 6 measurable, 90 nonmeasurable, 90, 91 c a t e g o r y , 33
50, 140, 149 neighborhood 2- f i l t e r , 141 r e a l Z - u l t r a f i l t e r , 153 cb- space, 163 i f and only i f , 164, 165
Cauchy
a r b i t r a r i l y small s e t ( i n uniform s p a c e ) , 140
i n v a r i a n c e of completeness, 2 1 0 , 212 p r o p e r t i e s o f , 164 v s . normal and countab l y paracompact, 164 weak cb-space, 163
archimedean ordered f i e l d , 144 Arens,
124
A r h a n g e l s k i i , 199
C-embedded s u b s e t , 30, 81, 86 compact s u b s e t , 31 d - d i s c r e t e s e t , 140 Hewitt-Nachbin subspace,
B a r t l e , 71 base
95
f o r closed s e t s , 6 f o r u n i f o r m i t y , 138 f o r % f i l t e r , 43 l o c a l base, 46, 49, 5 1 ,
i f and only i f , 3 1 , 111 normal space, 3 1
*
52, 57
normal, 57 bimorphism, 34 B l a i r , 80, 85, 109, 110, 111, 112, 114, 115, 116, 117,
118,,120, 126, 127, 128, 175, 176, 178, 179, 180, 181, 182, 187, 210, 211, 212, 227,
Blefko, 1 7
Z-filter,
*
v s . C -embedded, 31 v s . P-embedded, 124, 125 v s . z-embedded, 111, 112
C -embedded s u b s e t ,
30
completely s e p a r a t e d subsets, 31 i f and only i f , 31, 110 products , 130 v s . z-embedded, 109, 112
@(X), 140, 141 H e w i tt-Nachbin complete-
n e s s , 145, 146
INDEX
262
140,
Ch (X),
141,
146
Wallman-Frink, 4 4 ,
Banach s p a c e , 1 2 1
compact s p a c e , 8,
31,
57
60,
87,
124
class
a d m i t s unique u n i f o r m i t y ,
of compactness, 2 3 of complete r e g u l a r i t y , 15,
143
d e n s e subspace, 1 7 6 E- compact , 2 3 Hewitt-Nachbin complet i o n , 123 u n i f o r m s p a c e , 141, 146
18
clopen s e t , 18, 29, 6 4 c l o s e d mapping, 1 7 4 i n v a r i a n c e o f completeness, 2 2 1 i s E-closed, 227 not fiber-compact, 1 8 4 vs. open, 2 2 0 z-closed, 174, 1 8 1 closure, 6
complete c o l l e c t i o n of c o n t i n u o u s f u n c t i o n s , 1 60
of open c o v e r i n g s , 157 c o m p l e t e l y r e g u l a r s p a c e , 7, 21,
uniform topology, 1 3 8 cluster point of a n e t , 7 0 of a & f i l t e r b a s e , 4 5 of a 3 - f i l t e r on X , 45,
51
c l o s e d under c o u n t a b l e i n t e r s e c t i o n s , 52, 56
complement g e n e r a t e d , 53
d e l t a r i n g , 53 d i s j u n c t i v e , 46, 49, 57 normal, 4 5 , 49, 52, 53, 57
r i n g , 4 2 , 53, 5 7 8- d i s j unc t i v e , 4 5 8
i f and o n l y i f , 1 2 4 subparacompact s p a c e , 1 68 130,
132,
completely s e p a r a t e d s u b s e t s , 31
complete uniform space, 1 2 , 140
c l o s e d subspace o f , 1 4 1 compact s p a c e , 1 4 1 Hewitt-Nachbin completen e s s , 150, 1 5 1 i f and o n l y i f , 141, 150
products of, 1 4 1 subspace, 1 4 2 completions
co 1l e c t ionwi se norma 1 s p a c e ,
Comfort, 1 2 1 ,
E- comple t e l y r e g u l a r , 1 5 , 16, 1 7 , 2 1 i f and o n l y i f , 19, 102, 139 uniform s t r u c t u r e , 1 3 9 22,
c o l l e c t i o n of s e t s
54,
58
i n Hewitt-Nachbin s e n s e , 98, 1 4 6 , 166 uX, 27, 38, 78, 8 6
150, 39,
153, 58,
76,
uniform s p a c e s e n s e , 12, 125, 133
126,
128,
commutative diagram, 33 compac t i f i c a t i o n , 8 E- compac t i f i c a t i o n , 2 5, 37, 3 9 , 4 0 one-point, 1 5 Stone-Cech, 1 2 , 57, 79, 82
142,
146,
150,
153
Completion Theorem, 101 c o n n e c t e d dyad, 18 con t r a v a r i a n t f u n c t o r , 3 6 c o r e t r a c t i o n , 35 Corson, 9 5
26 3
INDEX
c o u n t a b l e i n t e r s e c t i o n prope r t y , 7 , 5 2 , 54, 56, 66 Z-ultrafilter,
60, 115
c o u n t a b l y compact s p a c e , 8, 2 18 necessary condition f o r , 176 v s . c b - s p a c e , 164 c o u n t a b l y paracompact s p a c e , 8 normal, 89, 164 v s . c b - s p a c e , 164 cozero-set,
z-embedded,
s p a c e , 23
E - C o m p a c t i f i c a t i o n Theorem, 25 functor, 37
e- complete (see Hewitt-Nachbin space) E-completely r e g u l a r s p a c e , 15 i f and o n l y i f , 16, 1 7 , 21 E-embedded s u b s e t , 2 3 0
19, 8 2 , 8 5 , 127
a- embedded,
E-compact
v s . c-embedded, 230 v s . c*- embedded, 230
117 112
Embedding Lemma, 10 E n g e l k i n g , 16, 24, 25, 2 8
d - c l o s e d s u b s e t ( i n uniform s p a c e ) , 139, 147 i n t e r s e c t i o n s o f , 140 i s a z e r o - s e t , 140 d- d i s c r e t e f a m i l y o f s u b s e t s , 139 d - d i s c r e t e s u b s e t , 139, 147, 149
i s C-embedded, u n i o n s o f , 140
140
E-normal,
23
E-open s e t , 2 0 E - p e r f e c t mapping, 227 i f and o n l y i f , 229 inverse invariance of E-compactness, 229 sufficient condition for, 231 when E = m , 232 epimorphism, 34
d e l t a r i n g o f s e t s , 53
e p i r e f l e c t i v e f u n c t o r , 40
complement g e n e r a t e d , 53 d i l a t i o n of a s u b s e t , 116, 118 Dilworth, 2 0 2 , 2 0 3
e v a l u a t i o n mapping,
lo
extremally disconnected space, 164
d i r e c t e d s e t , 69 d i s c r e t e f a m i l y of s e t s , 7 d i s c r e t e s p a c e ( o f nonmeasurab l e c a r d i n a l ) , 92, 124, 1 5 1 d i s j u n c t i v e c o l l e c t i o n , 46, 49, 57 Dykes, 166, 196, 206, 2 0 8 , 2 1 0 , 2 2 2 , 223, 224
E-closed set, 20,
21
i f ' a n d o n l y i f , 175 inverse invariance of 'compl e t e n e s s , 197 open b u t n o t a WZ-mapp i n g , 219 z-closed implies c l o s e d , 181 f i b e r - c o u n t a b l y compact mapp i n g , 173, 176
E-closed mapping, 2 2 7 v s . closed, 227 weakly E - c l o s e d , 228
f i b e r - c o m p a c t mapping, 173, 222
227,
i n v a r i a n c e of completen e s s , 191, 193, 2 1 2 , 213
INDEX
264
zero- s e t preserving imp 1i e s hyper- rea 1,
/
Glicksberg- F r o l i k Theorem, 120
212
z-open b u t not f i b e r compact, 184 z-open implies z e r o - s e t preserving, 1 8 1 f iber-Hewi t t-Nachbin mapping, 173, 187
fiber-paracompact mapping, 173 f i b e r - pseudocompac t mapping, 173
f i b e r - r e l a t i v e l y pseudocompact mapping, 173, 216 f i b e r - s t r o n g l y p o s i t i v e map ping, 216, 2 1 7 , 218 f i l t e r ( s e e Bourbaki f i l t e r , Z - f i l t e r , or 8 - f i l t e r ) f i n i t e intersection property, 7,
44,
f o r g e t f u l f u n c t o r , 36
193,
127,
131
Henriksen, 1 1 2 H e r r l i c h , 30 Hewitt, 3,
32,
61,
63,
85
68,
Hewitt-Nachbin completion 27, 155, 156
UX,
a s a space of measures, 156
a s a universal repell i n g object, 38 C- embedded subspace, 86 i f and only i f , 78 i n pX, 76 l o c a l l y compact, 130, 132
124
F r i n k , 96 192,
128,
not a k-space, 133 not normal, 94 P- embedded subspace,
140
F r o l l k , 82,
Hager, 111, 1 1 7 , 126,
158, 159,
160, 161,
194
products, 1 2 1 , 126,
127,
123, 1 2 5 , 129, 130
82, 113 0 f u l l subcategory, 35
pseudocompact space, 12 1 r e f l e c t i v e f u n c t o r , 39 Wallman-Frink type, 58,
f u n c t i o n a l l y closed ( s e e Hewitt-Nachbin space)
weak
F -set,
7,
lo2
Hewitt-Nachbin
f u n c t o r , 36
G - c l o s e d s e t , 79, 85
6
79,
80,
102,
67, 80, 223
161,
168,
n o t normal, 66, 9 5 n o t paracompact, 66, 95 p r o p e r t i e s o f , 84, 85, 82,
84,
117
-dense, 79, 1 0 2 , 111 6 G - s e t , 7, 85 G
6
Gillman and J e r i s o n , 19, 56, 59, 61, 64, 67, 76, 77, 78, 84, 85, 86, 87, 88, 121, 139, 140, 141, 143, 144, 145, 146, 148, 149, 1 5 1 , 1 5 3 , 179, 1 9 7 , 198
space, 2 3
i f and only i f , 61, 64,
c o n t r a v a r i a n t , 36 epireflective, 40 f o r g e t f u l , 36 r e f l e c t i v e , 38
G -closure,
cb-space, 166
91, 95, 142, 147, 176,
92,
115,
188
q u o t i e n t o f , 92 v s . almost realcompact space, 166 v s . weak cb-space, 166 v s . zero-dimensional m-compact space, 2 8 Horne, 89,
164
hyper-real i d e a l , 6 0 hyper- r e a l mapping, 2 1 1 s u f f i c i e n t condition for, 211
26 5
INDEX
v s . zero-set preserving,
Johnson, 112, 1 3 1 , 164, 165, 166,
212
204
202,
V
i d e a l , 59
Katetov, 81,
fixed, 6 0 free, 60 hyper- r e a l , 6 0 maximal, 59, 144 prime, 59 r e a l , 60, 61, 144 I d e n t i f i c a t i o n Theorem, 2 1
Kelley, 10, 142 Kenderov, 89
Imler,
125
induced mappings f
i
k-space,
129,
152
130,
199,
206,
223
and
133
irX,
l i m i t point and f s ,
of a n e t , 7 0 of a & f i l t e r b a s e , 45 of a 3 - f i l t e r on X,
2 04
infimum ( o f two f u n c t i o n s ) , 9
45,
i n f i n i t e l y l a r g e element, 144
51
L i n d e l s f space, 8,
interior, 6 68
i n v a r i a n c e (of a t o p o l o g i c a l property), 1 7 1 almost realcompac t space,
225
i f and only i f , 64, 104 v s . Hewitt-Nachbin space, 65, 94 z-embedded, 1 1 2
i n t r i n s i c topology f o r a chain,
l o c a l base, 46,
49,
51,
84,
52,
57
194
H e w i tt- Nachbin space, 191, 206, 213,
85,
192, 208, 221,
196, 210, 223
199, 211,
loca l l y bounded f u n c t i o n , 1 6 3 l o c a l l y compact space, 8, 98,
normal and countably paracompact space, 1 9 1
129,
130,
57,
199
i f and only i f f o r
ux,
132
i n v e r s e i n v a r i a n c e (of a topological property), 1 7 1 almost realcompact space,
128,
product w i t h cb- space, 164
l o c a l l y f i n i t e family, 7
194
lower semi-continuous funcE-compact space, 229 t i o n , 163 Hewitt-Nachbin space, 187, 224,
225
pseudocompact space, 2 1 1 i n v e r s e morphism, 34 I s i w a t a , 199, 200, 219,
220,
221,
214, 222
isometry, 36 isomorph i s m a l g e b r a i c , 63 c a t e g o r i c a l , 34
Mack, 89,
164,
165,
166,
204 215,
maximal n e t , 50 measurable c a r d i n a l , 9 0 measure, 9 1 metacompact space, 168 m e t r i z a b l e space, 152 Michael, 169, 209 minimal mapping, 202
202,
266
INDEX
monomorphism, 34 Moore p l a n e , 95
one-poin t compactif i c a t i o n , 1 5 , 98
Morita, 191
o r d i n a l s p a c e , 68, 92, 1 5 2 , 167, 1 7 0 , 219
morphism, 33 bimorphism, 34 epimorphism, 34 isomorphism, 34 monomorph i s m , 34
paracompact s p a c e , 8, 66 a d m i t s uniform s t r u c t u r e , 151 i m p l i e s H e w i t t- Nachbin space, 152 i n v a r i a n c e under p a r a p r o p e r mapping, 1 7 2 subparacompac t , 168
Mrdwka, 10, 16, 2 1 , 24, 25, 2 8 , 80, 81, 85, 88, 92, 189, 229 M-spate,
168, 169
p a r a m e t r i c mapping, 10 IN
(the positive integers), 6
p a r a p e r f e c t mapping, 174
Nachbin, 3, 150
i n v a r i a n c e o f paracompactness, 172 i n v e r s e i n v a r i a n c e of pa racompa c t n e s s , 1 7 2
Nachbin-Shirota Theorem, 150 IN-compact s p a c e , 2 8 , 64 p e r f e c t image o f , 191 N e g r e p o n t i s , 1 2 1 , 125, 126 n e t , 69, 7 0 maximal, 50 s e q u e n t i a l l y bounded, 7 2 s u b n e t , 69 universal, 70 & u n i v e r s a l , 72
p a r a p r o p e r mapping (see parap e r f e c t mapping) P-embedded s u b s e t , 124, 125 p e r f e c t l y normal s p a c e , 8, 99 z- embedded s u b s e t s , 109
p e r f e c t mapping, 174, 2 2 7 f a i l s t o p r e s e r v e comp l e t e n e s s , 189 i f and o n l y i f , 225 i n v a r i a n c e o f completeness (special cases), 192, 194, 196 i n v e r s e i n v a r i a n c e of completeness, 187 minimal mapping, 2 0 2 open implies z e r o - s e t preserving, 182 open imp1 ies z- open, 181, 194 p r e s e r v e s almost r e a l compactness, 194
Niemytzki p l a n e , 95, 189 nonmeasurable c a r d i n a l , 90, 91, 124, 126, 1 2 8 , 1 3 0 , 133 normal b a s e , 57 s t r o n g d e l t a normal b a s e , 99 normal c o l l e c t i o n of s e t s , 45, 49, 52, 5 3 , 57 normal f u n c t i o n , 203 s e m i - c o n t i n u o u s , 203, 204 normal s p a c e , 8, 31, 8 7 , 206 cb- s p a c e , 164 c o u n t a b l y paracompact, 89, 164, 191, 199 i f and o n l y i f , 1 1 2 , 113, 215 n o t Hewitt-Nachbin comp l e t e , 94
Nyikos, 185
power s e t , 6 , 37 prime i d e a l , 59 prime
8 - f i l t e r , 51,
54
p r o p e r mapping (see p e r f e c t mapping) pseudocompact s p a c e , 8, 131, 215, 2 1 8 i f and o n l y i f , 121
267
INDEX
i n v a r i a n c e of completeness, 196 i n v e r s e i n v a r i a n c e of completeness, 211 maximal i d e a l s i n , 61 re l a t i v e l y pseudocompa c t subspace, 1 7 3 v s . cb-space, 164 vs. Hewitt-Nachbin space, 68 vs. s t r o n g l y p o s i t i v e , 2 1 6 vs. weak cb-space, 164 ps e ud ome t r i c , 1 2 3 uniformity, 1 3 9 pseudo-m -compact space, 134 1 p s p a c e , 169 P-space,
168, 169
r e f l e c t i v e f u n c t o r , 38 r e f l e c t i v e subcategory, 38 regular closed set, 7 r e l a t i v e l y pseudocompact subs e t , 173 r e p l e t e subcategory , 3 5 r e s i d u a l s e t , 69 r e t r a c t i o n , 35 r i n g of sets, 4 2 ,
53,
57
s a t u r a t e d space ( s e e Hewi t tNachbin space)
s e m i - continuous f u n c t i o n , 203 normal, 2 0 3 s e p a r a b l e space
Q ( t h e r a t i o n a l numbers), 6 , 106, 144 Q - c l o s u r e (see G6-closure)
s e p a r a t i o n axioms, 7
q - p o i n t , 169, 209
s e q u e n t i a l l y bounded, 7 2
q-space,
s e q u e n t i a l l y compact, 8 , 69
169, 2 1 0
Q- space ( s e e Hewitt-Nachbin
S h i r o t a , 86, (the constant function), 9
IR ( t h e r e a l numbers), 6 IR ( t h e non- n e g a t i v e r e a l numbers), 6 IR - compact (see H e w i t t-Nachbin space) real +
i d e a l , 60
2-u l t r a f il t e r , 60 &ultrafilter,
99, 118
real- c l o s e d ( s e e Gb-closed) realcompact ( s e e Hewitt-Nachbin space)
95
Shapiro, 1 2 0 , 1 2 4 , 1 7 2 s h a r p mapping (f# 1 ,
space)
-r
Hewitt-Nachbin, m e t r i c , 65
56
150
o-compact space, 8 v s . Hewitt-Nachbin space, 65, 82, 94 Sorgenfrey space, 66, 169 S- s e p a r a t e d s e t s , 109
vs
.
completely s e p a r a t e d , 109, 110 v s . z-embedding, 109
S t e i n e r and S t e i n e r , 105 Stone, 94 V Stone- Cech compactifica t i o n , 1 2 , 5 7 , 79, 82, 102
realcomplete ( s e e Hewitt-Nachbin space)
pseudocompact space, 1 2 1 r e f l e c t i v e f u n c t o r , 39,
real-proper mapping, 227
uniform completions, 146 universal repelling o b j e c t . 38
refinement, 7 r e f l e c t i o n , 38
40
268
INDEX
Wallman-Frink t y p e , 57, 98 S t o n e topology, 63
compact s p a c e , 143 e x t e n s i o n s , 142 i f and o n l y i f , 139 uniform s t r u c t u r e , 137
Strauss, 202
necessarily implies complete r e g u l a r i t y , 139 p r o d u c t , 138
s t r o n g d e l t a normal b a s e , 99, 102, 103 L i n d e l o f s p a c e , 105 s t r o n g l y p o s i t i v e s u b s e t , 216 v s . pseudocompact, 216
s t r o n g l y zero-dimensional,
29
uniform s u b s p a c e , 140 uniform t o p o l o g y , 138 union of Hewitt-Nachbin s p a c e
s t r u c t u r e space, 6 3 subbase f o r the closed sets, 6 f o r uniform s t r u c t u r e , 138
w i t h Hewitt-Nachbin s p a c e , 92, 190 w i t h L i n d e l o f s p a c e , 94 w i t h paracompact s p a c e , 94 w i t h o-compact s p a c e ,
s u b c a t e g o r y , 35
v s . z-embedding,
f u l l , 35 r e f l e c t i v e , 38 r e p l e t e , 35 s u b n e t , 69
94
115
u n i v e r s a l n e t , 70 u n i v e r s a l r e p e l l i n g o b j e c t , 37 u n i v e r s a l u n i f o r m i t y , 140, 1 4 1 compact s p a c e , 143 paracompact Hausdorff space, 1 5 1
subparacompact s p a c e , 168 supremum ( o f two f u n c t i o n s ) , 9 T i e t z e Extension Theorem, 31
upper semi-continuous funct i o n , 163
t o p o l o g i c a l space, 6
+embedded
s u b s e t , 116, 1 2 0
cozero- s e t , 117 i f a n d o n l y i f , 118, 126 v s . z-embedded, 1 1 7 , 118
t o p o l o g i c a l sum, 188 t o t a l l y o r d e r e d f i e l d , 143 T s a i , 2 2 7 , 2 2 8 , 229, 231, 232
Urysohn E x t e n s i o n Theorem, 31
Tychonoff p l a n k , 164, 184, 185, 219
Urysohn M e t r i z a t i o n Theorem, 11
u l t r a b o r n o l o g i c a l , 155 uniform isomorphism, 13
vague t o p o l o g y , 156
uniformity, 137
Wallman-Frink c o m p a c t i f i c a t i o n , 44, 57, 9 7 , 1 0 2
a d m i s s i b l e , 138 g e n e r a t e d by a f a m i l y o f f u n c t i o n s , 140 Hausdorff, 1 3 8 p s e ud ome t r ic , 139 u n i v e r s a l , 140 uniformly continuous f u n c t i o n , 138
Wallman-Frink c o m p l e t i o n , 99, 102 weak
cb-space,
163
and t h e Hewitt-Nachbin c o m p l e t i o n , 166 i f and o n l y i f , 165, 2 0 2
269
INDEX
i n v a r i a n c e o f comple ten e s s , 1 9 6 , 206, 208,
normal s p a c e , 112 p e r f e c t l y normal s p a c e ,
product with l o c a l l y compact s p a c e , 1 6 4 v s . almost realcompact s p a c e , 166 v s . cb-space, 1 6 4 v s . pseudocompact s p a c e ,
v s . C-embedded,
2lo
164
weakly
E-closed mapping, 227
v s . E-closed, 228 Wenjen, 82 WZ-mapping, 214, 223, 227 i f and o n l y i f , 2 2 0 i n v e r s e i n v a r i a n c e of completeness, 224 n o t z-open, 2 1 9 open b u t n o t z - c l o s e d ,
lo 9
112,
vs.
218 6
z ,
59
z - c l o s e d mapping, 1 7 4 , 227 f i b e r - compact i m p l i e s c l o s e d , 181 i m p l i e s WZ-mapping, 200, 215
i n v a r i a n c e o f completen e s s , 212, 213 inverse invariance of c o m p l e t e n e s s , 187, 225
not closed, 184 n o t zero- s e t p r e s e r v i n g , 185
v s . f i b e r - s t r o n g l y posit i v e , 218 v s z- embedded f i b e r s ,
.
183
v s . z-open, 180, 184 8- d i s j u n c t i v e , 4 5 z- embedded s u b s e t , 108 F -set, CT
113
G -closure,
6
117
i f and o n l y i f , 109, 114
C -embedded, 112
109,
Zenor, 168, 215 zero-dimensional,
8,
28
D- c o m p l e t e l y r e g u l a r , 17 lN-compact, 64 strongly, 29
z e r o - s e t , 19, 46, 57,
77,
52, 53, 112, 153,
102,
56, 216
z e r o - s e t f i l t e r , 43, 54,
56,
59,
64,
44, 50, 6 7 , 76
Cauchy, 140, 153 t r a c e , 114, 1 1 5
219
v s . closed, 215 v s . m - p e r f e c t , 232 v s . z - c l o s e d , 215, 217,
*
111,
183
zero- set p r e s e r v i n g mapping, 174
i f and o n l y i f , 1 7 8 implies z-closed, 174 n o t open, 1 8 4 v s . h y p e r - r e a l , 212 v s . Z-open, 181, 184 2 - f i l t e r (see z e r o - s e t f i l t e r ) 8 - f i l t e r , 42 b a s e , 43 c l u s t e r p o i n t , 45, 5 1 converges, 4 5 f i x e d , 44, 51, 9 1 free, 44 l i m i t p o i n t , 45, 5 1 neighborhood, 46, 50, 52,
141
prime, 51, r e a l , 52
54
z-open mapping, 1 7 4 i f and o n l y i f , 179, 1 8 2 i m p l i e s open, 1 7 4 i n v a r i a n c e o f completen e s s , 193 n o t z-closed, 185 v s . open and c l o s e d , 1 8 2 v s . open and z - c l o s e d , 180,
184
v s . open p e r f e c t ,
181
INDEX
270
v s . zero- s e t p r e s e r v i n g , 181 8 - - u l t r a f i l t e r , 43, 47, 48, 49, 51, 5 2 2-universal n e t , 72
