Continuous Model Theory. (AM-58), Volume 58 9781400882052

Citation preview

Annals of Mathematics Studies Number 58

ANNALS OF MATHEMATICS STUDIES Edited by Robert C. Gunning, John C. Moore, and Marston Morse 1. Algebraic Theory of Numbers, by

H erm ann W

3. Consistency of the Continuum Hypothesis, by 11. Introduction to Nonlinear Mechanics, by N.

eyl

Ku r t G od el

Krylo ff

and N.

B o g o l iu b o f f

20.

Contributions to the Theory of Nonlinear Oscillations, Vol. I, edited bu S.

21.

Functional Operators, Vol. 1, by

J ohn

L

efsc h etz

Neum an n

von

24. Contributions to the Theory of Games, Vol. I, edited by H. W. 25. Contributions to Fourier Analysis, edited by A. A. P. C a l d e r o n , and S. B o c h n e r

and A. W.

Kuhn

W.

Zyg m u n d ,

28. Contributions to

the Theory of Games, Vol. II, edited by H. W.

30. Contributions to

the Theory of Riemann Surfaces, edited by L.

T ra n sue,

Kuhn

M

o rse,

et al.

Contributions to the Theory of Partial Differential Equations, edited by L. n e r , and F. J o h n

34.

Automata Studies, edited by C. E.

and J.

T u ck er

and A.W.T u c k e r

Ah lfo r s

33.

Sh a n n o n

M.

B e r s ,S. B o c h ­

M cC arthy

38. Linear Inequalities and Related Systems, edited by H. W.

and A. W.

Kuhn

39. Contributions to the Theory of Games, Vol. Ill, edited by M. and P . W o l f e 40. Contributions to the Theory of Games, Vol. IV, edited by R.

T u ck er

D resh er, A .

D uncan L

uce

W.

T u ck er

and A. W.

T u ck er

41. Contributions to the Theory of Nonlinear Oscillations, Vol. IV, edited by S. 42. Lectures on Fourier Integrals, by S.

43. Ramification Theoretic Methods in Algebraic Geometry, by S. 44. Stationary Processes and Prediction Theory, by

46. Seminar on Transformation Groups, by A. Sm

H. F

u rsten berg

B o rel

J.

N. E .

51. Morse Theory, by J. W . M i l n o r 59, Advanpps in Gamp Theorv. edited bu M.

St e en r o d ,

T oda

written and revised by

53. Flows on Homogeneous Spaces, by L.

D r e s h e r . L. S h a p le y .

A u sla n d er,

54. Elementary Differential Topology, by J. R. 55. Degrees of Unsolvability, by

G.

E.

a­

G u n n in g

D. B.

p s t e in

56. Knot Groups, by L. P.

L

et al.

49. Composition Methods in Homotopy Groups of Spheres, by H. 50. Cohomology Operations, lectures by

C e s a r i,

u llya n

48. Lectures on Modular Forms, by R. C.

E

efsc h etz

A bhyankar

45. Contributions to the Theory of Nonlinear Oscillations, Vol. V, edited by L. S a l l e , and S. L e f s c h e t z 47. Theory of Formal Systems, bu R.

L

B o ch n er

L.

Green ,

F.

and A. W . H

ahn,

et al.

M u n kres

Sa cks

N e u w ir t h

57. Seminar on the Atiyah-Singer Index Theorem, by R. S. P a l a i s 58. Continuous Model Theory, by C. C. C h a n g and H. J. K e i s l e r 59. Lectures on Curves on an Algebraic Surface, by D a v id M u m f o r d 60. Topology Seminar, Wisconsin, 1965, edited by R. H. B in g and R. J.

T ucker

B ean

A.

CONTINUOUS MODEL THEORY BY

Chen Chung Chang AND

H. Jerome Keisler

PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS

1966

COPYRIGHT © 1966, BY PRINCETON UNIVERSITY PRESS ALL RIGHTS RESERVED L. C. CARD: 66-17698

PRINTED IN THE UNITED STATES OF AMERICA

To M arjorie and Lois

PREFACE

This monograph contains a study o f th e o rie s o f models w ith tru th valu es in compact H ausdorff spaces

X.

This

c o lle c t io n o f re s u lts

might be c a lle d a theory o f model th e o rie s . I t i s w e ll known to those w ith in te r e s t s in the two-valued model theory that there has been a tremendous development in the f i e l d in the past f i f t e e n years.

W hile the r e s u lts which led to th is development

range over a reasonably wide spectrum, there has been a d is c e rn ib le c e n tra l theme among them.

This theme concerns the determ ination o f the exact r e l a ­

tion sh ip between the a lg e b ra ic p ro p e rtie s that a cla ss es and pu rely s y n ta c tic a l d es crip tio n s o f sets by which

K

low ing type:

may be ch a ra cterized . fo r a cla ss

K

A

the compactness theorem.

o f models possess­

o f f i r s t - o r d e r sentences

Thus many o f the r e s u lts are o f the f o l ­

to be closed under c e r ta in a lg e b ra ic r e la tio n s

i t is necessary and s u ffic ie n t that the sentences o f to those o f a c e rta in form.

K

may be r e s t r ic t e d

A

A b a sic t o o l needed fo r a l l o f these r e s u lts i s

More r e c e n tly , a uniform method o f p ro o f

fo r these

re s u lts has been given in terms o f s p e c ia l models. In th is monograph we have c a rrie d out a study culm inating in s ev era l r e s u lts along th is theme fo r cla sses ated w ith a space models in

X

X

to be a non-empty set R

in to

In the case that

X = Co, 1 ),

w i l l turn out to be the ordinary two-valued models.

g e n e r a lly , in case

of

o f tru th va lu es.

3Vf o f models which are a s s o c i­

X.

i s an a r b itr a r y s e t, a model R

and a sequence

A

in

o f fu n ction s mapping n -tu ples

I f we assume, however, that

compact Hausdorff top olo gy, then w ith continuous fu n ction s on r o le o f connectives and w ith continuous set fu n ction s on v ii

More

i s understood

As can be expected, not too much can be done i f

not carry any a d d itio n a l stru ctu re.

the

X

X

X

does

X

has a

p la y in g the

p la y in g the r o le

PREFACE

viii o f q u a n t if ie r s , we can

f i r s t d e fin e a s e t

the n o tio n s o f a model

A

E o f sen ten ces.

&K and o f

in

a sen ten ce

Once we have

q>

i n E,

we can p r o ­

ceed t o in tro d u c e and study the s a t i s f a c t i o n fu n c tio n which a s s ig n s a tr u th v a lu e

cp[ A ]

in

X

t o each p a ir

g iv e s r i s e to a fu n c tio n then c o n s id e r

[

]

[A ]

cp, A.

in

XE

The tr u th v a lu e d e fin e d by

[

]

i t s e l f as a fu n c tio n mapping

induces a n a tu r a l to p o lo g y on

to p o lo g y on

n a t u r a lly

[ A ] (cp) = cpCA]. in t o

the u su al p rod u ct to p o lo g y coming from the to p o lo g y on t io n

s“ (X) = {Y :

Y C

X

and |X-Y| < a )

H

Y

H

b e lo n g in g to

i s denoted by G

and

i s a s e t and H

H [x ]

t o fu n c tio n s

(...

x ...)

in t o

(...

f,

g.

f

A fu n c tio n

If

f

and

(HH,

G c H;

r e s p e c tiv e ly .

se e , f o r exam ple,

x e Y)

H [(x )]

x H y

when

. x

and no con­

< x , y > e H.

fo r

f

w ith domain f:

X

i s denoted by

b y X . Thus

Y )f(y ).

nj_ej Y^,

th e e x p re s s io n s

co n sid ered as X

w ith

f € X

which maps each X

and

and

f:

Y -► X

< x >

n ote the s e t o f fu n c tio n s

w ith

f e X05

x € X

i s any s e t , Y1

to the

X

same

F or a f i n i t e C a rte s ia n power, we

In p a r t ic u la r , i f

x.

Y

have the

X x X x . . . x X n -tim e s , so the elem ents

ord ered n - tu p le s . and

X

and the C a rte s ia n power o f Y

Y

The e x p re s s io n (x.x e X)

X -► Y.

The C a rte s ia n prod u ct o f s e ts

meanings and may be used in te rc h a n g e a b ly . Xn

and range in clu d ed in

i s a fu n c tio n w ith domain

TY = ( X y c X H

Y

That i s ,

by

The domain and range

and the r e l a t i v e p ro d u ct,

f o r some

fo r

We sometimes w r it e

x ...).

i € I

w ith

H

s h a ll den ote the fu n c tio n w ith domain

we d e fin e

X1

X.

as s e ts o f ord ered

i s a r e l a t i o n , we l e t

sometimes g r a p h ic a lly p re s e n te d as

id e n t ify

.

Fu n ction s a re d e fin e d as s p e c ia l b in a ry r e l a t io n s in the u su al u Thus, th e n o ta tio n s £2 f, (ftf, f , f o g , and f [ X ] may be a p p lie d

manner.

w ith

H

y >€ H

I t w i l l be con ven ien t t o w r it e

or power,

,

X a re d e fin e d

w i l l be denoted by £2H

H [Y ] = ( y : < x ,

Y

a set

x, y

or com p o sitio n , o f two r e la t io n s

is

,

( 0}

and |Y| < cc)

The con verse o f a r e l a t i o n

fu s io n can r e s u l t .

X

o f a (b in a r y ) r e l a t i o n

If

,

Y C

o f elem ents

K e lle y [ I 9 5 5 ] .

Y C X)

= {Y :

B in a ry r e la t io n s o v e r p a ir s

The c a r d in a lit y ,

We l e t

X* = S(X) Sa (X)

c a r d i­

n = 1,

The s p e c ia l n o ta tio n such th a t

f

X°°

o f Xn

may be

then wei d e n t i f y i s used to d e­

i s e v e n t u a lly co n sta n t.

D-PRODUCTS X°° = { f € X05:

f o r some

An enumeration o f a s e t whose range i s

1 .2 .

X

n,

fo r a l l

m > n)

i s a fu n c tio n whose domain i s an o r d in a l and

D -p rod u cts is

a filte r

s a id t o he

(i)

D C S ( I ) and

(ii)

t io n o f f i l t e r , A filte r I

D

and f i l t e r s

over

- X € D.

I

then

X

Y€ D

;

n Ye D

o ft e n dropped from the d e f i n i ­

flD € D.

fo r every

X C I,

X e D

e ith e r

a re a ls o known as u l t r a f i l t e r s .

A filte r

We assume th e r e a d e r i s f a m ilia r w ith the

elem en ta ry p r o p e r t ie s o f u l t r a f i l t e r s .

For in s ta n c e , he should know th a t

u l t r a f i l t e r s e x i s t o v e r any non-empty s e t S (I)

is

then

in the above sense a re c a lle d p ro p er f i l t e r s .

c a lle d maximal i f

is

;

X C Y,

D ^ S (I)

Maximal f i l t e r s

is p rin cipa l i f

and

whenever X, Y € D,

In the l i t e r a t u r e th e c o n d itio n

o v e r th e s e t I i f :

0 ^ D ^ S (I)

whenever X € D

(iii)

D

= f(m )

X.

D

or

f(n )

3

I.

In f a c t ,

any subset

E

w hich has the f i n i t e i n t e r s e c t io n p r o p e r ty (t h a t i s , no f i n i t e

s e c tio n o f

elem en ts o f

filte r

o v e r the s e t

D

E

i s empty) S^d)

in t e r ­

can be exten ded to an u l t r a f i l t e r . s a id t o be r e g u l a r i f

is

of

fo r every

A

j € I,

we have { i e S J I) : j e i ) The

phrase " r e g u la r f i l t e r

the

form

S ^ I).

over

j"

e D

.

i s m ea n in gfu l

o n ly when

J

i s as e t o f

Q u ite o b v io u s ly , r e g u la r f i l t e r s e x i s t o v e r any s e t

S (I)

because th e fa m ily o f s e ts {{i has th e f i n i t e

€ S J I)

: j e i)

: j e 1}

i n t e r s e c t io n p r o p e r ty .

I n the f o llo w in g s e r ie s o f e x e r c is e s we m ention some o th er s p e c ia l kinds o f u l t r a f i l t e r s w hich have a r is e n in th e l i t e r a t u r e and which, l i k e r e g u la r u l t r a f i l t e r s , EXERCISE 1A. filte r

o ver

S (I)

is

If

I

is

fin ite ,

the p r in c ip a l f i l t e r

EXERCISE 1B*.

re g u la r i f

a re o f i n t e r e s t when

L e t us

th e re i s a fu n c tio n

f:

D -p rod u cts are s tu d ie d .

then th e o n ly r e g u la r u l t r a ­ g en era ted by

c a l l an u l t r a f i l t e r I -►S ^ I )

(I). D

over

such th a t f o r a l l

I

weakly j e I,

k

TOPOLOGICAL PRELIMINARIES f[{i (a )

e Sm( I )

: j e i ) 3 e D

E very r e g u la r u l t r a f i l t e r o v e r a s e t

S^J)

i s w eakly

r e g u la r . (b )

An u l t r a f i l t e r th ere e x is t s

D

o ver a s e t

E CD

(a )

i s w eakly

such th a t

s e c tio n o f any i n f i n i t e EXERCISE 1C.

I

|E| = |l|

subset o f

E

r e g u la r i f f and the i n t e r ­

i s empty.

E very u l t r a f i l t e r o ver a f i n i t e

set is

w eakly r e g u la r . (b )

An u l t r a f i l t e r

D

r e g u la r i f f i t EXERCISE 1D. if

uniform

fin it e

|X| = |l|

o ver a cou n tab le s e t

i s w eakly

i s n o n - p r in c ip a l.

An u l t r a f i l t e r

fo r a l l

I

X € D.

D

D

over a set

I

is

i s u niform i f f

s a id to be

S ^ tl)

s e t w ith more than one elem ent has no u n iform u l t r a f i l t e r s

However, e v e r y w eakly r e g u la r u l t r a f i l t e r over an i n f i n i t e

set

CD.

A

over i t .

I

i s u n i­

form . EXERCISE 1E. over

J,

and

le t

Dj

L e t I = U j£ j

Ij,

le t

be an u l t r a f i l t e r o ver

E

Ij

be an u l t r a f i l t e r f o r each

j e J.

Then

the s e t D = {X C I i s an u l t r a f i l t e r o v e r

: {j

e J : X n I

e D .) J

e E}

J

I.

We s h a ll now g iv e the d e f i n i t i o n o f a D -p rod u ct. non-empty s e t , Let

D a filte r

R* =

le n t t o

Ri . g)

if

F or

(i:

over

f,

f(i)

I,

g € R’ ,

= g (i)} e

always used r e l a t i v e to some f i l t e r v e r i f y th a t

~

f ~ = { g € R ': (X i € I ) R ±

we say th a t

f ~ g

(re a d

D

g iv e n by c o n te x t. R '.

F or

be a i eI.

f i s e q u iv a ­

D. We remark th a t the symbol

~

is

I t i s ea sy to f e R!,

le t

We then d e fin e the D-product o f the fu n c tio n

by D-prod XiR± = { T :

S ince

I

and R^ a non-empty s e t f o r each

i s an e q u iv a le n c e r e l a t i o n o v e r f ~ g }.

Let

I

is

determ ined by

D,

stead o f

xi

€ I

same s e t

R,

the D -product o f

in our above

i s denoted by D-prod R.

f € R '}

th e re i s

no co n fu sio n in w r it in g

n o ta tio n .

If

each

( x i € I )R

is

c a lle d the D-power o f

Xi

in

R^c o in c id e s w ith the R

and

D-PRODUCTS

5

In the l i t e r a t u r e , D -produ cts a re c a lle d reduced d i r e c t p ro d u cts, or reduced p ro d u c ts , and D-powers a re c a lle d reduced ( d i r e c t ) case

D

pow ers.

In

i s an u l t r a f i l t e r , D -p rod u cts a re c a lle d u ltr a p r o d u c ts , o r prim e

(red u ced d i r e c t )

p ro d u c ts , and D-powers a re c a lle d u ltra p o w e rs , or prim e

(red u ced d i r e c t )

pow ers.

There i s a la r g e c o l l e c t i o n o f theorem s, and even

some open problem s, con cern in g the c a r d i n a l i t y o f D -p rod u cts. o n ly two v e r y weak r e s u lt s w hich we s h a ll need l a t e r on.

We s h a ll p ro ve

However, a number

o f o th e r r e s u lt s in t h is d i r e c t i o n w i l l be in d ic a t e d i n the s e r ie s o f e x e r ­ c is e s a t th e end o f t h is s e c t io n . LEMMA 1 .2 .1 .

F or any f i l t e r

any non-empty s e t PROOF.

R,

r € R

in t o the e q u iv a le n c e

{r },

i s a o n e -to -o n e fu n c tio n on R

w hich maps

Then

|D-prodR| > |l|.

R

i s an i n f i n i t e

s e t,

and D i s a r e g u la r u l t r a f i l t e r o ver

We may assume th a t co n ta in s

range

in t o D -prod R.

Suppose th a t

= I,

R

and

c la s s o f the con sta n t fu n c tio n w ith

S^tJ)

may a ls o assume th a t

I

|R| < |D-prod R |.

each

PROOF.

over a se t

f = (\ r t R ) ( ( \ i e I ) r ) ~ ,

The fu n c tio n

LEMMA 1 .2 .2 .

D

|l| > cd,

an i n f i n i t e

so

th a t

I.

|l| = |J|.

We

sequence

a0> a 1' a 2> * *•> am> o f d i s t i n c t elem en ts, and th a t For each

j e J

and

fjd )

Thus, f o r each Since

D

i € I,

=

{

i

if

j

am

if

j

i, is

the m -th elem ent in th e o r d e r in g induced on i . bj =

= {i

€ I :j 1, j 2 € i ) € D

we have f .

(i)

^ f. ( i ) .

J-l

(i: and

b. ^ b. . J1 J2

Suppose

the s e t

e I ’,

in somefa s h io n .

we d e fin e

f^ e R1 . L e t

I 1 Wow, whenever

has been sim p ly o rd ered

aQ

j € J,

i s r e g u la r ,

J

f, (i) J1

^ f. (i)) J2

T h is p ro v e s th e lemma.

J2

€ D

Hence ,

th a t

j 1 ^ j 2.

6

TOPOLOGICAL PRELIMINARIES EXERCISE 1F.

R

or

I

is

fin ite ,

|D-prod R | = |R |. ir 1 i

Suppose

D

i s an u l t r a f i l t e r o v e r

then |D-prod R| = |R|.

If

D

If

I.

D i s p r in c ip a l,

i s any u l t r a f i l t e r o v e r

I,

then

I f e it h e r then

|D-prod R|


|l|.

a u n iform u l t r a f i l t e r o v e r an i n f i n i t e

H in t: g iv e n a sequence o f

c o n s tru c t a new fu n c tio n

g: I

I

|l|

fu n c tio n s

which i s n ot e q u iv a le n t to

any o f them. EXERCISE 1H*. th a t If

|D-prod R| > R

|l|.

i s an i n f i n i t e

J,

then

Indeed one can p rove the f o llo w in g s tro n g e r r e s u lt :

s e t and

|D-prodR|

g (j) ^ h (j)

and

afu n c tio n

f

T

f*

on

onto

D-prod R

on T

R

onto

if

(R )

,

and

R01

then

f*

is

in such a way th a t, when­

|D-prodR|

in turn induces a fu n c tio n

and

D

is

R

set I

= |D -prodR |“ .

f"

on

co u n ta b le.

sequence o f members o f

sa id to be oountably incom­ such th a t

n E 4 D.

i s cou n ta b ly in c o m p le te .

Any No

cou n tab ly in c o m p le te . If

D

i s a cou n tab ly in co m p lete u l t r a f i l t e r

i s an i n f i n i t e

s e t,

then

i s a cou n ta b ly com plete u l t r a f i l t e r and S

E -prod S

Ex. IB;

induces in a n a tu r a l way a fu n c tio n

An u l t r a f i l t e r

EXERCISE 1K*.

E

onto

f : J -* be as i n

g '( i ) ^ h '( i ) .

|R| = |R|“ ,

on an i n f i n i t e

p r in c ip a l u l t r a f i l t e r i s

If

we have

If R

g ' e R^

a cou n table subset E C D

w eakly r e g u la r f i l t e r

I,

a fu n c tio n

le t

(D -prod R )a .

th e re e x is t s

o ver a s e t

H in t:

qj

EXERCISE 1J.

p le te

i s a w eakly r e g u la r u l t r a f i l t e r o v e r a s e t

j e f(i),

EXERCISE 11*. H in t:

D

= |RJ |.

g e R^

a s s o c ia te w ith each ever

Under the same hypotheses as in 1 . 2 . 2 , p rove

H in t: D

|D-prod R| = |D-prod R|^. is

co u n ta b le, then

show th a t th e re i s a cou n tab ly d e c re a s in g

whose in t e r s e c t io n i s empty, and then argue as

in E x e r c is e 11.

1 .3 .

Compact H a u sd o rff spaces We assume th a t the re a d e r i s f a m i l i a r w ith a l l o f the elem en tary

p a r ts o f K e lle y [1 9 5 5 ], p a r t i c u l a r l y w ith ch a p ters one through f i v e . s h a ll fo llo w

We

the t o p o lo g ic a l n o ta tio n o f th a t book, and where we d i f f e r from

COMPACT HAUSDORFF SPACES it

7

o r where we in tro d u c e new n o tio n s , we s h a ll so in d ic a t e a t the a p p r o p r i­

a te p la c e s . Let ef.

Let

e?0

X

be a compact H a u sd o rff t o p o lo g ic a l space w ith to p o lo g y

be an open b a s is f o r

We s h a ll i d e n t i f y

X

c?; f o r con ven ien ce, we assume (X , S )

w ith the p a ir

h e ld f i x e d throughout the monograph. open s e ts o f

X;

n o te open s e t s .

the l e t t e r s

c?q

Y

of

X

X

compact and H a u s d o rff.

C (Y ,

Z)

pact

and

Z

is

sometimes w ith s u b s c r ip ts ,

Y.

F or each

n

continuous fu n c tio n s o f f i n i t e l y v a lu e s in

le t

Y

Y

many v a r ia b le s

and

denote the

Each

Y

and If

is

Xn Z,

is le t

Y

i s com­

c lo s e d

i s a c lo s e d s e t o f

C = Un € Xn) (x m) 2)

Xn

if

t io n ( \z e Xm) g ( f 1 ( z ) ,

X

[19553*

the c lo s u r e o f &

1]

pages 115

end

under composition

the m -th p r o je c t io n fu n c tio n

b e lo n g s to

. .., fR

. ..,

con tin u ou s.

SF C C,

m < n,

fn( z ))

We alw ays have fu n c tio n s i s

f : Y -*■ [0 ,

is

f f 3 £F such th a t:

onto f 1,

(S ee K e lle y

are d i s j o i n t

U, V

the u n it i n t e r v a l on the r e a l l i n e . )

Whenever on

i s a normal space and

and f[ V ] = { 1 } .

G iven a subset

ff it s e lf.

Xn

Z.

then e v e r y member o f C°(Y, Z)

(?n = C(Xn , X ) , and l e t

le t

s h a ll de

The c lo s u re

w ith th e u su al prod u ct to p o lo g y .

sense th a t th e image o f each c lo s e d s e t o f each

n

continuous fu n c tio n s f :

H a u s d o rff,

g?q are

the c o l l e c t i o n o f a l l

F or a r b it r a r y t o p o lo g ic a l spaces

be the s e t o f a l l

and

a re r e f e r r e d to as b a s ic open s e t s

i s denoted by

n - f o l d prod u ct space o f

is

X

a re c a lle d b a s ic c lo s e d s e t s .

complements o f elem en ts o f o f a subset

S

Thus,

U, V , W,

Elem ents o f

and assume th a t

X e

ff C

e f f n Cm

(\ < x 1 ,

...

ffi andg e & r\ ?n ,

b e lo n g s to

then

the fu n c­

ff.

C, th a t i s , a com p osition o f continuous

M oreover, the c lo s u r e o f

Q’ under com p osition i s

8

TOPOLOGICAL PRELIMINARIES

We l e t 1) open in e f * ;

be the le a s t top ology on the set

Whenever

V

i s open in

X, the set

U

i s open in

X,

whenever

such th at:

{Y € X*: Y C V)

the set

is

{Y € X*: Y n U / 0)

gF*.

When we speak o f the to p o lo g ic a l space (X *, ef*) .

X*

and 2)

i s open in

e?*

An open basis fo r the space

X*

X*,

we mean the space

i s the fa m ily

g?q*

o f a l l sets

o f the form {Y € X* : Y CV1 where

V1, . . . , V"n e

example, i f

The

Y e X*, then

neighborhoods; indeed, does. of

X,

U. . . U VR

and

On the other hand, i f

X* i s in gen era l not H ausdorff.

Y

Y

and

Z

X*

of

(?(X*, X)

Y € X*.We d efin e

i f and only i f

Thus i f we r e s t r i c t

we have a Hausdorff space.

X*

i s compact.

i s closed and,

Since

X

i s H ausdorff,

fur-thermore, f ( Y )

=f ( Y )

We s h a ll now prove an easy lim it theorem fo r the space

if that

U = {Z

L et

Z1, Z2 e Y,

Y e X**

Then

closu re o f

Y

PROOF. L et

U Y e U e c?q*;

ym e Zm fo r each Z1 U. . .U Zn C Z.

Z e U n Y,

where Y

such is

the

X*. we must show that

fo r m < n.

m < n.

and

Z n Vm ^ 0

Choose sets

U n Y ^ 0.

fo r m < n)

Z1, . . . ,

By hypothesis, there i s a set

Then we have

Z n Vm ^ 0

fo r

Z C LfY C V1 U ...U V Hence

Z^ € Y

V 1, . . . , Vn e ^^ such that

e X* : Z C V1 U ...U V

Let ym e U Y n Vm

U Y e Y,

in the space

X*.

have the p rop erty th a t,

then there i s a set

Z1 U Z2 C Z^.

Choose b a sic open sets

As we

2 =