**U C L**

r - I

**B orel sets and cr-fragmentability**

**o f a B anach space**

**Luis O n cin a D e lte ll**

### Thesis Submitted for the Degree of

### Master of Philosophy

### University of London

**ProQ uest Number: 10046157**

**All rights reserved**

**INFORMATION TO ALL U SE R S**

**The quality of this reproduction is d ep en d en t upon the quality of the copy subm itted.**

**In the unlikely even t that the author did not sen d a com plete manuscript**

**and there are m issing p a g e s, th e se will be noted. Also, if material had to be rem oved,**

**a note will indicate the deletion.**

**uest.**

**ProQ uest 10046157**

**Published by ProQ uest LLC(2016). Copyright of the Dissertation is held by the Author.**

**All rights reserved.**

**This work is protected against unauthorized copying under Title 17, United S ta tes C ode.**

**Microform Edition © ProQ uest LLC.**

**ProQ uest LLC**

**789 East E isenhow er Parkway**

**P.O. Box 1346**

**A b stra ct**

**C on ten ts**

A b s t r a c t ... 2

A ck now ledgm ents... 5

I n tr o d u c tio n ... 6

**1 ** **cr-fragm entability ** **10**
1.1 Some definitions and rem arks ... 10

1.2 cr-fragmentability of a m etric space ... 13

1.3 C ountable cover by special sets of small local diam eter . . . . 15

1.4 Borel s e t s ... 20

1.5 D ecom position of discrete f a m i l i e s ... 27

**2 ** **O n e-to -o n e m aps ** **31**
2.1 In tr o d u c tio n ...31

2.2 d. cr-d. m aps ... 32

2.3 Inverse m a p p i n g s ... 39

2.4 Some topological invariants ... 41

2.5 T he **Co-sum **of some Banach s p a c e s ...42

**3 Spaces w ith a countable cover by sets o f sm all local d iam eter 48**

3.1 T he space c o ( r ) ... 48
3.2 P rojectional Resolutions of I d e n t i t y ...50
3.2.1 d. *a-d.* m aps into co(T) using P R I ... 53
3.2.2 W CD spaces, duals of Asplund spaces and C (K ) spaces 64
3.3 M arkushevich basis... 67

**A cknow ledgm ents**

Sincere th an k s to m y supervisor Professor J.E . Jayne whose ideas and sug gestions I tried to develop in this thesis. All the problem s we solved were suggested by him .

T he research was fully funded by a grant from CAM, C aja de Ahorros del M editerraneo, under th eir joint program w ith th e B ritish Council, “Becas p ara am pliacion de estudios en el Reino Unido de G ran B retan a e Irlanda del N orte” .

I would also like to th ank Dr. B. Cascales and Dr. J. O rihuela for th eir support and discussions from which I greatly benefited.

The encouragem ent I received from m y family throughout these two years, and especially from my parents, has played a very im p o rtan t p art in the re alisation of this thesis. Imm ense thanks to my girl-friend, Ju an a Inès, whose constant support, from th e very beginning, m ade it possible.

**Introduction**

T hroughout this thesis two notions will be constantly used. B oth of th e m were introduced by Jayne, Namioka and Rogers in [12, 13],

Let (X , r ) be a Hausdorff space and let p be a m etric on *X* not necessarily
related to th e topology on *X .*

T he space *X* is said to be *a-fragmented* by th e m etric *g* if, for each e > 0,
it is possible to w rite

**OO**

*X = \ J X i ,*
i=l

where each set *X{* has the property th a t each non-em pty subset of *X{* has a
non-em pty relatively r-o p en subset of ^-diam eter less th a n *e.*

We shall say th a t (X , r ) has a countable cover by sets of small local

g-diam eter if, for each c > 0, X can be expressed as a union

**OO**

**X = u **

**(0.1)**

**1 = 1**

each non-em pty *X{* having th e property th a t each of its points belongs to some
relatively non-em pty r-o p en subset of g-diam eter less th a n *e.*

A lthough our aim is to find results on Banach spaces we give th e statem en ts, when possible, in term s of topological spaces and m etrics defined on them .

equivalent concepts when th e topology on th e space is generated by a m etric.
We also prove th a t if a topological space * { X , t )* has a countable cover by sets
of small local diam eter and th e m etric involved is lower sem icontinuous for th e
topology

**T,**th en th e sets in equation (0.1) can be tak en to be differences of

**r**

-closed sets. W hen applied to a B anach space we obtain th a t if it has a countable
cover by sets of sm all local diam eter, th en it has a countable cover by differences
of weakly closed sets of small local diam eter. This condition was shown to be
sufficient, by th e authors above, to have *B o re l{X j\\* • ||) = *B o r e l[ X ,w e a k ) ,*

see [9], p. 215 and [11], Lem m a 2.1. We prove som ething stronger, we show
th a t if a B anach space *X* has a countable cover by sets of small local diam eter,
then b o th *X* and any || • ||-closed subset of X are Borel subsets of (X**,iu*),
and this im plies th e coincidence of th e Borel subsets of *X .* We finish this
C hapter characterizing both th e property of having a countable cover by sets
of small local diam eter and th e cr-fragmentability in term s of decom positions
of II • ||-discrete families into a countable num ber of relatively weakly discrete
families.

C h ap ter 2 is concerned w ith one-to-one m aps betw een B anach spaces (or
ra th e r betw een topological spaces w ith m etrics defined on th em ). *l i T : X * *Y*

cr-decom posable families in *{Y,* || • ||y). (Such a m ap will be said to be d. c7-d.).
Moreover, if we denote by T||.||y th e m etric given by

^ ||- ||y K ^ ) = 11^"^ - *T* *v\\y* *,u,v* G *X ,*

we show th a t (X , T||.||^) has a countable cover by sets of additive class *a* (for
th e topology 7]|.||y) of small local || • ||%-diameter if and only if *T~^* is of Borel
class *a* and *T* is d. cr-d.

We also show th a t (i) cr-fragmentability (first proved in [19]), (ii) to have a
countable cover by sets of small local diam eter, and (iii) th e coincidence of th e
weak and norm Borel sets on a Banach space are each topological invariants
for th e weak topology. We prove too th a t th e Co-sum of B anach spaces which
have countable covers by sets of small local diam eter (resp. cr-fragm entability)
has a countable cover by sets of small local diam eter (resp. cr-fragm entability),
(th e (T-fragmentability case is Theorem 6.1 in [9]), and apply this to th e case
of *C { K )* spaces w ith th eir weak topologies w ith *K =* U{ür„ : *n* G *N } ,* w here

*K , K ij K2, ...* are com pact spaces, when each *{C (K n ),w e a k )* has either one of
th e properties, [15].

p articu lar cases: W CD spaces, duals of A splund spaces and *C { K )* spaces w ith

**C hapter 1**

**(j-fragm entability**

**1.1 **

**S o m e d efin itio n s and rem arks**

We s ta rt by giving two definitions introduced by Jayne, Namioka and Rogers which can be found in [9].

Let (X , r ) be a Hausdorff space and let p be a m etric on *X* not necessarily
related to th e topology on *X .*

We define th e *local g-diameter* of a non-em pty subset *A of X to* be th e
infim um of th e positive num bers *e* w ith th e property th a t each point *x oi A*

belongs to a non-em pty relatively open subset of *A* of g-diam eter less th an *e.*

D e fin itio n 1.1 .1 *The space X is said to be a-fragmented by the metric g if,*
*fo r each* c > 0, *it is possible to write*

X = u t=i

*Chapter 1: a-{ragm entability* **11**

*where each set X i has the property that each non-empty subset o f X{ has a*

*non-empty relatively open subset o f g-diameter less than e.*

D e fin itio n 1. 1.2 *We shall say that the fam ily B is a cover o f X by sets o f*
*small local g-diameter i f each point o f X belongs to sets o f B having arbitrarily*

*small local g-diameter.*

We shall usually say th a t *X* has a countable cover by sets of small local
diam eter when b o th th e original topology and th e m etric on *X* are clearly
distinguished and it does not create any confusion.

N ote th a t *X* has a countable cover by sets of small local diam eter, if and
only if, for each g > 0, % can be expressed as a union

**OO**
**^ = u**

1=1

each non-em pty *X i* having th e property th a t each of its points belongs to some
non-em pty relatively open subset of g-diam eter less th an *e.*

W hen X is a B anach space and the m etric is the norm m etric we shall talk of th e local diam eter of a set rath er th an its local norm -diam eter.

We shall give some more definitions which will be of interest later on in this chapter. T he definitions as well as some properties concerning th em can be found in [5, 6].

**D efin ition 1.1.3**

*C hapter 1: a -fragm entability* **12**

called *e-discrete* (or m etrically discrete w ith separating distance *e)* if

*d { y i,y2) > £* whenever *yi* G *A s , y2* G *At* and *s ^ t .*

ii) A fam ily *A =* of subsets of a topological space *X* is called *discrete*

if for each æ G X there exists an open neighbourhood of *x* which intersects
at m ost one elem ent of th e family *A .*

iii) A fam ily *A =* of subsets of a topological space *X* is called
*a-discrete* if it can be w ritten as a countable union of families each of
which is discrete.

iv) A fam ily *A =* of subsets of a topological space *X* is called *dis*
*cretely a-decomposable* (d. cr-d., for short) if for each *m £ N* there exists
a discrete fam ily such th a t

*A . = y]* S t” ), for all a e 5.

m=l

**R em ark 1.1.4**

1) If *A —* is a discrete fam ily of subsets of a B anach space X , we
see th a t it is cr-decomp os able into a countable set of m etrically discrete
families as follows. Define

*C hapter 1: a-fragm entability* **13**

for all n > 1 and *s G S.* Then

**= Û **

**for all Æ 6 5.**

n=l

And for each n G AT th e fam ily : 5 G 5 } is (^)-d iscrete in X.

2) We will m ake use of th e fact th a t in a m etric space if a fam ily is discrete
in its union, then it is d. *a-à.* in the whole space.

3) It is also well known th a t in a m etric space any open cover of the space has a <7-discrete open refinem ent (see for exam ple [16], p. 234).

**1.2 **

**cr-fragm entability o f a m etric sp a ce**

Our first result shows th a t having a countable cover by sets of small local diam eter, although apparently stronger th an a-fragm entability, tu rn s out to be equivalent to th e la tte r when th e topology on the space is given by a m etric. The ideas in th e proof are from [9], Theorem 2.4. It reads as follows:

P r o p o s iti o n 1 . 2.1 *Let ( X , d ) be a metric space and g he another metric de*
*fined on X* . *The following conditions are equivalent:*

*i) ( X , d ) is a-fragmented by g;*

*ii) ( X , d ) has a countable cover by sets of small local g-diameter.*

*When the sets in i) can be taken to be differences o f d-closed sets (or more*

*C hapter 1: a -fragm entability* **14**

**P roof. **ii)=^ i) Is clear by definition.

i)=^ ii) Given e > 0 there is a decom position of *X* given by th e c-fragm enta-
bility of th e space

*X = [ J C i.*

**1 = 1**

Fix 2 Ç i\T. Because of th e cr-fragmentability, there exists a fam ily of d-open sets, : 0 < a < /x}, covering C, such th a t

Ci n [/• \ U and e-diam (Ci n % \ U *U'^) < ^*

0</9<a 0</3<a

for 0 < a < /X. Define

= {x e X : *d{x, X \ U i ) >* - } and = ( Q H \ U «7* ).

^ 0</3<a

It is clear th a t ^ -d ia m (if” J < *e.* Now *ior * *(3* th e sets i f ” ,- and iZ^^, when
non-em pty, are separated by d-distance at least i . So for each n G TV the
fam ily { if ” j : 0 < a < /x} is discrete in (%, d).

Set i f ” = U { if” ,- : 0 < a < /x}. We have

**U **

**= U U **

**= U U **

*{Oir\K.i\*

** U **

*=*

n=l n=l *0 < o t< n* n=l 0<a</i 0</3<a

**= U (C,n % \ U I7*) = C',,**

*0<ot<fj. * *0< j3 < a*

and therefore

**X = u U**

*C hapter 1: a-fragm entability* **15**

Let us see th a t for each *n , i G N* th e set i f " has local g-diam eter less th a n

*e.* Take *x* G if " . We have th a t for some a , x G if^^. Since th e fam ily

*{ H ^ i* : 0 < O' < ;/} is discrete in (X , *d)* th ere m ust be a d-open neighbourhood
y of X such th a t

*V* n if ^ j = 0 for *P ^ CL.*

So

g -d iam (y fl i f f ) = g -d iam (y fl i f f J < *e.*

Now if th e *Ci* are differences of d-closed sets, and since th e are d-
closed, we have th a t th e i f f are differences of d-closed sets. So each i f " is
th e discrete union of sets which are differences of d-closed sets. Since d-open
sets are iV -sets, it follows th a t th e i f f ’s are also f^-sets. ■

**1.3 **

**C o u n ta b le cover b y sp e c ia l se ts o f sm a ll**

**lo c a l d ia m eter**

*C hapter 1: a -fragm entability* **16**

T h e o r e m 1 .3 .1 *Let { X , r ) he a topological space and g he a lower semicon*
*tinuous metric on X . I f [ X , r ) has a countable cover hy sets o f small local*

*g-diameter, then { X , r ) has a countable cover hy differences of r-closed sets o f*

*small local g-diameter. Moreover, i f the g-topology is stronger than r , then the*

*sets can he taken to he g-closed.*

P r o o f . Let e > 0 be any positive num ber. We shall show th a t *X* has a
countable cover by differences of r-closed sets of local diam eter less th a n *e.*

Let { [/” *: a E A , n * *N }* be a cr-discrete refinement of a cover of *X* by balls
of diam eter less th a n *e.* We can suppose th a t each {Cf” : a G *A }* is m etrically
discrete w ith separating distances *6n >* 0.

For *n Ç: N let { C ^ : m* G *N }* be a countable cover of *X* w ith local diam eter
of each (7” < T hen for each *n , m * *N* th e fam ily {Î7” H *C ^clç^a* is discrete
in (Each point in (7^ has a non-em pty relatively open neighbourhood
w ith diam eter less th a n *8n-* This open subset of C ” can intersect at m ost one
of th e *U2,* n (7” ’s.)

For each œ G Î7” fl (7^ th ere exists a r-o p en neighbourhood of *x,* say
such th a t

n ( U : n*c z )* f 0 ,

*C hapter 1: a -fragm entability* **17**

Set

### o r = u

xg£/jnC“ is a r-o p en set w ith

**C " 3 **

*( u :*

** n C")**

**and**

n ( a ; n c : ) = 0 for

Set = ( % n % ) n GS'” - It is clear th a t

^ = U U ( U K '" *

)-77% n *a e A*

We notice th a t, since *g* is r-low er semicontinuous and

diam(C7'” fl C ” ) < e,

we have

d ia m (% H % "") < £.

We now show th a t

U ( 1% n c : ] " n G D = U ( % n % ) ^ n U © r •

aG^ a 6^ otE^

Since fl (C/jg fl C ” ) = 0 for /5 ^ a , it is clear th a t

U [ ( % n G : ) " n G : - ] = U ( % n G : ) " n U *G’c*

*C hapter 1: a-fragm entability* **18**

Now, we always have

### U (% n

### " c U (% n %) '

*aeA * *aeA*

and so we m ust check th a t

### U(%nC:)"n[U

**G^nc**

**G^nc**

### U (% n%)"n[U O

*-aeA * *aeA * *aeA * *aeA*

Set *Ba =* n and *Ga =* (j”’”' and suppose th a t th ere exists

such th a t

### aJ G U

### n [ (J (ja],

*aeA * *aeA*

### U ^»"n[U (?„].

*aeA * *aeA*

T hen, since

### X e U Ga,

*aeA*

th ere m ust be C/a;, a r-o p en neighbourhood of *x,* such th a t *Ux* C *Gao* for some

« 0 G A and

**% n ( U B .) # 0.**

*aeA*

Thus *Ux* n *Bao* ^ 0, since *Gao* H = 0 for /? ao- Suppose now th a t there
exists a r-o p en neighbourhood *Vx of x* such th a t

*Vx*

** n **

*Bao*

** = **

0-Take 0 7^ C/x H 14 C *Gao* • Since *U xC V x is a* r-o p en neighbourhood of *x* and

**æ e U **

**.**

*Chapter 1: a-fragm entability* **19**

we have

**(% n %,) n (u„B„) **

*yi*

** 0,**

which implies th a t Ui

**n **

Vi **n **

*B a ^ ^*0. This is a contradiction. Now we show th a t for each

*n , m Ç: N*th e set

**U **

**n(U G.)**

*cxE A * *O.Ç.A*

has local diam eter less th a n *e.* Consider

® 6 U n ( U G„) = U ( ^ a " n Ga).

*aeA * *aeA * *oc*

Then th ere exists *ao* such th a t *x* G *Gaa* and

diam (

### (J

H### ( (J

*Ga)*fl Gao) = diam (

### |J

*{Ba*fl Ga) fl Gao) =

*aeA * *aeA * *aeA*

*=* diam (5ao H Gao) < diam(Bao ’”) < 6.

So we have

^ = U U ( U K '™ ),

*neN meN aeA*

where for each *n , m Ç: N* th e set

U *K ' ”'*

*aeA*

*C hapter 1: a-fragm entability* **20**

As a corollary of th e previous result we have:

T h e o r e m 1 .3 .2 *Let X he a Banach space. I f [X, w ea k) has a countable cover*
*by sets o f small local norm-diameter, then { X ,w e a k ) has a countable cover by*

*differences o f weakly closed sets o f small local norm-diameter and also by norm*

*closed sets o f small local norm-diameter.*

P r o o f . Note th a t by Theorem 1,3.1, for each *e* we have

**OO**

*X = \ J { F i r \ G i ) ,*

»=1

where th e *Fi* are closed and th e *Gi* are open in the weak topology, and therefore
in th e norm topology. Now each *Gi* is a 1^-set, say *Gi —* : n G *N } ,*

w ith th e *Hi^nS* being closed sets. So we have

i=l n=l

and clearly for each *i , n G N* th e set *Fi* fl *Hi^n* has local diam eter less th a n *e.l*

**1.4 **

**B o r e l se ts**

Let *( X ,* II • II) be a Banach space. The norm || • || is called a *Kadec* norm if
th e weak and norm topologies agree on {æ G % : ||cc|| = 1}. It was shown by
E dgar th a t if a Banach space X adm its an equivalent K adec norm then:

i) *B o r e l{ X ,* || • ||) = *B o r e l{ X ,w e a k ),* ([3], Theorem 1.1);

*C hapter 1: a-fragm entability* **21**

On th e other hand, Jayne, Namioka and Rogers proved, ([9], Theorem 2,3), th a t a B anach space th a t adm its an equivalent Kadec norm has a countable cover by sets th a t are differences of weakly closed sets of sm all local diam eter.

We improve E d g ar’s result by showing th a t a Banach space w ith th e JN R p ro p erty verifies ii) above. Moreover, any norm-closed subset of th e space is also a Borel subset of its bidual when considered w ith th e weak* topology, from which we get i) above as a corollary.

R e m a r k 1 .4 .0 It was proved in [9], p. 215, and [11] th a t a topological space
(X , r ) which has a countable cover by differences of r-closed sets of sm all local
p-diam eter, for some m etric *g,* any g-open set can be w ritten as a countable
union of differences of r-closed sets and therefore th e Borel stru ctures for b o th
the r-topology and the p-topology agree.

T h e o r e m 1 .4 .1 *Let X be a Banach space and suppose that (^X^weak) has*
*a countable cover by sets of small local diameter. Then X is the countable*

*intersection o f countable unions of differences o f w*-closed sets in X** and*

*therefore X* G *Borel(X**,w*). Moreover, any closed subset o f X is o f the*
*same type.*

P r o o f . Let *n* ^ o: G G *N }* be a cr-discrete refinem ent of a cover of

*C hapter 1: a-fragm en tab ility* **22**

For *n , p G N* **let ** : m G *N }* **be a countable cover of ***X* **w ith the**
**local diam eter of each ** *< 6n,p.* Then **for each ***n , m , p* G *N* **th e fam ily**

*{ ^ a , n * *^ n , m } c c e A* **IS Weakly discrete in**

U K » n C ^ , m ) ,

*a e A*

in fact it is discrete in (7^^. So for each *x* G H there exists a iy*-open
neighbourhood of *x* in *X**,* say such th a t

**n **

**{Ul„**

**{Ul„**

** n **

**C I J**

**C I J**

** # 0,**

n ( % . n = 0, for # a .

Set

_ I I *T T n,m ,p*

a,n **U ** *^x,a*

**is clearly a u;*-open set with**

**D ([/;,. n c ;,„ )**

and

G” ? n n *C ^ J =* 0, for /3 5^ a .

Set = (tfS,„ n *c i m )* n
We show th a t

### 7

**^ — n U U{ U **

*^a,Tn,n,p}’*

*p m n aeA*

N otice th a t, since diam(C/J„ fl *< -,* we have

diam(Z7a,n fl *Cn,m* ) < - .

*C hapter 1: a-fragm entability* **23**

Let

x - e X - n f l U U U

*p m n o te A*

For each *p Ç: N* there exist *n , m E N* and a G A such th a t

SO th e re exists *Xp E* D such th a t ||cc** — ccp|| < Thus we have

• 11- lim *x„ = X*

' *p -* o o* *p ~*

and therefore *x*** G *X .*

We now show th a t

### u

( ( % " n c i m ) n =### U

n CK.,n) n (J**ct£A ****ot£A ****a £ A**

We have

U ( ( % " n CS,„) n G ^,j) = U (GJ,„ n CS,„) n U GT,?.

*cxE A * *a £ A * *a £ A*

since H „ D = 0, for *P ^ a. So* we m ust check th a t

U ( % " n CS,„) n [ U G” ^] c U ( % . n CS,„) n [ U

*olÇA * *ot^A * *ol^A*

Set B a — LI *Gnjn* and suppose th a t th ere exists

such th a t

**U ** **n [ U G„]**
**O.Ç.A ****a £ A**

**X ^ u ** **” n [ U G„]**

*C hapter 1: a -fragm entability* **24**

T h ere m ust be a it;*-open neighbourhood *Ux* of cc, since

*X G [J Ga,*
*aeA*

such th a t *Ux* C *Gaa* for some *œq G A* and

*aeA*

**T h us ****Ux**** n *** BaQ* 7

**^ 0, since**

**Ga^****fl**5 / 3

**= 0 for /3**7

**^**

ao-Suppose now th a t there exists a *w*-open* neighbourhood *Vx oi x* such th a t

*Vx* n *Bao =*

0-Choose *Ux* w ith 0 7^ C/x D C *Gao ■* Since *UxHVx is a* tu*-open neighbourhood
of *X* and

we have

**X 6 U **

**>**

*aeA*

( % n i / ) n ( U a . ) f 0,

*aeA*

which im plies again th a t t/x fl 14 fl *Bao* 7^ 0- This is a contradiction.
If 5 is a norm closed subset of *X ,* consider the families

*{ F* n n *c ^ J a e A*

and follow th e proof. In this case th e vectors *Xp^s* belong to *F* and, since *F* is

closed, th e lim it also belongs to *F.* ■

*C hapter 1: a-fragm entability* **25**

C o r o lla r y 1 .4 .1 *Let ( X,* || • ||) 6e a *Banach space with a countable cover by*
*sets o f sm all local diameter, then B o re l{X ,* || • ||) = *B o r e l[ X ,w e a k ).*

P r o o f . Let A be a norm-closed subset of *X .* By T heorem 1,4.1, *A* G

*B orel{X **,w eak*),* i.e. there exists *B* G *X**, B* G *B orel(X **,w eak*)* and

*A = B n X .* B ut th e weak topology on *X* coincides w ith th e restriction to *X*

of th e weak* topology on *X*** and therefore A is a weak-Borel subset of X . ■
A t th is point we need to give another definition which can be found in [16],
pages 345-346, as well as some basic properties.

D e f in itio n 1 .4 .1 Let *Fq* be th e fam ily of closed sets of a m etric space. Suppose
th a t for an ordinal num ber a , we have defined the families for *^ < a.* So
th e sets of th e fam ily *Fa* are countable intersections or unions of sets belonging
to *F^* w ith *^ < a* according to w hether *a* is even or odd (th e lim it ordinals are
u nderstood to be even). It is known th a t the transfinite union of this families
gives us th e fam ily of all Borel sets.

We can also do th e same using open sets. Set *Go* to be th e fam ily of open
sets. Suppose th a t for an ordinal num ber *a,* we have defined th e families
for ( < CK. So th e sets of th e fam ily *Ga* are countable unions or intersections
of sets belonging to *G^* w ith ( < according to w hether *a* is even or odd.

*C hapter 1: a-fragm en ta bility* **26**

a fam ily will be said to be of *multiplicative class a.* Similarly, th e families *Fa*

w ith odd indices as well as th e families *Ga* w ith even indices are additive and
form th e *additive class a.*

O ur last result of this section is a generalization of the one m entioned above from [11]. T he ideas for th e proof also come from there. It will be used later on in C h apter 2.

L e m m a 1 .4 .1 *Let ( X ,t) be a topological space such that any open set is an*
*Ff^-set. Let g be a m etric on X and suppose that X has a countable cover by*

*sets o f additive class a o f sm all local g-diameter, then each g-open subset o f*

*X is o f additive class a*..

P r o o f . Let G be a g-open subset of *X .* Let m > 1, be a countable
cover of *X* by sets of additive class a of small local diam eter.

For n > 1 set

M (n ) = {m *Ç: N \ m > l* and *Dm* having local diam eter less th a n —}.

T hen for each n > 1 th e fam ily *{Dm* • G *M ( n ) }* covers *X .*

W rite

*Gn = { x ^ X : {y e X : g{y, x) C G }}.*

For each n > 1 and each m > 1 in M (n ), we consider th e points cc G Gn H

*Chapter 1: a-fragm entability* **27**

T hus *X* G *U{x)* C *G.* Hence, for m > 1 and m G *M ( n )* th e set

*Un,m =* U{C/(œ) *: x e GnH Dm}*

**is **a **relatively open subset of ***Dm-* So **for every ***n ,m ^ N* **there exist ***Vn,m,*

**T-open subsets of ***X ,* **such that ***Un,m = Dm* H V^,m- Now we have

**OO**

### = u

*i = l*

where th e sets are r-closed in *X .* Hence

**OO**

**1 = 1**

and therefore th e sets *Un,m* are of additive class *a.* They contain (7„ fl *Dm* and
are contained in *G.* Hence

U{C/n,m : 71 > 1, 771 G *M {ti)}*

is a set of additive class a th a t coincides w ith *G.* ■

**1.5 **

**D e c o m p o sitio n o f d isc r e te fa m ilies**

*C hapter 1: a-fra gm enta b ility* **28**

P r o p o s i t i o n 1 .5 .1 *Let { X , r ) be a topological space and g he a m etric on X .*
*Then the follow ing conditions are equivalent:*

i) *{ X ^t) has a countable cover by sets o f sm all local g-diameter;*

ii) *There exists a decomposition o f X ,*

**OO**

*X = \ J C i*

*i = l*

*such that i f A = * *is a g-discrete fa m ily o f subsets o f X , then fo r*

*each s Ç: S there is a decomposition*

*A . = \ j*

1=1

*with A \* C *Ci such that each fam ily * *is discrete in*

P r o o f . i)=> ii) For each *n Ç: N ,* consider
**OO**
X = u c r ,

**1 = 1**

w here th e sets *C ^* have local g-diam eter less th a n *K*

Let *A =* {A,}ag5 be a discrete fam ily in (X , g). Then there exist ^ -d iscrete
fam ilies such th a t

A ,

### = IJ

AT", for*s e S.*

**771 = **1

W rite {A*’^ } = A ^ H *C ^ ,* for *m , i E N* and *s Ç: S* and fix z ,m G *N .* Take

*X* G *Cf^.* T h ere exists a T-open neighbourhood of *x,* say *U,* such th a t

d ia m fy n C f ) < — .

*C hapter 1: a-fragm entability* **29**

Therefore *U* m eets at m ost one elem ent of th e family So
is discrete in r ) and we have

**^ = U **

**= U U^mncr = **

**U4'"-m=l ** **m=l t=l ** **t,m**

Set = *Fn* and So we have

**OO**

*A , = \ J B : , B :* C Fn,

**n=l**

and is discrete in **( F n ,r ) .**

ii)=> i) Given c > 0, let { [/” : a G F} be a a-discrete open refinement of
an open cover of *X* by balls of radius less th an | . Let *{Cm}m=:i* a countable
cover of *X* such th a t for *n , m Ç: N ,*

**y n ^ **

### 0

**^n,n.**

**m=l**

and is discrete in *{Cm^T)-* W rite

### c = u

a er Obviously,

**% = U c .**

*n , m*

We now show th a t for each G W, th e set has local diam eter less
th a n *e.*

Take *x £ F ^ .* T hen *x* G *B^'J^* and therefore there exists a T-open neigh
bourhood *U OÎ X* such th a t

*«o-Chapter 1: a-fragm entability* **30**

So diam(C/' D *F ^ )* = diam(C/' fl *<* diam (j5”’”^) *< e.* ■
We give an analogous statem en t for th e case of the space being
cr-frag-m ented. We ocr-frag-m it the proof because it follows th e sacr-frag-me line as th e previous
one.

P r o p o s iti o n 1 .5 .2 *Let ( X ^ r ) be a topological space and g be a m etric on X .*
*Then the following conditions are equivalent:*

i) (%, r ) *is a-fragm ented by g;*

ii) *There exists a decomposition o f X ,*

**OO**
*X =* *\ J C* *i ,*

**1 = 1**

*such that i f A = * *is a g-discrete fa m ily o f subsets o f X , then fo r*

*each s Ç: S there is a decomposition*

### ^ = U

### A: c

**Ci**

**Ci**

**1= 1**

*such that each fam ily * *satisfies the following condition:*

**C hapter 2**

**O n e-to-on e m aps**

**2.1 **

**In tr o d u c tio n**

Suppose we have two B anach spaces *X* and *Y ,* and a one-to-one m ap *T* :

*X* *— > Y .* If we assume th a t *Y* has a countable cover by sets of small local
diam eter, w hat kind of condition do we have to im pose on T for *X* to have
such a cover as well? W hat can we say about T ” ^? We shall give some answers
to these questions in this chapter, b u t first of all we have to fix some notation.

If Ti and T2 are two topologies on a topological space üT, we shall say th a t
Ti is stronger th a n Tg, denoted by Tg T%, if any T2-open subset of *H* is also
Ti-open.

Let (X ,T i) and T2) be topological spaces and pi , p 2 be m etrics defined
on *X* and y , respectively. If T : X *— >* Y is a one-to-one m ap, we define: *Tr^*

*C hapter 2: O ne-to-one m aps* **32**

**to be the topology on ***X* **given by the family ***{T~^( U) : U* **Tg-open} and ** **to**

**be the topology on ***X* **associated to the metric ***T o Q2,*

**i.e, the family**

*{ T~^{U)* : *U* **p2-open **}. **Note that if ***T* **is **t i -T2 **continuous (respectively ***gi -Q2*

**continuous), then ** **(resp. ** **Tg^ ■;< g i ) .**

D e f in itio n 2 . 1.1 *We shall say that a map T , as above, is discretely a-decom *
*posable, d. a-d. fo r short, fo r the pair* (p i, P2) ^ 2^ *is one-to-one and it m aps*
*gi-discrete fam ilies o f subsets o f X into g2~d. a-d. fam ilies o f subsets o f Y .*

These m aps were used by Hansell, see [6], where we refer for fu rth er proper
ties, w ith o u t im posing on th e m ap th e condition of being one-to-one. It is easy
to see from R em arks 1.1.4 th a t p2-d. cr-d. families in *T { X )* are also ^2-d. *a-d.*
*i n Y .*

**2.2 **

**d. **

*a - d .*

** m ap s**

We now prove a lem m a which will be used in Theorem 2.2.2 below.

L e m m a 2 . 2.1 *L et { X,* t i ) *and (Y,* T2) *be topological spaces and gi,* g2 *be m etrics*
*defined on X and Y , respectively. Suppose that there exists a d. a-d. map*

*T* : *X* *— y Y fo r the pair* (g i, ^2)- (Y, T2) *has a countable cover by sets o f*
*sm all local g2-diam eter, then there exists a decomposition o f X ,*

*X = \ j E i ,*

**1**** = ****1**

*Chapter 2: One-to-one m aps* **33**

*then there is a decomposition*

**OO**

**= **

### [J

**A i ,**### 4

**C**

**Ei,**t=l

*such that each fa m ily* ^ *discrete in the space *

**(^Ei^Tr^)-P roof. ** Consider th e m ap *T* **: ***X* **— > ***Y.* T hen th e fam ily T A is d. cr-d.
Therefore th ere exit discrete families p G such th a t

**GO**
*T A , =* U

p=l

By Proposition 1.5.1 there exists a decom position of *Y ,*

*Y = l ) C „ ,*

n=l

and discrete families *{B^'^}aç.s* in th e space (Cn,T2), w ith

**OO**

•Bf = U BI"'", and B "'’’ C C„ for p e iV, a e *S.*

n=l

Since T is trivially T^^-Tg continuous it follows th a t th e family

T - ' ( { B n . e s )

is discrete in th e space ( T “ ^(7n, TT2). It is clear th a t

00 00 *00*

^ = U U = u

p=l n=l **1 = 1**

*Chapter 2: One-to-one m aps* **34**

We now give th e analogous lem m a for <j-fragmentability.

L e m m a 2 .2 .2 *Let* (X ,T i) *and* (y ,T 2) *be topological spaces and let gi,Q2 be*

*m etrics defined on X and Y , respectively. Suppose that there exists a d. a-d.*

*map T : X* *— > Y fo r the pair* (p i, ^2)- / / (Y, T2) *is a-fragm ented by Q2, then*

*there exists a decomposition o f X,*

00

*X = \ J E , ,*

*i = l*

*such that i f A = * *is a gi-discrete fam ily o f subsets o f X , then fo r each*

*s E S , there is a decomposition*

**A s = \ J A ) , A \**** C ****Ei ,****1 = 1**

*such that each fa m ily * *satisfies the following condition:*

*fo r any i E N , i f* : 5 € 5} ^ 0, *then fo r every non-em pty*
*subset A *

*o f*U{A* : 5 G 5 }

*there exists a r-open subset U o f X*

*such that U f)*

**A**C**A],**fo r exactly one s E S .T h e o r e m 2 .2 .1 *Let* (X ,T i) *and* (Y,T2) *be topological spaces and let* pi , P2 *be*
*m etrics defined on X and Y , respectively. Suppose that there exists a *

*one-to-one map T : X* — > Y. *Then the following conditions are equivalent:*

i) *T is d. a-d. fo r the pair* (pi, ^2)/

*C hapter 2: One-to-one m aps* **35**

iii) *(X, Tg^) is a-fragm ented hy gi.*

**P roof. ** ii) Given e **> ** 0, let {C/” **: ** a **G ** F} be a cr-discrete open
refinem ent of an open cover of *X* by balls of pi-radius less th a n | .

For every *n G N* th e family { T (f/” )}aer is d. *a-d.* in *Y .* T hus we have
**OO**

**T{u:)**

**T{u:)**

** = u**

m=l
w ith {B2'”*}e<gr being gj-discrete.

Define E J ’"* = C *U^.* It is clear th a t

n,m *a*

We show th a t for any *n , m Ç: N* th e set

**U **

*K’”'*

»er has local pi-diam eter less th an e.

So take *x* **G ** Then Tcc **G ** and so th ere exists a ^2-open set,
say *V ,* such th a t *V* fl = 0 for a *œq.* **Now**take *G = T~^{V)j* which is

Tgj-open, and we have

gi-diam (G fl ( *\J * *= gi-diam {G* fl *E^ f ^ ) <* ^i-diam(C/^’"") < *e.*
*aer*

ii)<=> iii) Proposition 1.2.1.

ii)=> i) Because of Proposition 1.5.1, any pi-discrete family, *A =*

can be decomposed in th e following way:
**OO**

*A , = \ J*

*C hapter 2: One-to-one m aps* **36**

w ith th e fam ily {A*}aG5 being T^j-discrete in its union. Since *Tg^* is a m etric
on *X ,* it follows th a t these families are *Tg^-d.* cr-d, in *X* and so *A* is also

*Tg^-d.* cr-d. Hence th e m ap *T* is d. cr-d. for th e pair (pi, ^2)- ■

**T h eorem **2 .2 .2 *Let* (X ,T i) *and* (Y,T2) *he topological spaces and let gi,Q2 he*

*m etrics defined on X and Y , respectively, with T2* d *9 2- Suppose that there*

*exists a one-to-one map T* : *X* —*> Y and that* (Y, T2) *has a countable cover hy*
*sets o f sm all local Q2-diameter. Then the following conditions are equivalent:*

i) *T is d.* cr-d. *fo r the pair (gi,* P2)/

ii) *( X, Tg^) has a countable cover hy sets o f small local gi-diam eter;*

iii) *(X, Tg^) is a-fragm ented hy gi;*

iv) *[ XjTt^) has a countable cover hy sets o f small local gi-diam eter;*

v) (X , TV2 ) ^5 *a-fragm ented hy g i*.

**P roof. ** i)=>iv) For each e > 0 and *p Ç: N , let Ap =* be discrete
families in X such th a t U{Ap : *p* G *N }* is a refinement of a cover of X by balls
of Pi-radius less th a n | .

By Lem m a 2.2.1 th ere exists a num ber of sets *C j ,* w ith

**^ = U**

i=i
and decompositions
*A l = \ _ } A l ' ’ , a € S , p € N , A ^-^C lC j*

*C hapter 2: One-to-one m aps* **37**

such th a t each fam ily is discrete in *{Cj^Tr^)’*

Now define

B? = U aes It is clear th a t

**OO ** **OO**

**X = u u**

J=1 *p-1*

We show th a t th e fam ily is a countable cover of *X* by sets of sm all local
gi-diam eter. Let *x* G Since the family is discrete in (C j’jTVj)
th ere exists a *Tr^* open neighbourhood y of æ such th a t

( y n C , ) n A r f 0,

for some *sq ^ S* and

(V n *Cj)* n = 0, for *s ^* 5q.

If V n *Cj* n = 0 for every 5 **G **5', we would have

y n c,- n ( U = 0, aes

and this would im ply th a t *x ^ B j ,* which is a contradiction.
So th ere exists 5q **G **5 such th a t

*V n C j H* A ÿ ^ 0 and y

**n **

*Cj*

**n **

*A ^ / =*0, for 5 ^ So.

Now, since *A^'J* is contained in a ball of radius less th a n | , pi-diam(Af^^) < *e.*

*C hapter 2: One-to-one m aps * *38*

g i-d ia m (y fl *B j ) =* p i-d ia m (y fl ( | J *=* g i-d ia m (y D *A ^ f ) < e.*
*ses*

iv)=>v) Obvious.

v)=4>iii) Obvious. ■

**T h eo rem 2.2.3 ***Let* **(X, Ti) ***and* **(T,T**2**) ***he topological spaces and let g i , g2 he*

*m etrics defined on X and Y , respectively, with T2* p2- *Suppose that there*

*exists a one-to-one map T* : *X* — *> Y and that* (Y, T2) *is a-fragm ented hy* p2-

*Then the following conditions are equivalent:*

i) *T is d. a-d. fo r the pair* (gi, ^2);

ii) (X , Tgj) *has a countable cover hy sets of small local gi-diameter;*

iii) (X , Tpj) *is a-fragm ented hy gi;*

iv) (X , *Tr2* ) *is a-fragm ented hy g i*.

**P ro o f. **i)=^iv) The same proof as above bu t using Lem m a 2.2.2.

iv)=>iii) Obvious. ■

*Chapter 2: One-to-one m aps* **39**

### 2.3

### I n v e r s e m a p p in g s

We sta rt by recalling a definition from [16].

**D efin ition 2.3.1 ***A mapping f* : *X* *— > Y is said to be o f Borel class a if,*
*fo r every closed subset F g Y , the set f ~ ^ [ F ) is Borel o f multiplicative class c l.*

*(Equivalently, f ~^ { G) is o f additive class a, fo r every open set G).*

O ur next result improves p a rt of a result in [14], Corollary 7.

**T h eorem 2.3.1 ***Let X , Y be two Banach spaces and f* : *X* *— > Y be a con*
*tinuous linear injection. Define ip = f~ ^* : /( % ) *— > X . Then the following*
*conditions are equivalent:*

i) *p is o f Borel class a and f is d. a-d.;*

ii) (X ,/||.||y ) *has a countable cover by sets o f additive class ex. (for the *

*topol-ogy fw-Wy) o f small local*|| •

*\\x~diameter.*

**P roof. **ii)=> i) By Theorem 2.2.2 we have th a t / is d. <7-d.

Now let *G he a* norm open subset of *X .* Since th e topology /||.||y in *X*

verifies th a t any open set is an *F^* set, we apply Lem m a 1.4.1 and ob tain th a t

*G* is of additive class *a* in (X , /||.||y)- Hence the set *p~^(G)* is of additive class

*a* in ( / ( X ) , II • ||y ) and therefore *p* is of Borel class *a.*

*C hapter 2: One-to-one m aps* **40**

discrete families *{B ^^}aeA* such th a t
**OO**

**b ; = u **

Bir-t=i

N ote th a t since th e sets C/” are open in *X* and *ip* is of Borel class *a,* th e
sets *B 2 =* are of additive class *a.*

Now fix 2, n G *N .* T he fam ily H *B2}aeA* is discrete in *f { X )* and its
sets are of additive class *a.*

Set

**%,» = U (%" n **

*B2).*

**ol****E A**

T h en is a discrete union of sets of additive class *a* and therefore is itself
of additive class *a.* (For a proof of this fact see [16], page 358, T heorem 1). It
is obvious th a t

*f ( X )* = U H .> .
*i , n*

Define The sets *Ci^^s* are of additive class *a* and they
form a countable cover of (%, /||.||y).

Now tak e *x* G and *f { x ) = y E * *So y E Ba^* fl *B^^* for only one
0=0 G *A.* Thus th ere exists an open neighbourhood *U oi y in Y* such th a t

*U* n *{ B T* n 5 ” ) = 0 for a ^ qq.

W rite *V = f ~^{U).* V is a /||.||y-open neighbourhood of *x* and

d ia m (y n *Ci,n) =* d i a m ( / - '( B fl H .>)) = d ia m (/- ^ (B n n <

*Chapter 2: One-to-one m aps* **41**

Therefore *Ci^n* has local diam eter less th an *e.* ■

### 2.4

### S o m e to p o lo g ic a l in v a r ia n ts

The following lem m a provides us w ith a useful tool to check when some m aps are d. cr-d. For a proof of it see [17].

**L em m a 2.4.1 ***Let X and Y be Banach spaces with norm s* || • *\\x cind* || • ||y,

*respectively. Let T : X* *— > Y he a one-to-one map such that fo r every hounded*

*sequence * *in X converging to some point x in the* T||.||y *topology we have*
*that X n *

*converges weakly (or pointwise in the case o f X being a C (K ) space)*

*to X . *

*Then T is d. a-d.*

**T h eorem 2.4.1 ***Let* (%, || • *\\x) and* (Y", || • ||y) *he Banach spaces. Suppose*
*that there exists an hom eom orphism (f) : [ X , w e a k )* — ^ *(Y^weak). Then the*
*following results hold.*

i) *( X j We a k ) has a countable cover hy sets o f small local* || • *\\x~diameter*
*i f and only i f [Y, weak) has a countable cover hy sets o f small local* || • ||y -

*diameter.*

ii) (X , lyeaA;) *is a-fragm ented hy* || • *\\x i f and only i f (Y, weak) is a-fragm en*
*ted hy* **II **• **| | y .**

iii) *B o r e l { X , w e a k )* = *B o r e l(X ,\\* • ||x ) *i f and only i f B orel{Y , weak )* =

*C hapter 2: One-to-one m aps* **42**

P r o o f . In this case it is clear th a t th e conditions in Lem m a 2.4.1 are
fullfiled and so our m ap * (f)* and its inverse are d. a-d. Moreover, since there
is weak to weak continuity, by R em ark 2.2.1 and Theorem s 2.2.2 and 2.2.3, i)
and ii) hold. To prove iii) assume th a t

*B orel(Y ^w eak)*=

*B orel{Y ,*|| • ||y).

D enote by R(x,||.||;^) th e closed un it ball of *{X ,* || • ||%). *B(^x,\\-\\x)* is a

w-closed subset of *X* and, since ÿ is a hom eom orphism , ÿ(jB(_y,||.||^)) is a iw-closed
subset of *Y .* Thus th e norm || • ||% is lower semi continuous on (X , ^||.||y). Since

*(j)* is d. a-d ., (X , ^||.||y) has a countable cover by differences of ÿ||.||y-closed sets
of sm all local || • ||%-diameter. So if G is a || • ||%-open subset of X , then, see
R em ark 1.4.0,

### G = U C"

**1 = 1**

where *Ci* is th e difference of two <ÿ||.||y-closed sets for every % E X .

Set *Bi = <!>{Ci).* T hen th e *B {S* are differences of || • ||y-closed sets and
therefore th ey are w-Borel sets. Thus *^~^[Bi)* = *Ci* are ly-Borel in X . We
conclude th a t G is a countable union of *weak-Bore\* sets and therefore is itself

a *w eak-B orel* sub et of X . ■

### 2.5

### T h e Co-sum o f so m e B a n a c h s p a c e s

**N o ta tio n : ** *Let A he a set. We shall denote hy \A\ the cardinality o f the set A .*

*C hapter 2: One-to-one m aps* **43**

*norm s* {|| - Un : G *N } . We denote hy* co{X„ : n G *N } the Banach space o f*
*all sequences x = * *with Xn* G *X n , fo r n £ N , and such that given* c > 0

*there exists m E N such that* ||aJ„||„ < *e fo r n > m . The norm in this space is*

In order to give our next result we need th e next proposition. The proof of p a rt i) can be found in [9], Theorem 6.1.

**T h eorem 2.5.1 ***Let {Xn,* **|| • ||n )^ i ** *a sequence o f Banach spaces. Set X =*

*co{Xn* : *n* **G ***N } . Then the following results hold.*

i) *I f fo r each n E N the space X n is a-fragm ented hy its norm, then the*
*space X is a-fragm ented hy its norm.*

ii) *I f fo r each n E N the space X n has a countable cover hy sets o f small*
*local*

### II

'*\\n~diameter, then X has a countable cover hy sets o f sm all local*

### II •

**\\oo-**

**\\oo-**

*diameter.*

**P roof. **For each n **G **iV, we can find a sequence *Bn = {^m)m=i* subsets
of *X n* such th a t each point of belongs to sets of *Bn* having arb itrarily small
local II • 11 n-diam eter. For *r , k E N* define

C* *= { x = {xn)Z=i* G *X : \{xn* : ||a;„||n > ^ } | = *r}.*

*C hapter 2: One-to-one m aps* **44**

For r > 1, A: > 1 define

^ n n ... n

for m.}< — ^771 j j ... j TTij*^ j dj. — ^cîj j ... ^ *dj-* ^ &nd

*■^o,k —* G *X* ; ||x|[oo ^ ^}*

I t ’s clear th a t

**OO ** **OO ** **OO**

### ^ = U U

### U

### U

### 4T“'U(U^0,.).

*k=l r=l* d^e{l r}^mrE{ji*,* *^=1*

So we have a countable collection of sets covering *X* and now we will show
th a t given *x =* G *X* and e > 0 th ere exist r, *k,* d^, n ir such th a t

**X e**

and th e local || • ||oo-diam(i4™j’’*’') < *e.*

Take *k E: N* such th a t ^ < f- T hen either ||x||oo < ^ or th ere exists

*r £ N* such th a t *x ^ * *In* th e first case, *x* G *Ao^k* and obviously th e local
II • 11 OO-diameter of *Ao^k* is less or equal th a n *e.*

So suppose th a t there exists *r E N* such th a t *x* **G **(7*, i.e., ||cci||,- > *\* for
% — *d*\ ,..., *dj-* « Let

**r ** **• r ll®»l|t ****~ k ****^ ****• ****J****J 1**

*0 = r m n {* — - , - : z = c ti,..., *dr}.*

Now for *i* **G **{d%,..., *dr}* we have th a t *X{* **G ***X{.* Thus we can find m*- **G ***N* and

*C hapter 2: One-to-one m aps* **45**

open neighbourhood of *X{* in *X{,* say *Vi,* such th a t

II • ||i-diam(K- n < 5.

Set

*P r \ V i ) .*

*i = l*

y is a weak open neighbourhood of *x* in *X .* We show th a t

II • ||oo-diam (y PI *< e.*

Take *y * *Cl* For *i = d i , d r ,*

l|y%||: > llk :||i - ll^i -2 /i||i| > p

SO for *i* 0 *{di, ...,d r},* ||i/i||i < Therefore

||®t- - 2/i||i < < I for z = d i , *dr.*

and

**1 ** **1 ** **2 ** **e**

ll^i - 2/*||i = - otherwise.

Hence || • ||oo-diam (y fl *< e as* required. ■

Following th e notatio n in Theorem 2.5.2 below, in [15], Kenderov and
Moors showed th a t if th e spaces *{C{Kn) , pt wi se)* are or-fragmentable, th en

*{ C { K ) , p t w i s e )* is also cr-fragmentable. We prove it for th e weak topology and
also in th e case of countable covers by sets of small local diam eter.

*Chapter 2: One-to-one m aps* **46**

i) *I f fo r each n ^ N the space { C{ Kn) , we a k ) is a-fragmentable, then the*
*space ( C { K ) , w e a k ) is a-fragmentable.*

ii) *I f fo r each n ^ N the space [C{Kn)j Weak) has a countable cover by sets*
*o f sm all local diameter, then the space {C i^K \w ea k') has a countable*

*cover by sets o f small local diameter.*

P r o o f . Define the m ap

*T* : *C { K )* — co(C (ir„), **II • H o c )**

by th e form ula

**T(/) = **

1.
By Theorem 2.5.1 and R em ark 2.2.1, since *T* is clearly weak to weak con
tinuous, we only have to show th a t *T* is d. cj-d.

So take *{ f m ) m = u f* ^ C'(R') and suppose th a t *{T{f m) )* converges to *T { f )*

in th e norm of cq, i.e., given *e > 0* there exists *m E N* such th a t for all Â: > m
we have

*m f k ) -* T ( / ) |U < e,

I.e..

*\\—f k \ K n* *f\Kn\\oo* ^ £ for all *n E N .*

*n * *n*

Now, *\{ X E K ,* th ere exists

*n E N*such th a t

*x E K n ,*and so

*\—f k M ~* ~ / ( ^ ) l — ^ for all *k > m .*

*C hapter 2: One-to-one m aps* **47**

Thus *fjn* converges to / in th e pointwise topology and by L em m a 2.4.1 we

**C hapter 3**

**Spaces w ith a countable cover**

**by sets o f sm all local diam eter**

We shall give some exam ples of spaces w ith countable covers by sets of sm all local diam eter as well as some w ith well behaved injections into Co(r) for some set

**r.**

### 3.1

### T h e s p a c e co(r)

D e fin itio n 3 .1 .1 *Let V be a set. We define* **co(F ) ** *to he the set*

c o (r) *= { x Ç: BF : fo r* a// £ > 0 |{i G F : |æ(i)| > s}| < oo}.

*When endowed with the swpremum norm,* **co(F ) ***is a Banach space.*

O ur next proposition is based on L em m a 3.2 from [22]. It also follows from

*C hapter 3: Spaces w ith a countable cover by sets o f sm all local diam eter* **49**

[9], Theorem 2.1 b), since Co(T) has an equivalent locally uniform ly convex
norm and on bounded sets in co(T) th e weak and pointwise topologies coincide.
P r o p o s iti o n 3 .1 .2 *{cq{T), p oin tw ise) has a countable cover hy differences o f*
*pointwise closed sets o f small local diameter.*

P r o o f . Given e > 0, for n G iV w ith ^ < | and *k Ç: N we* pu t

*= {y ^* Co(r) : |{t E r : *\yt\ >* - } | = *k}.*
*n*

We show th a t is the intersection of an open and a closed set in th e pointwise topology. Set

. . . , *CLm)* = {2: **G **Co(r) : > - , 2 = 1, . . . ,m } , *a{* **G **T.

and

### (cti am)

6### r»"

is open and clearlyWe now show th a t th e sets *B ^* have local diam eter less or equal th a n e.
Let *y'* **G ** and let {t **G **F : *\y[\* > ^} = *{ t i , . . . ,tk}-* T hen there exists
0 < (^ < I such th a t *\y’f*..| — <^ > F ,% = 1, . . . , A;. The set

**U = {y G Bk**

**U = {y G Bk**

### : |î/f^ ~

2### /fi I < • • • ?

**\yt^ ~ Vth\**

**\yt^ ~ Vth\**

is an open neighbourhood of *y '* in *Bk* in th e pointwise topology.
Take *x G U.* T h en for 2 = 1, . . . , A;

*Chapter 3: Spaces w ith a countable cover by sets o f sm all local diam eter* **50**

A nd since *x* G and *\xn\ >* = 1, . . . , A:, we have *\xt\ < ^ i o r t ^ ti.*

So take *x , y £ U.* L et’s show th a t **||a; **— y He» < For z G { 1 , . . . , A;},

* \^u — Vti\* <

*~ 2 / t J +*

**\^ti***“ 2/ t i l*

**\Vti**

**^ S -h S = 26 < e.**A nd for t G F w ith *t* ^ *ti,*

1 1 2

*\xt - yt\ < \xt\* + jytj < - + - = - < £ .

*n * *n * *n*

**3.2 **

**P r o je c tio n a l R e s o lu tio n s o f I d e n tity**

D e fin itio n 3 .2 .1 Let X be a Banach space. The *density character o f X,*

denoted by *dens(X),* is th e smallest cardinal num ber of a dense subset of *X .*

D e fin itio n 3 .2 .2 Let X be a Banach Space. We denote by *p* the smallest
ordinal such th a t its cardinality *\p\ =* d en s(X ). A *projectional resolution o f*
*identity, P R Iio r* short, on X is a collection {P« *: ujq < a < p \ oi* projections

from X into X th a t satisfy, for every *a* w ith *(jUq < a < p,* th e following
conditions:

i) l|Fa|| = l;

ii) PaoP/3 = P/goPa = FL if Wo < a < < ft;

*Chapter 3: Spaces w ith a countable cover by sets o f sm all local dhmet(er* **51**

iv) U{-f/3+ i(-^) *: CÜQ < /3 < a } is* norm dense in

v) = *I d x .*

T he following lem m a lists some properties of th e *P R F s* tha; we will need
later on.

L e m m a 3 .2 .1 *Let X be a Banach space and {Pa : ujq < a < p- h a P R f on*
*X . We pu t Pa+i — Pa = Ta> foT ljq < CL < p. Then the followinj r^.sults hold.*

i) *For every x* G *X , i f a is a lim it ordinal, ojq < a < p, we havi*

*Pa{x)* = II • II — lim f^ (æ ).

**/ 3 < a**

ii) *For every x £ X ,* {||Th(a:)|| • *ol* G *[ujq, p )} belongs to cq{[ljo,p)).*

P r o o f , i) Let a be a lim it ordinal, *< cl*

*< p,*and let

*x £ ï . l î*

^ e U

say * X £ P ^ q ( X ) ,* th en

*Pfi{x) = x*Thus, in this case, wehcve

**i f j3**>**^ q .**II • II — lim *Pj3{x) = X =* Pa(aj). (1)

If X G * P a { X ) ,* by (iv) in Definition 3.2.2, given

*e >*there exsti

**0***C hapter 3: Spaces w ith a countable cover b y sets o f sm all local diam eter* **52**

such th a t 11 a; *— y \\ <* | . For this c > 0 there m ust *he f3o < a* such th a t for any

*l3, /3o < /3 < a ,* we have

Thus

I l f . W - P;3(x)|| < ||P „(x ) - f .W I I + ||P „(ÿ ) - *P0(y)\\ +* ||P^(ÿ) - *Pp{x)\\ <*

< ll-P a lllk -y ll + - f . ( y ) || + ||-P^III|œ -ÿ|| < E,

whenever *(3 > po.*

Finally for *x £ X* and *P < a we* have

*P^{x)* - Pa(x) = *Pfi{Pcc{x)) - Pcc{x)*

and thus (1) holds for every *x Ç: X .*

ii) If th e assertion is false, there exists æo G X , £ > 0, and

Wo < 0=1 < #2 < . . . <

such th a t for every i > 1, ||7hX^o)|| > Set

*a* = sup{«i : i > 1}.

It is clear th a t

lim *Pfi{xo)* ^ *Poc{xo)y*

*Chapter 3: Spaces w ith a countable cover by sets o f sm all local diam eter* **53**

**3.2.1 **

**d. cr-d. m aps into co(F) using P R I**

The next theorem is a straightforw ard ad ap atatio n to th e case of *P R F s* from
th a t in [22].

T h e o r e m 3 .2 . 1. 1 *Let p be a class o f Banach spaces such that the following*
*results hold:*

i) *fo r every X G p there exists a P R I on X , {Pa : ujq < a < p }.*

ii) (P„+i - P .)(% ) 6 p.

*Then fo r every X G p there exists a one-to-one hounded linear map*

*T : X ^* co(r),

*fo r some set* T, *such that fo r every discrete fam ily A = * *o f subsets o f*
*X , the fa m ily T A = { T Aa]aç.s is d. a-d. in* co(r).

P r o o f . Let *X G p.* We proceed by induction on *densi^X).*

W hen *X* is separable, we have th a t is m etrizable and separable.
So let *{ fn* : n > 1} be a dense subset of *{Bx*)W*)* and define

T : X ^ c o ( i V ) b y r ( x ) = **.**
*/ n = l*

*T* is clearly a linear m ap, and it is one-to-one because *{fn)^=i* is dense in

*C hapter 3: Spaces with a countable cover by sets o f sm all local diam eter* **54**

Let *A =* be a discrete family in *X* and let be an open
base for th e topology on *X .* For each 5 **G *?, **pick **a, G ***Ag.* T hen th ere exists

*Ug^* open neighbourhood of such th a t *U g r\A t —* 0, for *t ^ s.* Now for each
5 **G **5 th ere exists **G ** AT such th a t **Ug**** G ****Bn^ C ***Ug.* Hence |6'| **< ** |JV| and
therefore *S* m ust be countable.

So let *A =* {An}neiv^ be a discrete family and define

£(n) ^ and g W = 0, for n 5^ m

Then for every *m & N* th e fam ily is discrete and
oo

*T{A n) =* I J for every *n e N .*

**771 = 1**

So th e fam ily *T A is* d. <7-d.

Let *X* he an uncountable cardinal and X **G p **such th a t *d e n s (X ) =* %.
Suppose th a t th e result is tru e for every *V £ p* w ith *d e n s (Y )* < %. Let *p* be
th e sm allest ordinal w ith cardinality *\p\ =* %.

Let *{Pa : LJo < a < p } he a P R I* on *X .* For any *ol^ Uq < a < p,* we set

*X a* = *{Bct+i — P a){X ).* T hen *X a* **G p **and *dens{X a) <* |o:| < *d e n s{X ).* Thus
there exist sets Fq and one-to-one continuous linear m aps

*Jcc'-Xa* ----^ Co(Fa)

*Chapter 3: Spaces with a countable cover b y sets o f sm all local diam eter* **55**

Since is separable, there is also *Jq* : *Puq[ X )* *— > Cq[N )* sharing
these properties. Set

*v = N c* U

and define *T* : *X* — > ^oo(r) by

*T {x ){n )* = Jo (fL o (^ ))W for *n ^ N ,* and

T (x )(7 ) = ij« (P « + i(a j) - fL (æ ))(7 ) for 7 E

T is clearly a linear m ap and is continuous. We have

||r x || = sup ||ij„ (P « + i(œ ) - <

*UQ<OL<H ^*

*< \* sup ||J a ||( ||P a + l(a ;)- P a (a j)||) < ^ SUp ( ||Pa+i || || CC || + || P^ || || æ || ) <
^ *UQ<ot<^x * *^ wo<a<fi*

< ^ sup (||æ|| + ||æ||) = ||æ||.

We show th a t *T { X )* C co(r). By Lem m a 3.2.1, ii), given *x E X* th e set

{||^a(a:)|| : « G [wo, /^)}

belongs to Co([wo,//)), where *Ta = Pa+i —*

Pa-So, given any e > 0, there exist a i , . . . , a „ G [wo, *p)* such th a t

||Tai(a;)|| > *e* and ||7]g(æ)|| < e for /5 0!i,z = 1, . . . , n.

For each *i E* {1, . . . ,n } , J a X ^ X ^ ) ) ^ co(FaJ. Hence th ere exist 71, . . . , 7m, G

*C hapter 3: Spaces w ith a countable cover b y sets o f sm all local diam eter* **56**

Now we prove th a t *T* is an injection. Take *x Ç: X* and suppose th a t
T(æ) = 0. We show th a t *P^{x) =* 0 for *loq < ( < p.*

For ( = Wo, we have *Jo{P^^q{x)) =* 0 and, since Jq is an injection, we

conclude th a t *P^oi^)* = 0.

Fix *ljq < ( < p,* and suppose th a t = 0 for Wo < a < (. If ( is a

non-lim it ordinal, then ^ = a + 1 for some *a < * *(.* Now

*- Pcc{x)) =* 0 = > *P^{x) =* 0,

because *Pa{x) =* 0 and is an injection.
If ^ is a lim it ordinal, we have

P^(x) = lim Pot(a3) = 0.

For *C — Pi* since *p is a* lim it ordinal, we have

*P^i{x) = X =* lim P a (x ) = 0,

hence *x = 0* and so *T* is an injection.

Now we introduce some notation. Set *Ya* = *Pa{X)* for *ujq < a < p* and

*Zcx = {y ^* co(F) *:yt = 0{oT t e* | J F J .

*a<C<H*

Define *ha* : co(F) *— > Za* C co(F) by

0 for t G U{F( *: a < ^ < p},*
*K { y )* = *y'* w ith *y[ = <*