O N CONTINUOUS IMAGES OF

R A D O N - N I K O D ~ COMPACT SPACES

MIHAELA IANCU

A dissertation submitted to the Faculty of Graduate Studies

in partid fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

Graduate Programme in Mathematics and Statistics

York University

Toronto, Ontario

June, 2001

National Library Bibliothèque nationak du Canada

Acquisitions m d Acquisitions et Bibliographie Services services bibliographiques

395 Wellington Street 395, rue Wellington Ottawa ON K I A ON4 Ortawa ON K1A ON4 Canada Canada

The author has granted a non- L'auteur a accordé une licence non exclusive licence dowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or seU reproduire, prêter, distribuer ou copies of this thesis in rnicrofom, vendre des copies de cette thèse sous paper or electronic formats. la fome de microfiche/nlm, de

reproduction sur papier ou sur f o m t électronique.

The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduit. sans son permission. autorisation.

On continuous images of RH compact spaces

by Mihaela Iancu

a dissertation submitted to the Faculty of Graduate Studies of York Universiq in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPWY

Permission h a s been granted to the LIBRARY OF YORK UNIVERSITY to lend or sel1 copies of this dissertation. to the NATIONAL LIBRARY OF CANADA to microfilm this dissertation and to lend or seU copies of the film, and to UNIVERSITY MICROFILMS to publish an abstract of this dissertation. n i e author reserves other pubLication rights. and neither the dissertation nor extensive extracts from it may be p ~ t e d or otherwise reproduced without the author's written permission.

Abstract

Applications to RN compactness through the metric characterization are introduced.

We show a necessary and sufficient condition for the invariance of the RN compact

spaces under continuous mappings. We observe a simple proof for this invariance in

the case of O-dimensional images. We apply the characterization theorem to condi-

tions of O-dimensionality and of rnetrizability of the closure of the set of nontrivial

fibers.

The K > O property is introduced. The RNK structure joins this property with

fragmentation- We present invariances and characterizations of RNK spaces, which

describe them as an intemediate class of the fragmentable and the RN compacts.

We show that RNK and RN compactness are equivalent properties in the realm of

O-dimensional compacts and so that RNK is a proper subclass of the fragmentable

compacts. We discuss extendibility of separating lower semicontinuous functions

and closing the RNK structure to F, operations.

Using extensions of nonexpansive functions, we show that compacts non-RN com-

pacts are not minimal in the sense that any compact non-RN compact space has

proper closed subspaces which are non-FtN compact.

Acknowledgement

I wovld like to take the opportunity to thank Professor Stephen Watson for the

support that he has oflered me throughout rny years of study ut York University.

Contents

Chapter 1. Introduction

1. Introduction

2. Preliminaries

Chapter 2. RN Compactness Through the Metric Characterization

1. Fragmentation, Lower Semicontinuity, Evaiuations

Along Finite Paths 12

2. Applications to RN Compactness 31

Chapter 3. The K > O Property

1. The I< > O Property, Invariances and Characterizations

of RNK Spaces

2. Other Properties Relating to RNK Spaces

Chapter 4. RN Compactness and Extendibility of Nonexpansive Functions 75

1. Nonexpansive Functions and Nonexpansive Extensions 75

2. RN Compactness through Functions of Variation under Almost

Neighbourhoods of the Diagonal

Bibliography

CHAPTER 1

Introduction

1. Introduction

The class of Radon-Nikodtjm (RN) compact spaces emerged from studies in

Functional Analysis since 20 years ago. In observing its historical context, we

first present the Radon-Nikod~ property.

Following the approach from (141, let X be a Banach space and (fi,F, p) a finite

measure space.

A mapping F : F + X is called an X-valued measure if for each painvise disjoint

sequence {An)nEN in 3 we have

F ( (J An) = C W*),

where the right-hand side of the equality is a series convergent ivith respect to the

norm topology on X.

I f SUP { CiQG II F(Ai ) II : Ai E F, Ai n Aj = I f OT i # j , n E N) is finite, we say

that F is of bounded variation.

F is absolutely continuous with respect to p if A E F and p(A) = O irnplies F(A) = 0.

For a measure space (Q, 3, p) and N : 0 -t X finitely valued mapping such that

- H ( s ) = x i # O for S E A, E F a n d i E 1,nand H ( s ) = O fors E O\UiEGAi we

define the pintegral of H by

A mapping G : Q -t X is Bochner integrable if there is a sequence {Hn)nEN of

finitely valued mappings which converges to G p-almost everywhere in such a way that

For A E F the Bochner integral of G over A is defined as

whcre X A is the characteristic function of A.

A Banach space X has the Radon-Nikodjim property (RNP) if for every ($1,3, p)

finite measure space and F : 3 -+ X an X-valued measure of bounded variation

which is absolutely continuous with respect to p, there is a Bochner integrable

function G : R -+ X such that F ( A ) = SA G d p for any A E 3.

A Radon-Nikod@ (RN) compact space was onginally defined as a space homeo-

morphic to a compact subset of (B', weak*), where B* is a dual Banach space with

the RNP.

The RN compacts were also introduced by Reynov in [17] under the name of

compacts of RN-type, which were there defined as Hausdorff spaces homeomorphic

to RN sets in dual Banach spaces. For K weak*-compact subset of a dual Banach

space, K is an RN-set if it has the property that each finite Radon measure on

(K, weak*) is supported on a nom-O-compact subset of K.

A Banach space is called Asplund if every separable Banach subspace of i t has norm-

separable dual.

An Eberlein compact space is defined as a space horneomorphic to a weakly compact

subset of a Banach space.

Namioka and Phelps investigated Asplund spaces and observed properties shared by

weak8-compact subsets of duals of Asplund spaces and Eberlein compacts in [15].

I t turned out that the class of RN compacts is also characterized as the class of com-

pact spaces homeornorphic with weak*-compact subsets of duals of Asplund spaces.

Later in [13] Namioka develops a self-contained extensive treatment for RN compact

spaces, where they are characterized as spaces horneomorphic to nom-fragmented

weak*-compact subsets of dual Banach spaces. Further on, one of the characteriza-

tions of RN compactness becomes the property of a compact space to admit a lower

semicontinuous fragmenting metric.

The notion of fragrnentability was explicitly defined by Jayne and Rogers in [Il]

and its application to RN compactness became an important tool in the study of

these spaces by freeing them of the Banach space context.

The question which remains open is whether RN compactness is preserved by

continuous images. In studying the continuous images of RN compacts, Reznichenko

defined according to [Il strong fragmentability, Fabian, Heisler and Matouskovg

defined in [8] countably lower fragmentability and Arvanitakis defined in (21 quasi

RN compactness. The relationship between these notions and what we cal1 in the

present work RNK cornpactness shall be described in work in preparation.

The present work uses the concept of fragrnentability as a property of R N compacts

and of their continuous images. We emphasize structural properties and develop

tools which we then use to clarify and simplify the problem and to extend existing

results.

2. Preliminaries

Let us now set u p the work environment by reviewing some definitions and results

related t o Our subject. We assume tha t Our working spaces have Hausdorff separation.

The unit interval is denoted by I.

DEFINITION 1.2.1. Let h : X2 + [O, CO) be a function on the square of a space X .

For A nonempty subset of X we define h-diam(A) = sup{h(a, a') : a, a' E A ) .

For Al, A2 subsets of X we define

i n f { h ( a c , a 2 ) : a l ~ A ~ , a î E & ) if Ai,&#@ h-dist(A1, AL') -

ot herwise.

A function h : X2 -t [ O , oo) is fragmenting if it has the property that for any A

(closed) nonernpty subset of X and for any E positive there is an open set O such

that A ri O # 0 and h-diam(A n 0) 5 E .

A space X is fragmentable if there exists a metric p : X2 + [O, m) fragrnenting X.

DEFINITION 1.2.2. For -41 and A2 subsets of a space X, a function h : X2 -+ [O, 00)

is separating Ai and A2 if h-dist(Al, A2) > O .

The function h is separating if h-'({O}) = A,. 5

A family of functions H = {hi : X2 -t [O, w))iEZ is separating if nisz h;' ({O)) = A,.

DEFINITION 1.2.3. A well ordered family U = {U,)F<6 of subsets of a space X is a

relatively open partitioning of X if the following conditions hold:

(i) UO = 0;

(ii) WC C X \ UV,€ Un and i t is relatively open in it for every O < E < Co ;

(iii) X = Ue<c0 Ue .

DEFINITION 1.2.4. Given U = {Ut}(<Fo relatively open partitioning of a space

x, by defining Wu = {Wc}o<F<Fo with kv' = Uncc Uq for every O < < 5 5,

we obtain what is called a regularly increasing family of open sets of X .

For 24' and u2 relatively open partitionings of X, U2 iS a refinement of U1 if the

regular increasing family WU, is contained in the regular increasing family Wu,.

A separating O-(refined) relatively open partitioning of a space X is a countable

collection {Un),,N of relatively open partitionings of X such that ( Un+' is a refine-

ment of Un for every n E W a n d ) for every two distinct points of X there is a n E N

such that they are separated in the partitioning Un.

NOTATION 1.2.5. For a function g : Y2 + [0,m) and no E N we dcnote by gAno

and by g A the function and respectively the pseudometric defined by

G

DEFINITION 1.2.6. For A, B subsets of x2 we denote by

A o B = { ( a 7 b ) ~ X 2 : t h e r e i S c ~ X s u c h t h a t ( a , c ) E A , ( c , ~ ) E B ) .

We denote the n-fold composition by A"", where A'' = A and A"("+~) = Aon O A.

A subset E of X2 is idempotent if Eo2 = E and is an almost neighbourhood of A,

if for any nonempty S C X there is O open in X such that S n O # 0 and

S2 n O2 c E.

For a space X we let U, be the family of al1 open symmetric neighbourhoods

of A, in X 2 .

We shall rnake use of following theorem from the theory of uniform spaces ( conse-

quence of 8.1.10 in [6] ) :

THEOREM 1.2.7. Let V = {V,),,, be a sequence of members of Ilx with V, = X2

and VI y for i E N. We let g, : x2 + I be the function

The property written as Lemma 6.5 in [13] will be applied:

LEMMA 1.2.8. Let X be a compact space and C , D closed subsets O/ x2 such that

Don C C . For U, open neighbourhood of C in X 2 there 2s an open neighbourhood U,

of D such that UT Ç U,. I f D is symmetric, then U, can be chosen to be symrnetric.

From the work of Ribarska [18] and Narnioka [13], we emphasize the following

characterizations of fragmentability and of RN compactness that we shall use:

THEOREM 1.2.9. For a topological space X the following are equiualent:

(i) X is fragmentable;

(ii) X admzts a separating O-(refined) relatively open partitioning;

(iii) X admits a jragmenting separating junct ion;

(iv) There is a sequence {AnInEN of almost neighbourhoods of 4, such that

An = A, *

THEOREM 1.2.10. For a compact space X the following are equivalent:

(a) X is RN compact;

8

(6) X admits a lower semicontinuous fragmenting metrie;

(c) There is a sequence {Cn),En of closed almost neighbourhoods of A, such that

nnrN C, = A, and C;:l C C, for n E N .

In the process of proving the latter equivalences, a direct proof for (c) +(b) is in-

ferred in 6.6 of [13] and we shall relate to it in what will follow through LEMMA 1.2.12.

DEFINITION 1.2.11. Let 2) = {D,),,, be a sequence of closed symmetric al-

most neighbourhoods of the diagonal A, of a space X such that Do = X2 and

02, ç Dn for n E w. A sequence V = {V,),,, of elements in U, is called adapted

to D if D, C Vn and Vnf, V , for n E W .

LEMMA 1.2.12. Let V = {D,),,, be a sequence of closed symmetric almost neigh-

bourhoods of the diagonal A, of a compact space X with the properties that

Do = X 2 , D:''I D, for n E w and nnE, Dn = A, .

W e denote by { V ( S ) ) ~ ~ ~ the family of al1 the sequences of elements in Li, which

are adupted to 2). For each s E S we let p,,,, be the continous pseudometric o n X

defined as in THEOREM 1.2.7.

The metric : X 2 -+ I defined b y pzP = sup { p,,., : s E S } ZS lower semicon-

tinuous fragmenting on X .

PROOF. The definition ensures that pFP is a lower sernicontinuous pseudometric.

separates al1 pairs of distinct points in X:

As the intersection of the closed sets D,, is the diagonal of X, for x and y distinct

points in X there is a no E N such that (x, y ) $ DnO.

We construct a sequence V = {V,},,, adapted to V such that (z, y) and (y, x) are

eliminated from the elements of the sequence at least from the index no:

V, := X2 and assume that we have constructed y E U, for i < n < no such that

Di C y fo r i < n and y*: C y for i < n - 1.

As 0" 3 DDl C Vn-i , there exists by LEMMA 1.2.8 a set V, E LI, such that

D, C Vn and n3 E Vn-,.

At the stage of defining VnO we follow the same procedure if (x, y) and (y, x) do not

belong to Vno-, . Othenvise we use LEMMA 1.2.8 to find a set WnO E LL, such that

Dno E Wno and W z , VnO-l

Then VnO := Wno \ ((2, y), (y,x)) E 11,. As (x, y) and (y,x) are not in Dno we have

Dno YO and as Ko C Wno, then also

The pseudometric p, will have p, (x , y) 2 & and so prP(x , y) distinguishes the

points x and y.

To show that p r P is fragmenting we first remark that each of the pseudornetrics

1 p, constructed as in THEOREM 1.2.7 will have the property that pv(x, y) 5 , for

( x , y) E V, and n E W .

As the pseudometrics pv which define pYP are adapted to V, we also have that

pv (x. y) 5 & for (2, y) E 0, and n E w and therfore prP(x, y) 5 $ for (x, y) E D,

and n E W .

For C nonempty subset of X and E > O we choose an no E W such that & 5 E.

D,,, is an almost neighbourhood of A,, so there is O open subset of X such that

O n C # 0 and O2 n C2 C Dno. Therefore / ip-d iam(0 n C) 5 1 sn0 - < E.

O

Orihuela, Schachermayer and Valdivia proved in [19] that Talagrand's example is

a compact fragmentable space which is not RN compact.

We outline the construction of this space, to which we shall refer.

EXAMPLE 1.2.13. TALAGRAND'S EXAMPLE

For each n E w we define a family Un of subsets of PIN as follows:

Uo = { { a ) : o E @) U 0 and for n E W

Un = { A fl : for a # IL E A we have a,, = pl, and q,+l #

Each Un is an adequate family of sets and UnEw Un is also such a family. For an

adequate family of subsets of a set S we have the property that the characteristic

functionç of their subset-elements are compact subsets of {O, 1IS.

For a family V of subsets of fl we let K(V) be the compact subset of {O, 1lN"

of the characteristic functions of the set-elements of V, K ( V ) = { X, : V E V ).

Tlien K = Un,, K(U,) is the compact space Talagrand's example.

CHAPTER 2

RN Compactness Through the Metric Characterizat ion

The first section motivates properties relating to fragmentation, lower semiconti-

nuity and evaluations along finite paths that we apply throughout the present work.

Applications to RN compactness through the metric characterization are introduced

in the second section of the chapter:

We show a necessary and sufficient condition for the invariance of the RN compact

spaces under continuous mappings.

We observe a simple proof for this invariance in the case of O-dimensional images.

We apply our characterization theorem to conditions of O-dimensionality and of

metrizability of the closure of the set of nontrivial fibers.

O ther applications and examples are presented.

1. Fragmentation, Lower Semicontinuity, Evaluations

Along Finite Paths

LEMMA 2.1.1. a) Let u : X 2 + [O, w) be a function defined on the square of a space

X a n d v positive value.

The function v-u zs fragrnenting i f l IL is fragmenting and i t is lower sernicontinuow

12

i f l u is lower semicontinuous.

b) Consider the functions ul, uz : X2 -+ [O, w).

If ul and u2 are lower semicontinuous, then so are ntax (ul, u2), min (ut, uz), ul - uz

and u1 + u2.

If ul and u2 are fragmenting, then max (u1,u2), min (ul, u2), ul u2 and u1+ uz am

fragm enting.

c) If { u ~ ) ~ ~ ~ is a pointwise bounded above family of lower semicontinuous functions

defined o n a space X , then the function S U ~ { U ~ : i E 1) i s lower semicontinuous.

Given two functions u, ut : x2 + [0,m) such that ut < u and u Zs fragmenting,

then so is ut-

I n the class of functions from x2 to I lower semicontinuity is closed under supremum

and fragmentation i s closed downwards.

d) Let be a countable family of functions defined from the pairs of points

of a space X to I .

If u, i s fragmenting for n E N, then &un i s a fragmenting function.

1 If u, i s lower sernicontinuow for n E IV, then CREN 28Un i s lower semicontinuous.

If there i s a countable separating farnily of lower semicontinuous fragmenting

pseudornetrics on X , then there is a lower sernicontinuous fragmenting metric o n X .

PROOF. a) The announced properties are immediate from the definition of frag-

mentation and lower semicontinuity.

b) Lower semicontinuity is a well known inherited property for al - u2, ul + uz,

max (ui , u2) and min (ui , u2)

F'ragmentation is also easy to obseme :

If C is a subset of X and E a positive value, we let 6 := min($, JE).

Then, as ui is fragmenting, there is O1 open in X such that C n O1 # 0 and

ul-diam(C n O,) 5 6. As uz is fragmenting, there is O2 open in X such that

(C n 01) n O, # 0 and u2-diam((C n Oi) n 02) 5 6.

Then C n (O1 n 02) # 0 and its diameter measured in terms of ui - u2, ul + u2, max (ul, u2) or min (ui, un) is smaller than or equal to E .

c ) I t is known that lower semicontinuity is closed under taking supremum and the

property that u' is fragmenting is immediate from the definition.

d) Lower semicontinuity and fragmentation are preserved by multiplication by a

nonnegative value, so each &un is lower semicontinuous fragmenting.

Looking at an infinite sum as a limit (supremum) of partial sums, we can complete

by c) the argument that c , ~ ~ & u , is lower semicontinuous when the functions un

are lower semicontinuous.

Assume that the functions un are fragmenting for n f N and let C be a subset

of X, E a positive value. We choose n, E PI such that & < E .

14

As ul is fragmenting, there is 0 1 open in X with the properties that Ol n C # 0

and ul-diam(Ol f~ C) 5 &.

For 0 1 n C use the property of u2 of being fragmenting to find O2 open in X such

that O2 n C l l n C # 0 and u2-diam(& n O1 n C) 5

Continuing recursively, for On, n ... n O1 n C we apply the property of une+l of being

fragmenting to find open in X such that n On, n ... n 0 1 fl C # 0 and

1 ~ , ,+~-d iam(O~,+~ n O,, n ... n Oi n C ) 5 m.

Letting now O = O,, +l n On, n ... n 0 1 be the open set which nontrîvially intersects

1 1 1 C, we find that Zn,, &'&-diam(OnC) I C1<i<nc+l Z~;TT 5 + 5 I &-

Let p, : X2 + [O, cm) be lower semicontinuous fragmenting pseudometrics such that

the farnily {pn InEN is separating. Then {p:},,~ given by dn := mi+,, 1) is a

separating family of lower semicontinuous fragmenting pseudometrics with values

in I and so CnEN & - pk is a lower semicontinuous fragmenting metric on X.

O

LEMMA 2.1.2. a) Consider the functions h, : X2 + [O, m), hy : Y2 + [O,CO), a, b

positive values and the function h : ( X x + [O, w) giuen by

If h, and h, are fmgmenting, then h is fragmenting.

I ' h , and h, are lower semicontinuous, then h is lower sernicontinuous.

b) Let h, : X n -+ 1 be funetions d e h e d on squares of spaces X,, for n E N.

For Z := nnEN Xn let q : + I be given by

If the famdy {hn}nEN C O M ~ ~ ~ of fragmenting f~nctions, then q is fragmenting.

If it consists of lower sernicontinuous funetions, then q is lower semicontinuous.

PROOF. a) Assume that h, and h, are fragmenting and let S be a nonernpty subset

of X x Y and E a positive value.

K, (S) is nonempty and there is O' open subset of X such that O' n T, (S) # 0 and

h,-diarn(Ot n rr, (S)) 5 &.

Further (O' x Y) ris is nonernpty, so for T, ((O' x Y) nS) there is O" open subset of Y

such that O"nr,((O'xY)nS) # 0 and that h , -d iam(O"~?r , ( (O 'xY)~S) ) ) 5 &.

Then (O' x O") n S # 0 and for (xl, y,), (x2, y*) E (O' x 0") n S i t is easy to see

t h a t if (x, y) E (O' x O") n S then (x, y) E (O'n n,(S)) x (O" nny ((0' x Y) n S))

and therefore h-diam((0' x O") n S) 5 E .

If h, and h, are lower semicontinuous functions , let h((xl, yl), (x2, y2)) = v > E

for (XI, yl), (x2, y2) E X x Y and E > 0.

By the lower semicontinuity of a- h,, we can find 01 and Oz open sets around xl and

respectively x2 sucli that a - h, ( O , , 02) > a h, (xi, x2) - Q for O , E OI and o2 E 02.

By the lowcr semicontinuity of b- h,, we can find Ul and LIÎ open sets around y1 and

16

respectively y2 such that b - h, (u l , uz) > b- h, ( y 1 , y2) - y for ul E Ul and u2 E U2.

Then for the points (ol, ul) in the neigbbourhood Ol x Ui of ( x l , y,) and (oz, 2 ~ 2 ) in

the neighbourhood 0 1 x LI2 of (z2, y2) we have h((o1, u l ) , (e, ~ 2 ) ) > E.

b) We define for n E N the spaces Zn = HiETiJi Xi and the partial functions

qn : 2: -t 1 given by qn (x, Y ) = CiEG $ hi (ri (z) ri (Y)) for z, Y E Zn-

We first use a) inductively to argue that the q,,-s are fragrnenting whenever {k)nE~

is a family of fragmenting functions and that the q,-s are lower semicontinuous

whenever {/L,),,~ is a family of tower semicontinuous functions.

Assume that {LInEN is a family of fragmenting functions.

For S # 0 subset of Z and m E N we can show that there is on open subset of Z which

nontrivially intersects S and leaves a trace on S of q-diarneter not greater than &: As Z,+1 is fragmented by q,+~, we can find O open subset of Zm+l such that

1 0 n r, m + l (S) # 0 and qm+l-diam(O n n,_+, ( S ) ) I m-

Then O I'Ii2m+2 Xi is an open subset of Z with (O x Xi) ri S # 0 and

1 1 q-d'am((Ox n i > , + 2 X i ) n S ) 5 q ~ + ~ - d ~ a m ( O n ~ ~ _ + , (S)) + I -

If {hn}nEn is a family of lower semicontinuous functions, let x, y E 2, and E positive

such that q(x, y) r E .

f There is no E N such that CiCG F - hi (xi(x), ri (Y)) > E.

As q,, is lower semicontinuous and qno(rzno (x), ?rZa0 (y)) > E , there are QI and O2

open sets around nZnO (x) and respectively n,, (y) in Zn, such that (ol, 02) > E

for any 01 E O1, 02 f 02.

Then 01 x ni,., Xi and O2 x ni,*, Xi are open sets around x, respectively y and for

a E O i x n i > n o X i a n d b E 02Xni>noXiWehaveq(a,b) L ~ , ~ ~ ( ~ ~ ~ ( a ) , ~ ~ ~ ~ ( b ) ) > E*

We remark that this lemma offers an alternative proof for the fact that countable

products of fragmentable spaces are fragmentable and that countable products of

RN compact spaces are RN compacts.

PROPOSITION 2.1.3. a) Any bounded below lower semicontinuow function on a n

open subset of a space extends lower semicontinuously to the whole space.

b ) If {uOi : Oi + R}isz is a pointwise bounded above family of lower sernicontznuous

functions defined on open subsets of a space, then the function u : UiE,Oi + R

given by u(x) = sup{uOi (x) : x E Oi) is lower semicontinuous.

c) Any lower semicontinuous fuvntion us : S -+ R defined on a subset of a space X -

admits a lower sernicontinuow extension, u, : S + W to the closure of S.

Further on, i f U S is Oounded above, then it adrnits a lower semicontinuous extension

18

to the whole space, i&, by mapping i ( x ) to a fized upper b o n d of 5 for x E X \S.

PROOF. a) Let u : O -+ [vo, CQ) be a lower semicontinuous function defined on O

open subset of X. We can extend it to u, lower semicontinuous function on X by

mapping al1 the points outside O to vo.

Then u, is lower sernicontinuous, as

b) If u ( x ) > v, then there is io E Z containing x such that uo, (x) > W. By the

lower semicontinuity of uoio there is U open subset of uoio(y) > v for any y E U.

Then u is defined for y E U and u ( y ) > v for y E U, open subset of UiEIOi.

c) Let ug : 3 + R be the function given by

u&) := sup {in/ {IL&) : s E Oz n S) : O, open neighbourhood of z).

Wc observe that u, is a lower

Ifu,(x) > r f o r z € S a n d r €

x such that in f { u,(s) : s E

open set in 3.

semicontinuous function:

IR, then by the definition there is O,, open set around

O, n S } > r and so u&) > r for any s E O, n 3,

To show that u, is also extending us let so be a point in S.

By the lower semicontinuity of us, for any n E N there is O) open set around so O

such that i n . { u,(s) : s E 0: n S ) 2 u, (so) - and therefore TL&,) 2 us (so).

As s, E O n S for any O open set around xoy we also have g ( s o ) 5 u,(so) and so

u&o) = u,(so) -

Extending u, to the whole space by mapping the points outside its closed domain

to a fixed upper bound of its range guarantees the extension % to remain lower

sernicontinuous.

PROPOSITION 2.1.4. a) Let K be a closed subset of a space X and let u, : K + B

and u, : X + P be lower semicontinuous functions such that u, 5 uxlK .

Then the function u : X + W defined by

is lower semicontinuous.

b) Let { K y U} be a partition of a space X into a closed and respectivety open set.

If u' : K2 -+ [O, CO) is a separating fragmentïng function and u" : U 2 + [O, 00) is a

separating fragmenting function, then the function u : X 2 -t [O, m) defined b y

20

PROOF. a) For u E P we have u-l ((-00, v]) = u;' ((-00, v ] ) U (uil((-cm, v ] ) \ K).

As u, 5 uX l K , then il;' ((-a, v]) n K C u;' ((-m, v ] ) and so u-' ((-m, v]) is

a closed subset of X.

b) Let C be nonernpty subset of X and E positive.

If U n C # 0, then there is O open set such that O n (U n C) # 0 and

u n - d i a m ( 0 n (LI n C ) ) = u - d i a n ( ( 0 n U) n C) 5 E.

If C E K , then there is O open set such that O n C # 0 and u r - d i a m ( 0 f~ C ) 5 E.

Then O LJ U is a n open set in X and u l - d i a m ( 0 n C) = u - d i a m ( ( 0 u LI) n C) 5 E.

The definition of u makes i t a separating function whenever u' and u" are separating.

O

PROPOSITION 2.1.5. a) If a space X admits a n increasing cover by a family of open

sets with UQ \ UBCa UB fragmentable for cr E <, then X is fragmentable.

6 ) Fragmentability i s Fu-closed: If a space X admits an increasing cover 6y a

sequence of closed sets, {Fn}nEw wzth Fo = 0 such that F, \ Fn-1 zs fragmentable for

21

n E N, then X is fragrnentable-

PROOF. a) Consider u, : (Ua\U,<- Ug)2 + [O, oo) for a E E separating fragmenting

functions and define the function u : X2 + [ O , CO) by

u is clearly separating and for fragmentation let C be a subset of X and E positive.

Choose a, E to be the first ordinal for which (U,, \ U UB) n C # 0. B<oc

For (U,, \ UBCPC LIo) " C there is relatively open subset O of (U,, \ (J UB) such B<ac

that (Uac \ UB<PC uB) " C " O # 0 and u,,-diam((U,, \ UBcae UB) " C n 0) 5 E.

Then O u U LID is an open subset of X such that (O U (J U,) n C # 0 and B<ac B <oc

u-diam((O U U B<OC LiB) n C) = na,-diam((U,, \ UBCac UB) n C n O) 5 E.

b) By b) of PROPOSITION 2.1.4 each Fn is fragrnentable, so the statement from this

part is indeed equivalent to having fragmentability Fm-closed.

We can assume without loss of generality that the functions that witness frag-

mentability are bounded above by 1, so let un : (F, \ Fn-1)2 t I be fragmenting

separating functions for n E N . We define u : X2 + 1 by

It is clear that u is separating and let us observe that i t is also fragmenting:

For C C X and E > O , if C C Fno for some no E W, let us assume that we named no

the last value such that C n (Fw \ FnOdl) # 0.

B y the fragmentability of Fm \ F,-l we can find O open set such that

O n C ri (Fno \ Fno-1) # 0 and u,-diam(0 n C i l (Fno \ Fn,-l)) 5 E.

Shen the open set O \ Fn0-* has nonempty intersection with C and

u - d i a m ( ( 0 \ Fno-l) n C ) = $ - h o - d i a m ( 0 n C n (Fno \ Fno-t)) 5 E.

If there is no last n E N such that C n (F, \ F,-l) # 0, then we let no E FI1 be

such that $ 5 E. The open set X \ F,,-l has nonempty intersection with C and

1 U-d iam((X \ Fno-1) n C ) 5 U-diam(X \ Fno-1) 5 ;, 5 E.

PROPOSITION 2.1.6. a) Let f : X + Y be a mapping from a space X ont0 Y .

Given a function u : Y 2 + [O, M ) , we define the function h : X2 -t [O, m) b y

h(x,x') := u ( f (x), f(x')) for (z,x') E X 2 .

If f i s continuous and u is lower semicontinuous, then h is lower semicontinuozls.

I f f is a closed mapping and u i s fragmenting, then h is a fragmenting funclion.

b) Let f : X -t Y be a mapping from a space X onlo Y .

Given a function u : X 2 + [O, cm), we define h : Y2 + [O, m) by

h(9,p') := il-dist( f -' (y), f - ' ( y f ) ) for ( y , y') E Y2. 23

If f i s a perfect mapping and u is fragmenting, then la is also fragmenting.

If f is perfect and u iS iower semicontinuous, then h is lower semicontinuous.

If f is continuous, X 2 is eountably compact and u i.s lower semicontinuous

separating, then h separates ail the pairs of disjoint closed subsets of Y and in

particular u itselj separates all the pairs of disjoint closed subsets of X .

PROOF. a) The lower semicontinuity of h follows from the fact that it is the com-

position of a lower sernicontinuous function with a continuous function, h = u O f2.

Assume now that u is fragmenting and f is closed.

Let C be a closed nonempty subset of X and E positive.

Then f (C) is nonempty closed subset of Y, so there is O open in Y such that

O n f (C) # 0 and u-daam(0 f~ f (C)) 5 E.

Then f - ' (O) i s o p e n i n x and f - ' (O)nC#@. As f ( f - ' ( 0 )nC)COn f(C),by

the definition of h we conclude that h-diam( f -'(O) n C ) 5 E .

b) Assuming that f is perfect mapping and u is fragmenting, we show that h is

fragmenting on Y:

Let C be a closed subset of Y and E > 0.

The mapping f 1 (cl : f -' (C) -+ C is defined on f -' (C) closed subset of X and so

the fibers f -' (c ) remain compact for c E C. 24

As f rl-~(C) is continuous with compact fibers, by 3.1.C.a of [6] there is a restriction

of frf - L ( ~ ) to a irreducible mapping ont0 C, fr, : D + C, where D is a closed subset

of f- '(C).

As u is fragmenting, there is U nonempty relatively open subset of D such that

u-diam(U) 5 r .

By the irreducibility of f ,,, f (D\U) # C and so O := C\ f (D\U) is nonempty.

O is relatively open in C as f is closed and also f i1 (O) 2 LI.

Then h - d i a m ( 0 ) = sup {u( f -' ( y ) , f - ' (y t ) ) : y , y' E O )

5 sup { 4 f ~ ' ( y ) , f ~ ' ( y t ) ) : Y,Y' E O ) 5 u-diam(U) 5 E-

Assuming that f is perfect and u is lower semicontinuous, we show that h is lower

semicontinuous:

Let y, y' E Y be such that h ( y , y') = v > E > 0.

Then u ( x , x') > for any x E f Z1 E f

By the Iower semicontinuity of p,, for x , x' fixed we can find O,, Oi neighbourhoods

of x, x' such that u(a, b) > y for any a E O,, b E O+

Using the compactness of f -' (y), we can find open sets U containing f -' ( y ) and

Uzl containing x' such that u(a, b) > 2 for any a E U, b E Cli and then, using

again the compactness of f - ' (y ' ) , we find open sets V containing f - ' ( y ) and V'

containing f-'(y') such that u(a, b) > 2 for any a E V, I> E V'.

As f is a closed mapping, we can find W and LV' open sets around y, rcspectively

y' with the property tha t fh l (W) C V and f-'(W') C V'.

25

Then u(w,w1) 2 2 > E for any w E W, W'E W'.

Assuming that f is continuous, X2 is countably compact and u is lower semicon-

tinuous separating, we show that h has the property that for any two closed subsets

Kt and K" of Y we have h-dist(K1, Kt') = O implies Kt n Ku # 0 :

We can find sequences {kk)nEN and {kn),,N of points from Kt and respectively K"

such that h(kk, ki) 5 -&- This implies that for n E W we have that f-'(k;) and

f -'(k:) contain points x; and respectively x: with u(xk, x:) 5 1. We denote by A the set {(xl,,~;) : n E W}. If IA l< No, then there is no E N

such that u(xn,, x;,) is arbitrarily small, hence u(xl,,, XE,) = O. As u is separating,

xno = .no and then f -' (Kt) and f -'(Ku) intersect and so do Kt and K".

If A is countably infinite, then by countable compactness it has a complete accumu-

lation point, (z', xt') E A \ {(xt, x")} C f -'(Kt) x f - ' (KM).

If u(xt, x") > $ for some no E N, then by the lower semicontinuity of u there is O

open set around (xf, x") such that u(a, b) > -& for any (a, b) E O. As only finitely

many of the pairs of points of A can be a t u-distance grcater than &, we conclude

that u(xt, xf') cannot be positive.

As u is separating, u(xt, x") = O implies that x' = x" and so that K t and K" intersect.

For the last part of the proposition it suffices to consider f the identity of X.

O

PROPOSITION 2.1.7. a) Consider a partition {A, B ) of a space Y and a function

g : Y2 -+ [O, 03). Fixing Si, S2, 53 E { A , B), we cal1 ( s l , s ~ , s3) E SI x S2 x S3 a

specific triple associated to the partition {A, B}. Let a and b denote a point from A

and respectiuely from B.

If gA = gAno for some no E W by global restrictions imposed through reducing

paths on specific triples associated to the partition {A, B}, then the reductions must

contain one of the folloun'ng three sets of triples:

i- {(a,a ,a) , (b, b, b)7(b,a, b), (b ,a ,a ) } or { ( b , b, b) , (a ,a ,a ) , (a , 44, (a, b, b));

a- {(a,a ,a) , (b , b, b)7 ( h a , b), (a, b,a)}*

b) Let {AnlnEN be a countable partition of a space Y. For r E Y we let n, E W such

that z E A,, . If g : Y2 -+ [O, 00) is a function with the property that

then gA(x, y ) = gAnz+nu-2 ( x , Y ) f o r x , Y ~ Y -

In particular, i f {Ak}I<C<R - - is a finite partition of Y and g : Y 2 + I is a function

with the property (A), then gA = gA2(n-l) .

c) Let Y be a space and g : Y 2 + [O, 00) lower ~ e r n i ~ ~ n t i n u ~ u ~ function unth the

property that gA = gAno for some no E W.

If YnO is countably compact and g is separating, then gA is a rnetric.

27

If Y is compact, then g A is a lower semiwntznuovs pseudometric.

PROOF. a) It suffices to look at paths of length above no of the following types:

For type (a, a, a, a, a, a, ...) we need to reduce the (a, a, a) triples.

For type (b , b, b, b, b, b, ...) we need to d u c e the (b, b, b) triples.

For t ype (a, b, b, a, b, b, ...) we need to reduce the (a, b, b) or the (b , a, b) tnples.

For type (b , a, a, b, a, a, ...) we need to reduce the (b, a, a) or the (a, b, a) triples.

For type (a, b, a, b, a, b, ...) we need to reduce the (a, b, a) or the (b , a, b) triples.

Combining al1 these conditions gives us either one of the dual sets of reductions from

i., either the reductions from ii.

b) Let x, y E Y with x E A,, and y E A,. for some n,, n, E H

All the paths from x to y which are using points from A, with n > max(n,, n,) are

reducible to shorter paths by finding a subpath given by a triple (a, b, c) such that

maz{n,, n,} 5 nb and therefore g(a, b) + g(b, c) can be reduced to measuring g(a, c) .

So al1 the paths from x to y can be reduced to paths which use intermediate points

only from the A,-s with n < max(nz, n,) .

Without loss of generality, assume that n, = max(n,,n,) and let us follow an

irreducible path ( x, zl, ZZ, ... , zk, y ) which uses distinct intermediate points only

from the A,-s with n 5 n,.

We have n,, # n, by the irreducibility of the subpath ( x, 2 1 , zî ) and so n,, < n,.

Further on we look a t the irreducible subpath ( t l , Z2r ..., zk, y ) , where rnax(n,, , n,) 5

max(n,, n+). We follow the same type of argument to conclude that either n,, =

mux(n,,,nJ and n,, < n,, or n, = maz(n,,,q) and n,, < %.

So we have now a irreducible subpath ( either (z2, 3, ..., zk, y ) or (q, 22, ..., zk ) ) of

which we h o w now that mm(n / irst point of the path t nlast point of the p t h ) < maz(nz 7 5)

For the case when the irreducible path is ( 22, ~ 3 , .-., zk, y ) we know that n, >

n,, > n, and for the case when the irreducible path is ( , z , . z ) we know

that n, > n,, and n,, < n,.

By continuing the process we find tha t for a irreducible path from x to y of the form

( x, zl, ~2~ ..., z k , y ) there is a 1 5 k such that

and so any path from x to y can be reduced to one containing a t most n, + n, - 2

intermediate many points.

In the case of {Ak}i<k<, finite partition of Y, we observe by al1 of the above that al1 - -

the paths betiveen two points can be reduced to paths which use at most 2(n - 1)

intermediate points and so gA(x7 y) = gA2(n-1)(x, y) for al1 x, y E Y .

c) By definition gA is a pseudometric. Let us see that if Y' is countabIy com-

pact and g is lower semicontinuous separating, then the pairs of distinct points not

separated by gA are on the diagonal of Y:

If y2) = 0, then for every n E N there are zr, e, ... z& E Y such that

1 !?(yl, z ï ) + g ( z ï , 2,") + - - - + g(z&~ ~ 2 ) 5 i-

If l { ( z Y , ~ , ... zEo) : n E N} 1 < No there is mo E W such that g(y1,z;"O) + g(z", z;"O) + ... + g(z,O, y2) is arbitrarily small.

Then g(yl,z;"O) = g(z;LO, z,mO) = ... = g(z,OO, y2) = O. As g is separating, we

conclude that y, = z" = z y ... = z z = y2.

If { ( z ï , z,", ... z&) : n E N) is an infinite set, then it has a complete accumulation

point (zl, z2, ... , a s YnO is countably compact. By the lower semicontinuity of

g we then conclude that g(yl, zi) = g(zl, z2) = ... = g(zno, y*) = O and g separating

is used again to conclude that yl = zl = z2 = ... = = y2.

For the Iower semicontinuity of gA when Y is compact and g lower sernicontinuous

we proceed by contradiction:

Let and be nets in Y converging to y1 and respectively y2 and

E positive such that gA(y;, y:) $ E for a E C , but gA(yl, y2) = v > E.

For o E C we can find z r , z;, ... z& in Y such that

By assuming that we have already passed to a convergent subnet, we can consider

that for each i E 1, no we have { Z ~ } , ~ Z converging to some zi E Y.

As g A ( y l , ~ 2 ) = 7 we have in particular g(y,,z~) + g ( z 1 7 ~ 2 ) + -- - + g ( . n O 7 f i ) 2 2).

Using the lower semicontinuity of g, we find open sets O,,, O,,, O,,, ... O,, and

O,, around y, , zl, ~ 2 ~ . - - ~ ~ and respectively y2 such that

Then we have for these points

As { y ; ) r E ~ > { ~ l ~ } r E ~ y -.. {znou)nE~, { Y ~ } U E E converge to y17 r ~ 7 - - - and

respectively y2, the latter relation cornes in contradiction with (2.1.7) and therefore

gA is a lower semicontinuous pseudometnc. 0

2. Applications to RN Compactness

Closures of RN Compactness t o Fu Structures

PROPOSITION 2.2.8. a) If a compact space X admzts a lower semiconlinuous metr ic p

and a countable increasing cover by closed s c b such that PIF,? zs fragmenting

31

for n E N and that the p-diameter of X \ Fn converges to O , then X is RN compact.

b) Let X be a compact space covered by an increaîing union of closed sets,

X = UnEN F, with the property that there is a unique limit point xo f o m e d along

{F,, \ Fn-l)nEN and that there is a no E N S U C ~ that z o E n,,,. F, \ F,-l. -

Also assume that there is a metn'c p on X such that for every n E W we have pl=

lower semiwntinuow fragmenting and that the p-diameter of X \ F, converges tu O.

Under these conditions X is RN compact.

c) Let X be a compact space which is a countable increasing union of closed sets,

X = UnEN Fn with the property that there is a unique limit point xo fonned along

{ F n \ Fn-i}nEN-

Assume that there zs a metn'c p : X 2 -t 1 such that for every n E N we have

lower semicontinuous fragmenting, that the p-diameter of X \ F, converges to

O and that there exists a seqvence of points { c ~ ) ~ ~ ~ fonned along {F, \ c-l}neN

( Say c, E Fn \ F,-I ) such that the p-distance between xo and c, converges to O .

Under tlzese conditions X is RN compact.

PROOF. a) The given metric p is also fragmenting:

For a closed subset C of X for which there is a n E N such tha t C C Fn we find a

relatively open set of small diameter by using the fragmentability of plpz.

For a closed subset C of X for which there is no n E N such that C C Fn we use

32

the fact that the pdiameter of the open sets X \ F, converges to 0: for a positive

E we can find n, E W such that p - diam(X \ Fn,) 5 E and so C n (X \ F,) is a

nonempty set of smail diameter.

b) By the same argument as in a) we know that p is fragrnenting.

Let xi # x2 be distinct points in X , different from xo. We can choose around them

open sets with dosure disjoint from xo and therefore there is a n E N and open

sets O1 and Os around XI and 2 2 which are included in Fn. Then by the lower

semicontinuity of plp; we can further reduce these open neighbourhoods such that

we witness the lower semicontinuity of p a t (xl, x2) globally on X.

Consider now xi such tha t XI # xo and p, (xo, xI) = v > a > 0.

Let no and n; E A be the values for which z o E Fnp \ Fnb-l and XI E Fni \ Fni-l-

Let n' E N be a value for which g d i a m ( X \ Fnr) < 7".

Let ni E PI be a value such that there is an open set around XI contained in F,,.

Finally let rn := max(nr, no, ni, no, ni). We use this value t o apply the Iower semi-

continuity of p [ ~ : + ~ by letting O. relatively open set in Fm+1 around xo and Oi open

set contained in Fm around xl such that p(a, b) > E + 3 (v - E ) for every a E O.

and b E O1.

Now we daim that for a, b points in the open sets O. u ( X \ Fm) and respectively

0, WC have p(a, b) > E . For the case in which a 6 O. we make use of the fact that

O. contains sorne point c E Fm+1 \ Fm. By construction ~ ( b , c) > E + $ - (v - E )

and p(a, c ) < y and so by the triangle inequality we conclude tha t p(a, b) > E.

Therefore the metric p witnesses the fact that X is RN compact.

c) For the lower semicontinuity of p we find in the same way open neighbourhoods

around pairs of distinct points both different from xo which are contained in F,, for

some n E W and we use the lower sernicontinuity of p , q

Let now X I be a point distinct from xo and p(xo ,x l ) = v > E .

We let nl E W be a value for which pd iam(X \ F,,) < 7.

We let n2 f N be a value for which XI has an open neighbourhood contained in F,,,

and both xo and xl are in F,, .

Let nt E N be a value for which p(xo, en) < 7 for al1 n 2 no.

We choose m = max(nl,n2, nJ) and use the lower semicontinuity of plpz to find O.

relatively open set around xo and Oi open set around xi contained in Fm and so

that p(a,b) > E + i - (u - E ) for a E O. and b~ 01.

Then O. U ( X \ Fm) and Oi are open sets around xo and respectively xi such tha t

p ( a , b ) > ~ f o r a n y a ~ O ~ ~ ( X \ F , ) , b ~ 0 ~ :

We are left to see this for the case when b E X \ Fm. Choose a point G, E

v-€ {cnInEN \ Fm. AS we know tha t p ( z o , ~ , ) < 7 and that p ( ~ , , a ) < 7 we

have by the triangle inequality p(xo, a ) < 7 ( our conditions actually ensure that

p(xo, X \ F,) converges to O ). Further on, having p(xo, 6) > E + 5 - (v - e), we

conclude by the triangle inequality that p(a, b) > E. O

AN EXAMPLE FOR a) OF PROPOSITION 2.2.8

We present an example of an RN compact space which has no isolated points and

is not an EberIein compact.

Let denote the closed subspace of [a, b]" whose points consist of the functions

f : n + [a, b] which change their values at most n many times ( with a t most n+l

f j f f many steps ), i.e. such that there are nf 5 n , vo, v , , . . .vLj E [a, b] with v, # v,,

f f f f v [ # v ~ / ,... Z ~ ~ , - ~ # V ~ a n d a , , a ,,... ai, E ~ w i t h O = c u ~ < a , < a ~ < . . . < a ~ j

such that

If K > wl and n > 1, then SE,q contains a copy of wi + 1 and therefore it is not

Eberlein compact by [3].

First we prove that Ta,,] is RN compact by arguing that the supremum norm is

fragmenting the space.

Letting C be a nonempty subset of Ta,bl, we show how we can trace a small diameter

35

through an open set which nontrivially intersects it.

For f E C we let - y f : {0,1 ,... n) + K bedefined by

The set l?, := {rf : f E C) has a minimal element -(, = (Bo, pi. - - - A) with

respect to the lexicographic ordering.

We let g be a point of C with the property that 7, = 7,.

Let vo, v l , . . .un, be the distinct steppings that g takes beginning with O = Bo7

,&, . . . and respectively /3,, .

For E > O we let := min

~i := min

E , , ~ := min

We now define the open set O around g by taking Oi := (vi - Ei ,v i + E ~ ) on the

coordinate pi for i E O, n,. This ensures that OnC # 0 and also that the supremum

n o m measures distances between the points of O f~ C not greater than E:

If there is a point h in O such that its distance from g is greater than f , then this

must be witnessed on a coordinate different from any f i j for i E O , n,. Let a be the

first such coordinate.

As Oi and Oi+* are disjoint, the i-th distinct value of h appears no later than pi for

i E 1, n,.

If pi-* < a < Pi with i 5 n,, this implies that h forms the i-th step before fi and

so y,, < yg = T~ and therefore g 6 C.

If a > n,, then in addition to the i-th distinct values of h appearing no later than

pi for i E 1, n,, h makes a (n, + 1)-th change of value not later than cr and again

~h < yg, inducing that g 4 C.

Therefore al1 the points in O n C are at a distance from g not greater than 5 and

so the supremum norm measures the diameter of O n C not greater than E.

We observe that the union of two such sets, S ; ~ , a i l ~ S ~ , w can be easily seen as an RN

compact by looking at i t as a t a closed subset of the RN compact s ~ ~ ~ ~ : : ~ ~ ~ , ~ ~ ( , ~

Also , by a) of PROPOSITION 2.2.8, countable unions with increasing number of steps

remain RN compacts when their heights converge to O ( for example UnEN yo,Inl is

RN compact).

Reducing the scattered part of a space

A reduction that we can apply when looking at the continuous images of RN

compacts is through the scattered part of the space.

For a space X we obtain the partition {S, D) into the scattered subspace S and

respectively the dense in itself closed subspace D by letting D be the closed dense

in itself set given by the union of the subsets A of X with each A dense in itself.

37

Dually the open set of isolated points of X can be defined as the maximal set of

points {% Lx such that for any a < a, me have x, isolated in X \ (x~}~,,-

PROPOSITION 2.2.9. a) F o r a space X the following are eqvivalent:

i) The 0-1 metn'c on X ts fragmenting;

ii) X zs scattered;

iii) The diagonal A, zs an almost neighbourhood of itself.

In particdur compact scattered spaces are RN compacts.

b) Let { S , D} be the partition of a compact space X into the scattered subspace S

and the dense in itself closed subspace D.

Then X is RN compact zfl D is RN compact.

PROOF. a) i) + ii) + iii) + i):

If the 0-1 metric is fragmenting, then by the definition of fragmentation we know

that any subset of the space must contain an isolated point.

If X is scattered, then for every nonempty subset C there is O open and x E X

such that On C = {x} and so 0' n C2 # 0 and O2 n C2 C A x , Le. A, is an almost

neighbourhood of 4,.

The 0-1 metric on X has the inverse image of [O, i] equal to A, for n > 1 and if A,

is an almost neighbourhood of itself, this induces that the 0-1 metric is fragmenting.

b) If X is RN compact, then D is also RN compact as it is closed subset of X .

If D is RN compact let p , : D2 + 1 be a lower semicontinuous fragrnenting metric

and define p, : X2 -+ 1 by

The definition above gives a fragmenting metnc:

As p, is fragmenting the closed set D and the 0-1 metric is fragmenting the scattered

set S, we conclude by b) of PROPOSITION 2.1.4 that p, is fragmenting.

The lower sernicontinuity of p, can be argued through a) of PROPOSITION 2.1.4.

O

PROPOSITION 2.2.10. a) V the dense in itself part Dx of a compact space X is

metrizable, then X is RN compact.

b) If f : X -t Y zs a continuovs mapping from a space X as in a ) onto Y , then Y

hos the the dense in itself part Dy also metrizable and therefore is R N compact.

PROOF. a) is immediate from b) of PROPOSITION 2.2.9.

b) Taking a irreducible restriction, flE : E -t Dy of f n - l ( D u ) , we observe that E

does not contain isolated points from X, and therefore E is a closed subset of the

metrizable subspace Dx of X. Then Du is metrizable and therefore Y is RN compact.

O

In particular we can argue that continuous images of the Alexandroff double circle

are RN compacts.

Characterizing RN compactness of images of RN compacts

through the closure of the points with nontrivial fibers

THEOREM 2.2.11. Let f : X -+ Y be a continuous mapping /rom X RN compact

space onto Y . Then Y is RN compact ifl there exist K closed subspace of Y with

{ y E Y : 1 f - ' ( ~ ~ ) l > 1 ) Ç K and p, : x2 + 1, pK : K2 + I lower sernicontinuous

f ragment ing rnetrics with the property that

PROOF. Assume first that Y is R N compact. There are p, : X2 --+ 1, py : Y2 -+ I

lower semicontinuous fragmenting metrics on X and respectively Y. 40

Let pz : X2 + I be the pseudornetric defined by p, (x, x t ) := p,, ( f (x), f (x')) for any

(x, x') E X 2 . By a) of PROPOSITION 2.1.6, the pseudometric p, is lower semicon-

tinuous fragmenting.

Let p, := max ( p , , pz ) . As pl is separating, p, is a metric which, by b) of LEMMA

2.1.1, is also lower semicontinuous and fragmenting.

We take K := Y, pK := p, and then condition (*) is satisfied as for y, y' E Y we have

P ~ ( Y , Y ' ) = ~ y ( f ( f - ' ( y ) ) , f ( f - ' ( ~ ' ) ) ) = ~ Z ( f - l ( ~ ) r fd'(i)) I P ~ ( / - ' ( Y ) ~ ~ - ~ ( Y ' ) ) -

Assume now that we have K closed subset of Y absorbing the points with nontrivial

fibers and that p, and pK are metrics satisfying (*).

The function h : Y* -+ I defined by h ( y , y') = p,-dist (f - ' ( y ) , f - ' (y t ) ) for y ,

y' E Y is lower semicontinuous fragmenting separating by b) of PROPOSITION 2.1.6.

By defining g : Y 2 + I as

we obtain a lower semicontinuous scparating function by condition (A) and a)

of PROPOSITION 2.1.4.

We further consider the pseudometric gA : Y2 + 1. As gA is under the fragmenting

function g, i t also fragments X.

We shall see that gA is also a Iower semicontinuous metric.

For this purpose we first observe that the formula for calculating gP reduces to an

infimum over paths passing through at rnost two intermediate points:

By having the function g defined through a metric on the points of K2, rneasuring

along paths in K can be reduced to measuring the distance between two points in K.

As gl(y\~)= also is a metric, measuring along paths in Y \ K reduces to measuring

the distance between two points in Y \ K.

We can further reduce measurings along paths by the following

REMARK 2.2.12. z) If y, y' E Y \ K and k E K , then g(y', y) + g(y, k) 5 g(yf, k);

ii) If k, kt E K and y E Y \ K , then g(k, y) + g(y, k') 2 g(k, kt)-

29 g ( k , y ) + g ( y , k ' ) = h ( k , y ) + h ( y , k ' ) = P,(f-'(k),f-'(~))+Px(f-'(Y),f-'(~k)

= in f { P X , ( y ) ) : x E f - ' ( k ) ) + inf { p , ( f - ' ( y ) , x f ) : z' E f-'(k')).

As px is lower semicontinuous, the distance between pairs of closed sets is attained.

We let a and b be two points in the closed sets f - '(k) and respectively f -'(kr) where

the distances p, (f -' ( k ) , f -' ( y ) ) and respectively p, (f -' ( y ) , f -'(k')) are attained,

so d k , Y) + S(Y, k') = P x h f - Y y ) ) + ~ x ( f - ~ ( ~ ) , 6 ) -

As I f - ' ( Y ) I= 1 we have P x @ , f - ' ( y ) ) + P x ( f -'(y), b) 2 Px (a, b) and then by (*)

PX (a, b) 2 px( f -w, f - l ( k t ) ) L d k , kt)-

O

By al1 of the above, we are satisfying a condition of type i. from a) of PROPOSITION

2.1.7 and so the formula for calculating gA reduces to measuring along paths with

at most two intermediate points.

Therefore g satisfies the conditions in c) of PROPOSITION 2.1.7 and so g A is also a

lower semicontinuous separating on Y, finally witnessing the RN compactness of Y.

We also observe that the lower semicontinuous fragmenting metric gA constructed

above is calculated in the end as

REMARK 2.2.13. (2 ) If (y, y') E (Y \ K ) 2 , then

(22) If ( y , y') E K x (Y \ K ) u (Y \ K ) x K, then

Applying the Characterization Theorem with O-dimensionality

It was recently claimed that there is a simple proof for the fact tha t extremally

disconnected images of RN compact spaces are RN compacts.

Indeed, such an image is homeomorphic t o a closcd subspace of its preimage, as one

can argue through irreducible mappings and the GIeason absolute.

There is also a simple proof of the fact that O-dimensional continuous images of RN

compact spaces are RN compacts and it ernphasizes property (*):

LEMMA 2.2.14. If f : X + Y is a continvous mapping from a RN compact X ontu

a O-dimensional Y , then Y is RN compact.

PROOF. For y, y' points of Y we let

PYPV' ={(A, B) C Y2 : {A , B } clopen partition of Y svch that y E A and y' E B . } Having p, : X2 + I lower semicontinuous fragrnenting rnetric we define p, : Y2 + I

by

This defines an ultrapseudometric in generd. In particular p, is an ultrametic

due to the fact that p, induces proper distance in between closed disjoint sets,

as remarked in b) of PROPOSITION 2.1.6.

For the lower semicontinuity of p, let p,(y, y') > E for y, y' E Y and E positive.

There is then { A o , Bo) clopen partition of Y with y E Ao, y' E Bo such that

p, (f -'(Ao), f -'(Bo)) > E , hence p, (a, b) > E for any (a , b) E A. x Bo open in Y2.

The function h : Y2 + I given by h ( y , y ' ) = p, (f -'(y), f - ' ( y f ) ) is fragmenting as

seen in b) of PROPOSITION 2.1.6. As p, (y, y') 5 p,( f -'(y), f - ' (yt)) for y, y' E Y,

we conclude that p, is also fragmenting. Cl

PROPOSITION 2.2.15. a) If f : X + Y is a continuous mapping from X RN compact

space onto Y such that {y E Y : 1 > 1) zs included in a O-dimensional closed

subspace of Y, then Y is RN compact.

b) Quotients wllapsingjinitely m a n y closed subsets of RN compacts are R N compacts.

PROOF. a) Let K be a closed O-dimensional subspace containing the the set of

points with nontrivial fibers, { y E Y : 1 f - ' ( y ) 1 > l}.

By LEMMA 2.2.14 we observe that having px lower semicontinuous fragmenting

metric on X ive can produce p~ lower semicontinuous fragmenting metric on K

which also has the property that p K ( k , k') < px (f -'(k), f -' (kt)) for any ( k , kt) E K.

4 5

As condition (*) is satisfied, we can invoke THEOREM 2.2.11 to motivate t h a t Y is

RN compact.

b) We observe that {y E Y : 1 f -' (y) 1 > 1) is fini te and therefore closed O-dimensional.

Applying the Characterization Theorem with Metrizability

For the analogue of b) of PROPOSITION 2-2-15 where we replace O-dimensionality by

metrizability we shall first prove the following:

PROPOSITION 2.2.16. Let pK : K2 -t I be a continuous pseudornetric o n I< closed

subset of a compact space X and Ex closed idempotent superset of A, svch that

Ex n K2 = pil({O}).

There ezists a continuous pseudornetric p, on X such that Ex C p;'({~)) and

P;'({OH n K2 = P;~({OI)-

PROOF. By the continuity of p K , p;l({O)) is a closed set which is Gd relatively

to K2, pil((0)) = nnE, Gn, where Go := K2 and G, := p i ' ([O, i)) for n E H.

As Ex is idempotent, EXn = Ex for n E N and EX" remains a closed set.

Therefore, by LEMMA 1.2.8, for any U E Il, such that Ex C U there exists a

V E LL, such thah Ex C V and Von C U.

We construct a sequence V = {VnInEN of open symmetric neighbourhoods of A, 4 6

with the properties that

a) Ex 2 Vn for n E w ;

b) Vnn K2 5 G, f o r n E w ;

c) V,$ 2 Vn for n E W .

Let Vo = X2 and assume that we have defined V;. E Li, for i < n such that Ex C

and Vin K2 C: Gi for i < n and that C for i < n- 1.

As EX3 tL',_, and Vn-i E U,, there is by LEMMA 1.2.8 a W, E fi, such that

Ex Ç Wn and W13 C Vn-1.

We now let Vn be the symmetric open neighbourhood of A, given by

This element of tl, has the properties that Ex C V,, VnnK2 C Gn and Vi3 C Vn-i.

Through the method of THEOREM 1.2.7, we can now construct p, continuous

pseudometric on X.

As Ex C Vn { ( z l , x 2 ) : p y ( z , y ) 5 $1 for n E N, the pseudometric p, has the

property that Ex E p;'({O)).

Also p, ,KZ has the property that p, (kl, kz) > O for ( k l , k ~ ) E UnEN(K2 \ Vn) 2

UnEN ( K 2 \ G,) = K2 \ ({O)) = K2 \ Ex, i.e. that pi1 ({O)) n K2 Ç Ex -

Therefore ive conclude that p ; ' ( { O ) ) fl K2 = EX n K~ = p;l({0)).

O

COROLLARY 2.2.17. a) For any continuous pseudometric pK defined on a closed

subset K of a compact space X there exists a continuous pseudometric p, on X 47

with the property that the pairs of points from K which are separated by p, are

ezactly the ones separated by p K .

6) If X is a compact space with a closed metrizable subspace K , there ezists a con-

tinuous metric on K which eztends t o a continuous pseudometric on the whole X .

c ) Let f : X + Y be a continuous mapping from the compact space X onto Y and

K closed metrizable subspace of Y .

There exists a continuous pseudometric p, : X 2 + 1 such that

(f2)-'(A,) c P;'({OH and p;l({O)) n (f2)-YK2) = (f2)-'(a,)-

The pseudometric p, : Y2 + 1 given by py(y17 y2) = p,(f-1(yl)7 f -'(y2)) for

(yl, y2) E Y2 2s continuous and it separates al1 the pairs of distinct points /rom K .

PROOF. a) Given pK continuous pseudometric on K we have that p;; ' ( {0)) is a

closed idempotent subset of K2.

Then Ex = &'({O}) U A, is a closed idempotent superset of A, for which we can

apply PROPOSITION 2.2.16.

b) Let pK be a continuous metric on K.

Then Ex = A, is closed idempotent with Ex n K2 = AK = p; ' ( {O) ) and we ap-

ply PROPOSIT~ON 2.2.16 to obtain a continuous pseudometric on X whose restriction

48

to K is a metric.

c) We first argue that the closed set f -'(K) admits a continuous pseudometric

p : ( f d ' (K) )2 -+ I with p-1 ({O}) = ( f 2)-' ( A n ) .

For this let pK : K2 -+ I be a continuous metric on K. We define the pseudometric

P = PK O f : ( 1 - 1 ( ~ 1 ) 2 which is continuous as composition of continuous functions.

We also observe that p-'({O)) = {(x1,x2) E ( f - l (K) )* : f (q) = f ( z2)} = (f2)-'(AK).

For the construction of p, we make use again of PROPOSITION 2.2.16.

We let the idempotent set Ex = (f 2)- ' (Ay) and then Ex Ti ( f - 1 ( K ) ) 2 = P-l ({O)).

Therefore there exists a continuous pseudometric p, on X with

(f 2 PY'({'}) and p i1 ({O}) ( f 2)- '(K2) = f 2 - 1 ( ~ K ) -

As p, does not distinguish distances between points on the same fiber, pv(yl , y*) =

pv ( f - l ( y l ) , f -' ( y 2 ) ) for yl, y2 E Y inherits the triangle inequality. For kl and k2 dis-

tinct points of K we have (f - ' ( k l ) x f - l(k2)) n (f 2)-1(AK) = 0 and so p, (kl, k2) > O

by the property of p, to distinguish al1 the (equal) distances between points on

distinct fibers in f - l ( I ' ) .

As p, is a pseudometric, i t suffices to show upper semicontinuity in order to c l a h

the continuity of p,.

Let y1 and y2 be points of Y with py ( y l , M ) = v < E , where E > O. For xl E f - '(y1)

and x2 E f -' (y2) we have p, (XI, x2) = v < E and using the continuity of p, and

finite covcrs for compact fibcrs we find O1 and 0 2 open sets containing f - l ( y i ) and

f -'(y2) such that p, (al, a2) < E for any al E O1, a2 E 02.

As f is closed, we can choose VI and V2 open sets around y, and y;! such that

f -' (VI) E 0' and f -'(fi) C 02- By the definition of p, we have then p, (vi, y) < E

for any vl E and 712 E VS.

O

2.2.18. If f : X -t Y is a continuous mapping from X RN compact

space onto Y such that { y E Y : 1 f-'(y) 1 > 1 ) is "cluded in a closed metrizable

subspace of Y, then Y is RN compact.

In particular, if X is an RN compact space and Ii is a closed metrizable subspace

of it, then X admits a lower semicontinuous fragmenting metnc whuse restriction

to K is a continuous metric.

PROOF. Let K be a closed rnetrizable subspace containing the the set of points

with nontrivial fibers, { y E Y : 1 f-'(y) 1 > 1 }-

We intend to produce metrics p, and pK on X and respectively I< which satisfy

condition (*) of THEOREM 2-2-11.

We first use the metrizability of K to produce the continuous pseudometric p, on

X with al1 the enounced properties from c) of COROLLARY 2.2.17.

We define pK := py lK2, where p, is defined as in c) of COROLLARY 2.2.17 and

so pK : TC2 -t I is a continuous metric with the property that pK(kl, k2) =

Let us now consider p,, lower semicontinuous metric on X RN compact.

50

We define Co = X2 and C, = pi: ([O, SI) for n E W closed almost neighbourhoods

of the diagonal with C:3 C Cn-1 for n E N-

p, \vas constructed as in PROPOSITION 2.2.16, using a countable sequence V =

{Vn)nEu of sets frorn f i x with = X2 and Vz3 c for n E PI-

As IlnEw C,, = Ax, for every m E w there exists an n(m) E w such that Cn(,) C Vm.

We can easily choose n(O) = O and n(m) such that n(m) > n(m - 1) for n E W.

We now consider for the sequence C := {Cn(m))mEu the lower semicontinuous

fragmenting metric p r P constructed as in LEMMA 1.2.12.

**P As the sequence V = {Vn)nEu is adapted to C, we conclude that p, 5 pc -

We let p, := P;'' and then condition (*) is satisfied as

~ , ( k l , k2) = ~ v ( f -w, f -l(k2)) I ~ x ( f -l(k1), f -l(k2))-

The lower semicontinuous fragmenting metric gA constructed in THEOREM 2.2.11

from g : Y2 + 1 given by

coincides with pK on the pairs of points from K by (iii) of REMARK 2.2.13. Therefore,

when we apply the above construction for the identity of X, we obtain the lower semi-

continuous fragmenting metric gA equal to the continuous metric pK on the pairs of

points from K. This gives us the argument for the second part of the proposition.

O

CHAPTER 3

The K > O Property

The separation properties observed in the previous chapter motivate us t o define

the K > O property. The RNK structure joins this property with fragmentation.

In the first section of this chapter we present invariances and characterizations of

RNK spaces, which describe them as an intermediate class of the fragmentable and

the RN compacts.

In the second section we show that RNK and RN compactness are equivalent

properties in the realm of O-dimensional compacts and so that RNK is a proper

subclass of the fragmentable compacts. We make use of separation properties

to extend separating lower semicontinuous functions, that we further apply in an

example of RN compactness obtainable through a partition by an open and a closed

subspace. We present a condition of closing the RNK structure under F, operations.

1. The K > O Property, Invariances and Characterizations

of RNK Spaces

DEFINITION 3.1.1. If h : Y2 + [O, m) is a function with the property tha t for any

two K I , K2 disjoint closed subsets of Y we have h-dist(Kl, I(,) > O, we Say that h

has the I< > O property.

5 2

We have seen in 6 ) of PROPOSITION 2.1.6 the property which now mites a s

COROLLARY 3.1.2. If X 2 zs a countably compact space and h : X 2 + [O, M) is a

iower semicontinuous separating function, then h has the K > O property.

DEFINITION 3.1.3. If a space Y admits a function h : Y2 + [O, m) which is frag-

menting and has the K > O property, we Say that Y is an RNK space.

PROPOSITION 3.1.4. a) RNK spaces are fragmentable.

b) RN compacts and continuous images of RN compacts are RNK spaces.

PROOF. a) We observe that a function with the the K > O property is a separating

function and so a fragrnenting separating function on the square of a space ensures

the fragrnentability of the space.

b) If X is RN compact and f : X + Y continuous mapping ont0 Y , there is p,

lower semicontinuous fiagrnenting metric on X. The function h : Y2 + I induced

by the p,-distance between the fibers of f is also lower semicontinuous fragrnenting

separating by 6) of PROPOSITION 2.1.6.

-4s X and Y are compact, by COROLLARY 3.1.2 tve conclude that the fragrnenting

functions p, and h also have the K > O property.

53

LEMMA 3.1.5. If f : X + Y is a continuous mapping of a space X onto Y and

h, : X* + [O,oo) is a function with the K > O property, then the function h,

gzven b y h, (y, y') = h, -dzst (f -' ( y ) , f -' (y')) for y, y' E Y has the K > O property.

PROOF. For K I , K 2 closed disjoint subsets of Y we have

Let v' = hy -dis t (Kl , 16) and u = h,-dist( f -'(KI), f - ' (K2)).

For n E N there are xy E f - '(Ki) and xz E f -'(K2) such that h, (x;, x:) 5 v + i. Then h,-dist(Kl, &) 5 hy -d is t ( f (xy), f (xz)) 5 v + for n E N, therefore v' 5 v.

A~SO h, - d i ~ t ( K I , K2) = ZR f { h y ( k l , A+) : kl E Ki, kZ E hT2 }

= in f{ h, -d is t ( f - ' (k l ) , f - ' ( k 2 ) ) : kl E K I , kz E K2) 1 h x - d z s t ( f - ' ( K i ) , f V 1 ( K 2 ) )

and so v' 2 v.

As h, has the K > O property, h , - d ~ s t ( f - ~ ( K ~ ) , f - ' ( K 2 ) ) > O and the latter

inequality shows that h, also has the K > O property.

O

THEOREM 3.1.6. RNK spaces are invariant under closed subsets, f i i t e sums and

perfect mappings.

If Y zs a countable product of regular RNK spaces with Y Z countably compact, then

Y is RNK.

I n particular, RNK compactness is invariant under closed subsets, finite sums,

54

continuous images and countable products.

PROOF. Closed subsets: The restriction of a fragrnenting function with the K > O

property to the square of a closed subset of the space preserves both these properties.

Finite sums: If hi : Yi2 --+ [O, 00) are fragmenting functions with the K > O property

- for i E 1, n, then h : (@,&)' + [O , m) given by

hi (Y 3 Y') if (Y Y') E Y*; h(y, y') :=

witnesses that eieirnY, is an RNK space:

For fragmentation we use the fact that each I: is clopen in eiG,Y, together with

the property of the hi-s of being fragmenting.

- By letting S := {i E 1, n : KI n Y, # 0 and K2 n k; # 0) we have

min{hi(KlnY,,KÎnY,) : & S ) if S f 0 ; h-dist (KI, K2) =

that we use together with the K > O property of the hi-s.

Perfect images: Let f : X + Y be a perfect mapping of a space X onto Y and

h, : X2 + I fragrnenting function with the K > O property.

55

The function h, : Y2 + I given by h, (y, y') = h,-dist( f f has the

K > O property as shown in LEMMA 3.1.5.

The function h, defined as above was also shown in b ) of PROPOSITION 2.1.6 to

be fragrnenting when f is a perfect mapping and therefore Y is an RNK space.

Countable products: Let be a countable family of regular RNK spaces with

Y := nnEN Yn having countably compact square and let hn : Y: + I be fragmenting

functions with the Ir' > O property for n E N.

We consider the function h : Y2 + I given by

We have seen in b ) of LEMMA 2.1.2 that h is fragmenting whenever the functions

h, are fragmenting.

We shall see that h also has the K > O property when the spaces Y, are regular,

the functions h, have the K > O property and Y2 is countably compact:

Let I< and K' be two closed disjoint subsets of Y and assume that h-dist(K, K') = 0.

For n E W there are k, E K and kk E Kt such that h(k,, kk) 5 i. The set {(k,, kk) : n E N) cannot be finite, as it would imply that it contains a

point (k,,, k;,) with h(k,,, k;,) = O and this brings in contradiction the disjointness

of Ir' and IC' and the Ii' > O property of each of the functions h,.

Having Y2 countably compact, we find a ( k , k t ) E K x K' complete accumulation

point of {(ka, k;)}nEN-

As K and KI are disjoint, there is a coordinate no E N on which k and k' are

distinguished, so h,, (rn0 ( k ) , rn0 (k t ) ) is positive and by regularity there are O and

Ot open neighbourhoods with disjoint closures of these points from Yno witnessing

some positive distance, h,o-dzst(O,O') = vo > 0.

Then for any points o E (rrno)-'(O) neighbourhood of k and O' E (~,&'(0') neigh-

bourhood of k' Ive have h(o, 0') 2 & vo, which contradicts h(kn, kk) 5 for n E N

Therefore h-dzst(K, KI) > 0 .

We further give characterizations of the RNK class of spaces which describe them

as an intermediate class of the RN compacts and the fragmentable spaces through

means comparable with those from THEOREM 1.2.9 and from THEOREM 1.2.10.

DEFINITION 3.1.7. A K-separating O-(refined) relatively open partitioning of a space

Y is a countable collection {Un},EN of relatively open partitionings of Y such that

( Un+' is a refinernent of Un for every n E N and that ) for any K , KI closed disjoint

subsets of Y there is a n E W such that any two points from K and respectively K'

are separated in the partitioning Un.

THEOREM 3.1.8. For a space Y the followzng are equiualent:

(a) Y is RNK;

(b) There zs a K-separating o-(refined) relatively open partitioning of Y ;

(c) Y adrnits a f~agmenting (uZtra)rnetric with the K > O property.

PROOF. Through Ribarska's interna1 characterization of fragmentability from (181

we have a method of producing a separating a-relatively open partitioning for a

space which adrnits a fragmenting separating function and a fragmenting ultra-

rnetric for a space which admits a separating O-relatively open partitioning.

This method also gives our characterization for RNK spaces.

(4 * (b ) :

If Y is an RNK space, let h : Y* + [O, cm) be a fragmenting function with the K > O

property.

For every n E N we construct a relatively open partitioning of the space Y, Un =

{Ut : < 5") with h-diam@) 5 for < < <" and such that un+' is a refinement

of Un for n E N.

Constructing U' :

We choose Ui = 0 and U: nonempty open subset of Y with h-diana(U:) < 1.

Having constructed {CI; : J < Co} with UF contained in Y \ Un<F UV, nonempty

rdatively open in it and of h-diameter not greater than 1 for 1 5 < < JO, we look a t

58

Y \ uF<Fo u ~ -

If Y \ UFcSo Ut = 0, then := 6 and the covering of Y by the members of U1 is

complete.

If Y \ UF<Fo Ci' # 0, then for Y \U,,So U, we apply the property t hat h is fragrnenting

to find CIco nonempty relatively open subset of it of h-diameter not greater than 1.

The process of constructing U' is completed a t a of cardinality not greater than

the cardinality of Y.

Constructing Un+' assuming that we have U' constructed for i 5 n:

In the same fashion as in the first step, we use the property that h is fragmenting

to construct Vn+' = {V:+' : I ) < ~ " + l ) relatively open partitioning of Y with

h-diam(V,ncl) 5 for 71 < qn?

Further on we define the relatively open partitioning

ordered lexicographically.

Then the regular increasing family WUn associated to Un is contained in the regular

increasing family Wun+i associated to u"+' and so Un+' is a refinement of Un by

1 sets of h-diameter not greater than ; ;~ i

We can see that {Un)nsN is K-separating:

For K, Kr disjoint closed subsets of Y there is a no E W such that h-dist(K, K r ) > &. The partitioning induced by the members of UnO consists of sets of h-diameter not

59

greater than $ and therefore any two k E K, k' E Kr are in distinct members of Uno.

( b ) + (c) : Assume that Y admits a K-separating o-relatively open partitioning,

{Vn)nsNy where Vn = {V: : 7 < f } .

We can inductively construct from it a K-separating a-refined relatively open par-

titioning, {Un)nEN by applying the refining technique as above:

Let U1 = V1 and assuming that we have constructed U' = {Ul : 5 < y) for

i 5 n, we let Un+' = {un+' . un+' = UF n V;+l, 6 < Cn, g < qncl) ordered (€d - ( € 9 ~ )

lexicographically. Un+' is then a relatively open partitioning of Y which refines Un.

{Un)neN remains K-separating as for any K and Kr closed disjoint subsets of Y

there is no E N such that any k E K and kt E K' are separated in the partitioning

VnO and so they are also separated in Un0.

For y and y' distinct i;oints in Y wve let n,,t be the first n E N for which y and y'

are separated in Un and we define the ultrametric p : Y2 -f 1 by

p is fragmenting:

Let C a nonempty subset of Y and m f N. We find an open set O such that

O n C # 0 and p d i a m ( 0 n C) 5 m. As in a) of PROPOSITION 2.1.5, we let < < cm be the first ordinal for which

U?.C#0.

The open set UVsp UT intersects C on U ' nC and the pdiameter of (UnSc Ll; n C)

is smaller than as no points of U r are separated by the members of Um and so

neither by the members of Ui with i < m due to having {UnInEN a sequence of

refined relatively open partitionings.

The ultrametric p also has the K > O property:

We observed that {Un}neN remained K-separating farnily of partitionings, so for KI

and K2 closed disjoint subsets of Y there is no E W such that any k E K and kt E Kt

are separated in Un. By the definition of p, we have then p ( K , K') 2 &-.

(c) + (a): A fragmenting metric with the K > O property is a witness of Y being

an RNK space.

a

THEOREM 3.1.9. We consider the following statements for a space Y :

( a ) Y zs RNK;

(b ) There zs a sequence {Dn)nEN of idempotent almost nezgh6ourhoods of Ay with

nnENIjn = &Y-

(c) There is a sequence of closed almost neighbourhoods of Ay such that

nnEN = A ~ ;

(d) There is a h : Y* i 1 lower semicontznuous fragmenting separating funetion.

T h e n (a) and Y regular imply (b), (6) implzes (c), (c) implies (d), (d) and Y 2

countably compact zmply (a).

I n particular the staternents above are equzvalent when Y 2s a compact space.

PROOF. (a) and Y regular zmply (b) : By THEOREM 3.1.8 there is p : Y* + I

fragmenting ultrametric with the K > O property. We define the symmetnc sets

Dn := p-'([0, LI) for n E N

{Dn)nEN is a sequence of idempotent sets:

If ( y l , M ) and (y,, y3) are in D,, then both p(y l , y2) and p(y2, m) are not greater than

and as p, is ultrametric we conclude that p(y l , y3) < i. Therefore DE2 C Dn- By n

definition the D,-s contain the diagonal A, and so D n > Dn-

{DnlnEN is a sequence of almost neighbourhoods of Av:

For A subset of Y and n E W we use the property that p is fragmenting to consider

O open subset of Y such that O n A # 0 and that p,-dzam(0 n A) 5 i. Then

O2 n A2 # 0 and by the definition of D, we also have O2 n A* 5 D,.

- Almost neighbourhoods of the diagonal contain the diagonal, so n,,, D. > Ay. For (y,, y2) point off the diagonal Av, we find by regularity O1, O2 open sets with

disjoint closures which separate y1 and y2. By the K > O property of p we have

62

p(Ol, 02) > $ for no sufficiently large. Then (yl, y2) is not in the closure of Dn, -

and so nnEN D, = A,.

( b ) + (c ) : Given {Dn)nEN sequence of idempotent almost neighbourhoods of Ay - -

such that nnEN D, = Au, we take Cn := D, for n E Pd and we obtain a sequence of

closed sets with intersection Ay. Almost neighbourhoods of the diagonal are closed

under taking supersets, so the Cn-s are also almost neighbourhoods of Ay.

(c) * (d) : Let {CnInEN be sequence of closed almost neighbourhoods of Ay with

on,, Cn = Au. As almost neighbourhoods of the diagonal are closed under finite

intersections, we can assume that {Cn)nEN is a decreasing Sequence of closed almost

neighbourhoods of the diagonal.

By letting Co = Y2, the sequence {Cn-1 \ CnInEN forms a partition of Y2 \ Ay. We

define h : Y2 + 1 by

Then h is a separating function which is fragmenting and Iower semicontinuous:

For fragmentation let C be a nonempty subset of Y and no E N.

As Cno is an alrnost neighbourhood of Ay, let O be an open subset of Y such that

O n C # a and 02nC2 C Cno-

By the definition of h we have then that h(a, 6) 5 $ for any a, b +E O n C and so

h-diam(0 n C) 5 E -

For lower semicontinuity we observe that h-'({O)) = Ay and for E positive we have

h-'([O, E ] ) = h-'([O, $1) = Cn, closed set, where n, E W is minimal such that & 5 E.

(d) and Y 2 countably compact imply (a) : A lower semicontinuous fragmenting

separating function h : Y2 + I also has the K > O property by COROLLARY 3.1.2

and therefore it witnesses that Y is RNK.

O

2. Other Properties Relating to RNK Spaces

Through Talagrand's example we know that in the realm of compact spaces frag-

mentability and RN compactness are distinct properties.

Therefore the intermediate class of RNK compacts will be distinct from at l e s t one

of these two classes of spaces.

P ROPOSITION 3.2.10. a ) O-dimensional RNK spaces admit lower semicontinuous

fragmenting rnetrics.

b) In the class of O-dimensional spaces RN compactness and RNK compactness

are equivalent.

64

PROOF. a) Let Y be a O-dimensional RNK space and consider h : Y* -t [0,m)

fragmenting function with the K > O property.

We proceed as previously and let

PY,d = {(A, B) C Y2 : {A, B) clopen partition of Y such that y E A and y' E B)

for y, y' E Y in order to define p, : Y* + [O, 00) by

sup{ h-dist(A, B) : ( A , B) E P,,,,) if y # y'; P y (Y Y Y') : =

On a O-dimensional Hausdorff space this defines a ultrapseudometric. In particular,

py is an ultrametric due to the K > O property, which induces proper distance in

between disjoint closed sets.

p, is fragmenting due to having i t defined under h fragmenting and it is lower semi-

continuous through the definition ( for y, y' E Y with p,(y, y') > E > O there is a

clopen partition { A , B) separating y from y' such that h-dist(d, B) > e ).

b) To what we have shown above we add property 6 ) of PROPOSITION 3.1.4.

We therefore conclude that the O-dimensional fragmentable compact space Tala-

grand's example is not RNK.

COROLLARY 3.2.1 1. RNK zs a proper subclass of the fragmentable compact spaces.

65

We have seen that RNK is invariant to perfect mappings and in particular we know

that a space covered by two closed RNK subspaces is RNK.

Similarly to the case of joining fragmentability for a space partitioned by a closed

and an open set, a space admitting such a partition by RNK spaces will also inherit

the RNK structure:

LEMMA 3.2.12. If a space X admits a partition into a closed set K and an open set

O such that K and O are RNK , then X zs RNK.

PROOF. Let u, : K2 + [O,oo) and u, : O2 + [O,cu) be fragmenting functions with

the K > O property.

As in b) of PROPOSITION 2.1.4, we define the function u : x2 + [O, CO) by

We have seen that u is fragmenting and we observe that it also has the K > O

property: For KI and K2 closed disjoint sets we have u(K1, K2) = 1 if one closed set is

in O and the other is in K , u(K1, K2) = uK ( K l n K, K2 n K) if not both sets intersect

O, u(K1, 16) = u0 (K1 n O, K2 no) if not both sets intersect K. If both sets intersect

In order to extend a lower semicontinuous separating function from the square of

a subset of a regular space to a lower semicontinuous separating function on the

square of the space we can explicitly use separation properties:

PROPOSITION 3.2.13. Let S be a subset of a regular space X and us : S2 -t I lower

semicontinuous separating function.

If us separates al1 the pairs of sets (S Cl, S n C2), where Cl and C2 are closed

-2 disjoint subsets of X , then there exists Y : S + I lower semicontznuous separating

extension of us.

In particular if us h a the K > O property, then there ezists a lower semicontinuous

separating extension of us.

Further on, us admits a lower semicontinuous separating extenszon to X2.

If in addition us zs null on A,, then the extension uF remains null on As and fur-

ther on uF extends to a lower semicontinuous sepamting function, ÜF : X 2 + 1,

by mapping al1 the points of A, to O and al1 the points outside A, U s2 to 1.

-2 PROOF. AS in c) of PROPOSIT~ON 2.1.3, we let uS : S + I be the function given

for x, y E S by

us(x, y) = sup{us-dist(O,nS, 0,nS) : O, open set aroundx, 0,open set around y).

We have seen a t that time that the definition above gives a lower semicontinuous

extension of us and we want to see that is separating under the given conditions.

By regularity, for x # y points of there are 0, and respectively O, open neigh-

bourhoods with disjoint closures. Therefore O, n S and n S are a t a positive

us-distance and so u,(x, y) 2 us-dist (O,flS, 0 , n S ) 3 us-dzst ( Z ~ S , G ~ s ) > O.

The existence of a loiver sernicontinuous separating extension of u, to X2 is a con-

sequece of c) of PROPOSITION 2.1.4, where we have any value not smaller than 1

available as an upper bound for uo.

Having started with us null on A,, uF remains null on A3:

uS(s,, s,) = s u p {us-dist (U SI V n S ) : U, V open sets arovnd s,) for s, E S\ S.

For any such sets U and V, U n V is an open set around s, point in the closure of

S and so there is a point z E U n V n S.

Therefore us-dist (U n S, V n S ) 5 us (2, z ) = O and so %(s,, s, ) = 0.

On X2 we have the 0-1 separating lower semicontinuous function defined as a metric.

Given uJ lower semicontinuous separating, we define üs : X 2 + I by

The lower semicontinuity of can be argued by having replaced as in a) of PROPO-

SITION 2.1.4 the lower semicontinuous function 0-1 with the smaller function u, on

-2 S closed subset of X 2 .

We want to create an example of an RN compact structure obtainable through con-

ditions reflected on a partition of a space through an open and a closed subset.

We first provc the following

LEMMA 3.2.14. Let O be an open subset of a regdar space X , u : O* 4 I coat in~ous

pseudornetric and let I< := X \ O.

The Zower semicontinuous extension of u, 5, : X 2 + I defined as in PROPOSITION

3-2-13 admits shortcuts for al1 triples of type (O, O, O ) , ( k , O , k) and ( k , O, O ) , where

O E O andk E K .

PROOF. ( O , O, O ) triples : If 0 1 , 0 2 , 03 are al1 points in O then we have üo(oi, 04 + Üo(o2, 03) 2 Üo(ol, 03) by the property of üo to coincide with the pseudometric u

when restricted to 02.

( k , O , k ) triples: WC first show that üs(k l , O ) + ÜT(o, k2) 2 üo(ki, ka) for kl, kZ E

O n K and O E O:

69

Let a = ïïo(kl, O ) , b = üo(o, k2) .

By the definition of Co for any open neighbourhoods V,, Vk,, Vk, of O, k l and res-

pectively k2 we have u-dist(Vcl n O , l/, n 0) 5 a and u-dzst(V, n O, vk2 no) 5 b.

By the continuity of ücj at ( O , O ) , for any n E N we have that O contains an open

neighbourhood Vt of O such that ü5-diam(Kn) < i. Then for any n and m E N and for any Vk, and Vk2 open neighbourhoods of ki and

respectively k2 there are sm, and sk2 E Vk, n O and respectively Vk2 n O such that

u(s\, ,s~,) $ a + A++ + b + $

Therefore u-dist(Vkl n O, Vk, n 0) 5 a + b for any Vk, and Vk, open neighbour-

Iioods of kl and respectively k2.

Then ü,(k~, k2) 5 a + b and therefore such ( k , O , k ) triples admit shortcuts.

The inequality iio(kl, O ) + ü&, k2) 2 %(kl, k2) is equivalent to üo(k l , O ) + 1 2 1

f o r k l ~ O n ~ , k 2 ~ ~ \ ~ a n d t o 1 + 1 ~ 1 f o r k l , k 2 ~ ~ \ ~ .

( k , O, O ) triples: Using similarly the continuity of u at (o l , o l ) and a t (02, 02) , we can

prove that iiT(k, O, ) + üo(ol , 0 2 ) 3 üo(k, 02) for any 01,02 E O and k E K n O and

for k E K \ O we observe that Uo(k, o l ) + Üo(ol, 0 2 ) 5 L ( k , 0 2 ) is equivalent to

1 + u(o l , 0 2 ) 2 1.

PROPOSITION 3.2.15. If a compact space X adrnits a partition in to a O-dimensional

RN compact K and a n open set O which admits a continuous metr ic separating al1

the pairs of subsets traced in O b y pairs of closed disjoint subsets of X , then X is

RN compact.

I n particular if K is O-dimensional RN compact and O admits a continuous metric

with the K > O property, then X is RN compact.

PROOF. Let po : O2 + I be a continuous metric with p, (Cl no, Cz no) > O for any

Cl, C2 closed disjoint subsets of X and let pK : K2 + 1 be a lower semicontinuous

fragmenting metric.

As in PROPOSITION 3.2.13, we construct the extension of po to the lower semicon-

tinuous separating function /50 : X2 -+ 1.

We further let h : K2 + 1 be the function h = min(& 1,, , p K ) and we observe

that it is lower semicontinuous separating as both Pz and pK are. Also h is

fragrnenting as it is under the fragrnenting pK.

This allows us to construct from h a lower semicontinuous fragmenting metric,

d, : K2 + I in the same way as in LEMMA 2.2.14, by using the supremum of the

h-distances measured between the clopen partitions which separate pairs of distinct

points.

We obtain in this way & lower semicontinuous fragmenting metric under 3,.

Therefore, by a) of PROPOS~TION 2.1.4, the function g : X2 + I defined by

71

which replaces on a closed set the lower semicontinuous fragmenting function &

through the smaller lower semicontinuous function d, is a lower semicontinuous

fragmenting function.

We can further show that g satisfies a condition of type i. from PROPOSITION 2.1.7

with respect to the partition of X into {O, K ) .

If x, y, z are points al1 in O or al1 in IC, then we have

g ( x , Y ) + g ( y , z ) 2 g ( x , z ) by the property of h to coincide with the metric p, when

restricted to O* and with the metric p; when restricted to K*.

We can show that g ( k l , O ) + g ( 0 , k2) 2 g ( k l , k 2 ) for ki, k2 E K and O E O:

By LEMMA 3.2.14 we have

g ( k 1 , o ) + g(o, k2) = & ( k ~ , O ) + p5(o, 2 p 5 ( k ~ , k 2 ) 2 ~ ' ( ~ 1 , kz) = g ( k l , k 2 ) .

For o l , o2 f O and k E K ive apply again LEMMA 3.2.14:

9 ( k , 01) + g ( o i , 0 2 ) = &(k, 01) + &(01 , 0 2 ) 1 &(k7 0 2 ) = d k , 0 2 ) -

Therefore by c) of PROPOSITION 2.1.7, the function g A : X2 + I obtained from

applying to g the infimum ovcr finite paths is a lower semicontinuous metric under

g fragmenting and so X is RN compact.

cl

We have seen in b ) of PROPOSIT~ON 2.1.5 that fragmentability is Fu-closed and we

know by Talagrand's example that the property does not extend to neither one of

the RNK or RN compact classes of spaces.

We can close the RNK structure to the Fm operation under conditions formulated

not stronger than the ones in the previous chapter when creating closures of the RN

compactness under F, operations within the reaIm of compact spaces:

PROPOSITION 3.2.16. Let X 6e a space covered by an increaszng sequence of close&

sets, {Fn),, with Fo = 0 and such that al1 the sequences of points which f o m along

{F, \ Fn-L}nEN have a unique limit point X.

If F, is RNK for n E N , then X is RNK.

In particular if Fn \ F,-l is RNK for n E IV, then X zs RNK.

PROOF. Let un : F: + I be fragmenting functions with the K > O property.

As in b) of PROPOSITION 2.1.5, we define u : X2 -t I by

We have seen that u is fragmenting and ive show that it also has the K > O property:

For any two Cl and C2 disjoint subsets of X there is a t most one which contains the

unique limit point in X formed by moving along F,+l \ F, and so we can assume

73

without loss of generality that there evists a no E Pd such that CI C F,, .

Then the u-distance between CL and C2 is

min min U,-~ZS~(CL n (Fn \ Fn-1), C2 fi (F' \ Fn-1)) : n E 1, no ), no-i . ( { AS U~-~ZS~(C~~(F~\F,-~), c ~ ~ ( F ~ \ F ~ - ~ ) ) 2 un-dist(C1nFn,C2nFn) for n E 1,no,

the distance above is positive by the K > O property of the un+.

As we have remarked in LEMMA 3.2.12, the existence of un : (Fn \ Fn-l)2 + I

fragmenting functions with the K > O property induces RNK structure for each of

the F, for n E N and so X is an F, space of RNK spaces for which we apply the

arguments above.

By this proposition we observe that if we construct from Talagrand's example the

quotient space which collapses the closed subspace Ko to a point, then we obtain

an RNK space. This quotient maintains O-dimensional structure and therefore it is

also RN compact by PROPOSITION 3.2.10.

CHAPTER 4

RN Compactness and Extendibility of Nonexpansive

Funct ions

In this chapter we show that compacts non-RN compacts are not minimal in the

sense that any compact non-RN compact space has a proper closed svbspace which

is non-RN compact.

We prove this property using extensions of nonexpansive functions, an alternative

approach which can also be applied to some instances of previously presented results.

1. Nonexpansive Functions and Nonexpansive Ex tens ions

DEFINITION 4.1.1. If p : x2 [O, cm) is an arbitrary pseudometric on a space X

and f : X -+ IR is a function with the property that 1 f (x') - f (x")l 5 p ( x f , x " ) for

every x', XI' f X , then we Say that f is nonexpansive with respect tu p.

THEOREM 4.1.2. Let p : X 2 -t [O, w) be a pseudometric on a set X . Gzven A

subset of X and f : A + R nonexpansive function with respect to Plaz, there exists

F extension off to A' which i s nonexpansiue with respect to p.

75

PROOF. Let us first observe that given a nonexpansive function, f : A + IR and a

fixed x E X \ A, there exists g : A U {x) + R nonexpansive extension of f .

In order to create such an extension, it suffices to define g(a) = f (a) for a E A and

g(x) such that lg(x) - g(a)l 5 d(x, a ) for any a E A.

Letting I, := [g(a) - d(x, a ) , g(a) + d(x, a)], the condition above writes as

We shall see that this intersection is nonempty.

First we have that I, n Iat # 0 for any a, a' E A:

This reduces to observing the inequalities g(a1) - d(x , a') 5 g(a) + d(x, a) and

g(a) - d(x , a) 5 g(a' ) + d(x, a') , which wnte as

But 9 (a) = f (a), g(a l ) = f (a') and 1 f (a) - f (a') 1 $ d(a, a') by having f non-

expansive and therefore, through the triangle inequality we can conclude (2).

Further on, a family of closed intervals with the property that 1. I.1 # 0

for any a , a' E A also has the property that

I,, n I,, n ... n I,, # 0 for any n E N, ai, a î , ..., a, E A.

So {la)oEA is a family of closed bounded intervals with the finite intersection property

and then na,, 1, # 0. Therefore the extension g exists.

Let now Ç := {g : Y + E 1 A E Y C X and g is a nonexpansive extension of f )

ordered by extension of functions ( for g, : Yl + R, g, : Y2 + B elements of B

define g, 5 9, iff f i _> Yl and g,,,, = g, ).

If {g, : Y, -t is a linearly ordered subset of 8, let g : UaSc YQ + P be the

function given by g(y) := gQ,(y) for some a, E < such that y E Y,,.

Then g is a well-defined function by linearity and a nonexpansive extension of f

which extends each g, for a E c.

Therefore, by Zorn's Lemma, contains a maximal element that will absorb al1 the

points of X inside its domain, so there is an F : X -t IR extension of f which is

nonexpansive with respect to p.

PROPOSITION 4.1.3. a) A function which zs nonexpansive with respect to a contin-

uous pseudometric is itself continuous.

b) Given a space X , a continuous pseudometric p : X 2 -t [O, CO) and / : A -+ R

a nonexpansive function with respect to ~ 1 . 4 2 , there is a continuous nonexpansive

extension of f to the whole space.

c ) Giuen a space X , a pseudometric p : X 2 -+ 1 and f : A + R a globally Lipschitz

funct ion w2th respect to ~ 1 ~ 2 , there is a n extension F : X + W o f f which is globally

Lipschitz with respect to p.

PROOF. a) We observe that 1 f (x,) - f ( x ) l i d(x,, 2) implies that {f (~,)),,r con-

verges to f (x) whenever {x~)~~= converges to x.

b) We just use a) when applying THEOREM 4.1.2.

c) It suffices to observe that a globally Lipschitz function with respect to p is a

nonexpansive function with respect to k - p with k > O and k - p is a pseudometric on

X for which we apply again THEOREM 4.1.2.

O

2. RN Compactness through Functions of Variation under Almost

Neighbourhoods of the Diagonal

DEFINITION 4.2.4. I f f : X + B is a function on a space X and {En}nE~ is a

sequence of subsets of X2 such that En E {(x, y) : 1 f (x) - f (y) 1 5 &} for n E Ni

we Say that f is of variation under {En)nE~-

PROPOSITION 4.2.5. A space X zs RN compact if and only if there are a sequence

{En)nEN of a h o s t neighbourhoods of A, and a separating farndy 3 of continuous

functzons on X of variation under {En}nEN-

PROOF. If a sequence of almost neighbourhoods and a family of continuous functions

with the above properties exist, we let then p : X2 -t [O, m) be the pseudometric

given by

P ( X , Y ) = SUP,,, I f ( 4 - f ( y ) I for (x,Y) E X2.

As the family of functions separates the points of the space, the pseudornetric is

separating.

The lower semicontinuity of p is derived from having defined it as the supremum of

a family of continuous functions.

En is an almost neighbourhood of A,, so if C is a closed subset of X and no N

there is an open set such t t a t O n C # 0 and that O2 f~ C2 C En,. The pdiameter

of O n C is not greater than & as for (x, y) E Eno we have 1 f (z) - f (y) 15 for

any f E 7, function of variation under {En)nE~-

Assume now that X is RN compact and let p : X2 + I be a lower semicontinuous

fragmenting metric on X.

By letting En := p- ' ( [O, hl) for n E w we have £ = {En},,, sequence of closed

almost neighbourhoods of A, with the properties that E::, E En for n E w and

79

nnEv En = A,

As in LEMMA 1.2.12, we consider the family { V ( S ) ) ~ ~ ~ of al1 sequences of open

symmetric neighbourhoods of A, which are adapted to E and then we let {pv,,,)ses

be the family of continuous pseudometrics associated to these sequences by the

method of THEOREM 1.2.10.

As each V ( s ) is adapted to E , we have p,,,(x, y) 5 & for (x, y) E En.

We let 3 := { f , , , where : X -t I is defined by f,,y(x) := p,,, (x, y)

for x E X.

Through the triangle inequality, the members of 3- are functions of variation under

{En}ne~-

We have also seen in LEMMA 1.2.12 that the family {pv(,l)sES is separating the

points of X, so for x # y there is a s E S such that p,,,,(x, y) > O. The function

fS,, will then separate x and y.

Therefore T is a separating family of continuous functions on X of variation under

{En)nEN-

LEMMA 4.2.6. Let p : X2 + 1 be a lover semicontinuous fragmenting metric and

K l , K2 closed disjoint subsets of a compact space X .

There i s a continuous pseudornetn'c p, on X such that pu-dist(K1, Kz) > O and

P-' ( [ O ? f 1) C Pl,-' ([O, $1) for E w -

PROOF. The sets C, := p-l ([O, $1) for n E w are closed, with the property that

nnEo cn = nx -

By the compactness of X there is a no > O such that Cm n ( K i x K2 U K 2 x K i ) = 0 .

As C:3,, 5 Cn for n E w, ive can choose by LEMMA 1.2.8 a sequence U = {Un)nzo

of symmetric open neighbourhoods of A, with C, E Un and Un:, C Un for n < no.

As Czt C there is by LEMMA 1.2.8 an open symmetric neighbour-

hood W of A, S U C ~ that Cno C W and Wo3 C

W \ (K1 x K2 U & x K I ) is an open syrnmetric neighbourhood of Ax which contains

C,, and so by letting Un, := W \ (Ki x K2 U K2 x KI) we have Cm E Un, and

We continue the process of constructing symmetric open sets Un around C, with

U,"il C un for n E w by applying LEMMA 1.2.8 and obtain a sequence U = {Un)nEw

adapted to {Cn)nEu.

Through the method of THEOREM 1.2.7 applied to U, we generate the continuous

pseudometric p, .

As ( K I x K2 U Kz x K i ) fi Un, = 0, we have by this theorem pu(kl, k2) 2 for

kl E K1 and k2 E K2 and therefore the pu-distance betwecn Kl and K2 is positive.

As pu was created from the sequence 2.4 adapted to {Cn)nEw, the property that

P - L ( [ O , $1) C pu -' ([O, &]) for n E w is also a consequence of THEOREM 1.2.7.

PROPOSITION 4-2.7. If XI and X2 are RN compact spaces &th the property that

X I \ X2 n ,Y2 \ X1 = 0, then Xi U X2 is RN compact.

PROOF. Consider pi : X': -t 1 and f i : XZ -t I lower semicontinuous fragmenting

metrics on XI and respectively X2.

On the free union of X1 and X2 we can define the lower semicontinuous fragmenting

metric d by

The Hausdorff metric Hd associated to d reflects on the points of the union of the

two spaces as

I l if none of the above,

which is a fragmenting metric on Xl U X2 by b) of PROPOSITION 2.1.4.

As Hd : (XI u X2)2 + I is fragmenting, En := H ~ I ([O, &] ) is an alrnost neighbour-

hood of for any n 3 0.

Hd is also separating, so {En},2o is a sequence of almost neighbourhoods of A,

with n n2a E~ = A ~ ~ ~ ~ ~ .

For n E W we mite En = En u En U E:, where we use the notation

E,! = { (GY) E ( X I \X2I2 : P I ( X , Y ) < + ) Y

En = (x, Y) E (&. \ P ~ ( x , Y ) 5 $1,

E," = { ( x , Y) E ( X I n X d 2 : max(p1, a ) ( x , Y) 5 $1 . Consider now xo, y0 distinct points in XI U X 2 .

If one of the points is not in X1 ri X2 we can assume without loss of generality that

xo E Xi\ X2-

Then { x o ) and ({yo) u X 2 ) n Xi are disjoint closed sets in X I . B y LEMMA 4.2.6 there

is p, : X: -t I continuous pseudometric with p' ( {xo) , ( { y o ) u X z ) n X I ) = a > O

and such that pl-' ([O, &]) C Pu-1 ([O, &]) for n E w.

Let g : { X O ) U U X 2 ) 2)n xi)+ B be the function which sends z a to a and

( { Y o ) U X2) n xi to O.

As g is nonexpansive with respect to pu, it admits by THEOREM 4.1.2 an extension

to h : Xi + W nonexpansive function with respect to pu. The pseudometric pu is

continuous and so h is also continuous.

As h is nul1 on the closed set XI n X 2 , we can extend it to a continuous function

f : Xi U X2 + IR by mapping al1 the points of X2 to O .

The function f is constructed to separate xo and y0 and now we show that for

n~ N w e have 1 f ( x ) - f ( y ) I 5 & when (x,y) E En:

1 l This or (x, Y) E E," we have max(pl(x, Y), P+, Y)) a , and so p1(x, Y) 5 F.

implies that pu (x, y) 5 & and again, as f is nonexpansive with respect to pu, we

conclude that 1 f (z) - f (y) 15 & for (x, y) E E,".

For (x, y) E E: we have f (x) = f (y) and our inequality is trivially satisfied.

For the case when both xo and y0 are in Xl n X2 we can choose to look at xo, which

is not in a t least one of X2 \ XI or XI \ X2. Without loss of generality xo $ X2 \ XI.

As {xo) and {yo} U (X2 \ XI nXl) are closed disjoint in X1, we find by LEMMA 4.2.6

a continuous pseudometric pu : X: -t I with pu ({xo), {y0) u (X2 \ Xi nXi) = a > O

and such that ([O, $1) pu -l ([O, $1) for n E W .

Shen g : {XO} u {yo} u (X2 \ XI n XI) + {O, a) which rnaps {yo} u (X2 \ Xl n X1)

to O and x0 to a is nonexpansive with respect to pu, so i t extends to a function

h : XI + R which is continuous and nonexpansive with respect to pu.

The restriction of h to X2 \ XI n X1 is null, so we can extend h to a continuous

function f : X1 U X2 + R by mapping al1 the points of X2 \ X I to O.

The function f separates xo and y0 and it is again of variation under {En)nEN by

using exactly the same arguments as in the previous case, when we showed that

Therefore there exists a continuous function f on XI U X z which separates xo and

y0 distinct points in X and of variation under {En)nEN

In conclusion the family from PROPOSITION 4.2.5 is satisfiable for X I u X2, which

makes it an RN compact space.

O

PROPOSITION 4.2.8. No compact non-RN compact space zs minimal, i.e. euery such

space has a proper closed subspace which is non-RN compact.

PROOF. It suffices to take two open sets, Ui and U2 with disjoint closures in a

compact space X.

If al1 the proper closed subsets of X are RN compacts, then X \ 4 and X \ U. 9 are

two of them and, by PROPOSITION 4.2.7, their union X is an R N compact space.

cl

Bibliography

[l] A.V. Arhangel'skii Encyclopaedia of Mathematical Sciences - General Topology II,

Springer-Verlag (1996) 103- 105.

[2] A.D. Arvanitakis Sorne Remarks on Radon-Nikodtjm Compact Spaces. To appear.

[3] KR. Bennett, D.J. Lutzer, J.M. van Wouwe Linearly Ordered Eberlein Compact

Spaces, Topology Appl., 12 (1981), 11-18.

[4] Y. Benyamini, M.E. Rudin, M. Wage Continuous Images of Weatly Compact Subsets

of Banach Spaces, Pacific Journal of Mathematics, 70 (1977), 309-324.

[5] J. Dieudonné Foundations of Modem Analysis, Academic Press Inc London (1963).

[6] R. Engelking General Topolog y, Heldermann Verlag (1989)-

[7] R. Engelking, K. Sieklucki Topology : a Geometric Approach, Berlin: Heldermann

Verlag (1992).

[8] M. Fabian, M. Heisler, E. Matuskovii Remarks on Continuous Images of Radon-

Nikody'm Conapacta, Comment. Math. Univ. Carolinae, 39(l), (l998), 59-69.

[9] M. Husek, J. van Mill Recent Progress in General Topology, Amsterdam, New York:

North-Holland (1992).

[IO] J.E. Jayne, 1. Namioka, C.A. Rogers Topological Properties of Banach Spaces, Proc.

London Matli-Soc. (3) 66 (1993), 651-672.

[ I l ] J.E. Jayne, C.A. Rogers Borel Selectors for Upper Semicontinuous Set- Valued Maps,

Acta Math., 155 (1985): 41-79.

86

[12] J.L. Kelley General Topology, Springer, New York (1975).

[13] 1. Namioka Radon-Nikodim Compact Spaces and hgmentability, Mathematika, 34

(1987), 258-281.

[14] 1. Namioka Radon-Nikodtjm Compacta. To appear.

[15] 1. Namioka, R.R. Phelps Banach Spaces which are Asplund Spaces, Duke Math.J. 42

(1975), 735-750.

[16] J.R. Porter, R.G. Woods Eztensions and Absolutes of Hausdora Spaces, Springer,

New York (1988).

[17] 0.1. Reynov On a Class of Hausdorff Compacts and GSC Banach Spaces , Studia

Math.,71 (1981), 113-126.

[18] N.K.Ribarska In ternal Characterization of Ragmentable Spaces, Mathematika,

34(1987), 243-257.

[19] J. Orihuela, W. Schachermayer, M. Valdivia Every Radon-Nikodcm Corson Compact

Space 2s Eberlein Compact, Studia Mathematica 98(2) (1991), 157- 174.

1201 K. Yoshida Functional Analysis Springer-Verlag Berlin Heidelberg New York (1979).